Informal Presentation of Geo-Spatial Analysis Project

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Casebook.org Forum 'Thread': "Informal Preview of Geo-Spatial Analysis Project"

I am attempting to move beyond the 'hodge-podge' of the "Informal Preview"; and so have begun the process of an "Informal Presentation".

Its narrative bears a few holes, and lacks the overall fluidity that I would want it to have. It even suffers from the occasional 'first-person' intrusion.

It lacks the necessary notes/references.

It is … "informal"!

The formal presentation will come to fruition … well, … when it comes to fruition: Someday!

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flickr Set: "Informal Presentation of Geo-Spatial Analysis Project"

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A question that is often raised in regard to the murders most widely attributed to 'Jack the Ripper', is whether the perpetrator of these crimes was 'local'. While the answer to this question is likely to remain forever elusive, a better conceptualization of the question itself is well within reach, and likely to enhance the focus of today's ongoing informal 'investigation'. In other words: While the location of the 'base', from which 'Jack the Ripper' operated, like his identity, will probably never be discovered, a better understanding of the circumstances surrounding the murders, for which he is believed to have been responsible, can be obtained by establishing a set of parameters that allow for a definition, in this case, of that which was 'local'.

Webster's defines the term 'local', accordingly: "Of, pertaining to, or characteristic of a particular place or a limited portion of space." It is therefore, imperative that a "particular place" and "limited portion of space" be established, in order to fully conceptualize the question of whether 'Jack the Ripper' was indeed 'local'. Put simply: The concept of a murder 'locale' having clearly defined parameters is a necessary component of the question itself.

For clarity's sake, the question being raised should be whether 'Jack the Ripper' was 'local' specifically to the area, in which his crimes were committed. Thus, the establishment of a "particular place" to serve as a focal point, and a "limited portion of space" to serve as a specified degree of vicinity, e.g. 'immediate vicinity', 'general vicinity' or 'broad vicinity', should be based on the locations of the murders that are most widely believed to have been his 'work'. Unfortunately, knowing which murders are most widely attributed to this 'phantom' killer is virtually impossible, as opinions vary and tend to be fraught with subjectivity.

Supposed 'canons' notwithstanding: It would appear that three of the so-called 'Whitechapel Murders' (those of Polly Nichols, Annie Chapman and Catherine Eddowes) are almost universally accepted as having been the work of a single killer; and that three of the remaining eight (those of Martha Tabram, Elizabeth Stride and Mary Jane Kelly) are preponderantly accepted as having been committed by the same hand. Of particular note: This set of six murders occurred in uninterrupted sequence within the series of eleven 'Whitechapel Murders', during a span of just ninety five days (7 August 1888 – 9 November 1888); within an area of less than one square-mile; and under certain circumstances, which were strikingly similar. It would seem likely therefore, that those directly involved in the contemporary investigations of these murders perceived at least some degree of correlation within the set, if not a common denominator in the form of a single perpetrator. So, while the inclusions of Tabram, Stride and Kelly in the 'Ripper's tally' are each debatable; the spirit of objectivity virtually dictates the factorization of their murder-sites in the establishment of a murder 'locale'.

Martha Tabram (7 August 1888) First-Floor Stairway Landing of George Yard Buildings, George Yard, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 18.45" West
Latitude: 51° 31' 0.60" North

Mary Ann 'Polly' Nichols (31 August 1888) Gateway to Brown's Stable Yard, Buck's Row, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 3' 37.53" West
Latitude: 51° 31' 12.14" North

Annie Chapman (8 September 1888) Back Yard of 29 Hanbury Street, Parish of Christ Church Spitalfields, County of Middlesex
Longitude: 0° 4' 21.40" West
Latitude: 51° 31' 13.67" North

Elizabeth Stride (30 September 1888) Gateway to Dutfield's Yard, Berner Street, Parish of St. George in the East, County of Middlesex
Longitude: 0° 3' 56.14" West
Latitude: 51° 30' 49.44" North

Catherine Eddowes (30 September 1888) Southeast Corner of Mitre Square, Parish of St. James, Aldgate Ward, City of London
Longitude: 0° 4' 41.06" West
Latitude: 51° 30' 49.35" North

Mary Jane Kelly (9 November 1888) Interior of 13 Miller's Court, Dorset Street, Parish of Christ Church Spitalfields, County of Middlesex
Longitude: 0° 4' 30.47" West
Latitude: 51° 31' 7.17" North

It would seem logical therefore, to establish a "particular place" as the focal point of the murder 'locale', on the basis of the 'central tendency' of the six murder-sites under consideration. A widely used measure of 'central tendency', which minimizes the aggregate distance to a set of observations, is the 'Median'.

Consider for example, the following set of six numbers: 3, 11, 12, 33, 36 and 55.

The median of the set is the point on the mathematical number-line that evenly divides the set; such that half of its observed quantities are below the median, while the other half of its observed quantities are above the median. As this particular set contains an even number of components, its median is the 'mid-point' between the two most 'central' observed quantities: '22.5'.

Again; this measure of 'central tendency' (i.e. the 'Median') minimizes the aggregate distance to a set of observations; such that in this particular case, the aggregate distance from '22.5' to each of the six components of the set, is the smallest attainable.

It should also be noted that the 'Median' is not affected by outliers. If for example, the sixth observed quantity in the above set, were '1,055', rather than just '55'; the median of the set would still be '22.5'.

This immunity to 'outlier-effect', unfortunately, renders the 'Median' (i.e. the 'Median-Center', in the case of a two-dimensional 'field' of murder-sites) of little use as a descriptive statistic beyond the establishment of a "particular place" to serve as the focal point of the murder 'locale'. In other words: The 'Murder-Site Median-Center' will not lead to the establishment of a "limited portion of space" to serve as a specified degree of vicinity, on the basis of murder-site dispersion around its location.

Put simply: If the 'Murder-Site Median-Center' is not affected by the extent of murder-site dispersion, then it cannot lead to the employment of such dispersion as a descriptive tool.

Another shortcoming of the 'Median-Center' is its relative degree of elusiveness.


Figure 1: Two 'Medians' (i.e. Two Points of Murder-Site Median-Center) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Figure 1, above, depicts two distinct examples of the murder-site median-center that result from different orientations of the two-dimensional 'field', in which the six murder-sites are located.

In each instance (i.e. 'white' and 'yellow'), the 'field' of six murder-sites is being evenly 'divided' by a set of Cartesian-Coordinate axes; such that both 'x-axis' and 'y-axis' divisions (i.e. 'lines of median separation') are being considered. The two-dimensional murder-site median-center, in each case, lies at the intersection of the respective 'lines of median separation' (i.e. the 'origin' of the respective Cartesian-Coordinate axes).

Murder-Site Median-Center (Example 'White') ('Y-Axis' Orientation: 0.00°, i.e. 'Grid-North')
Interior of Combined Set of Common Lodging Houses, 11-15 Thrawl Street, Parish of Christ Church Spitalfields, County of Middlesex
Longitude: 0° 4' 19.93" West
Latitude: 51° 31' 3.88" North

Murder-Site Median-Center (Example 'Yellow') ('Y-Axis' Orientation: 324.47°, i.e. the orientation used later in this project, for construction of the 'Standard Deviation Ellipse')
Southern Exterior (i.e. Rear Wall) of Dwelling, 76 Wentworth Street, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 14.84" West
Latitude: 51° 31' 1.63" North

There is in fact, a distinct murder-site median-center for every conceivable orientation of the murder-site 'field'. But, only one of these points (i.e. the 'Center of Minimum Distance') possesses the quality of a true 'Median'; that of minimizing the aggregate distance to the set of six murder-sites.

Unfortunately, there is no formula for the determination of a two-dimensional 'Center of Minimum Distance'. It can however, be estimated through iterative measurements of aggregate distance. But, where does one begin; and just how much iteration might be necessary for the attainment of a reasonable estimate?

*** I have been experimenting with a process, which I believe should serve to 'isolate' the murder site 'center of minimum distance', if carried through a sufficient number of iterations. I will graphically depict a few such iterations of the process, in order to arrive at a relatively broad estimate of this most elusive measure of 'central tendency'. ***


Figure 2: Isolating the Murder-Site 'Center of Minimum Distance' (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

The process of isolating the murder-site 'center of minimum distance' begins with the construction of a 'Convex Hull' (white); i.e. the smallest convex polygon (in this particular instance; an irregular pentagon), in which the six murder-sites are contained. Its construction is simply a 'connection of the dots', in which the Tabram murder-site is by-passed in order to maintain convexity. It is analogous to wrapping a rubber band around an arrangement of push-pins depicting the murder-site locations on a bulletin-board map; in as much as the centrally located push-pins (e.g. the Tabram murder-site) would not come into contact with the rubber band, and hence would not affect its polygonal 'shape'.

Of Note; The Area of the 'Convex Hull' (White): 782,065.96 Square-Yards, i.e. 0.25 Square-Miles

*** At this point, I should acknowledge the fact that all measurements of distance/area, as well as all coordinates of longitude/latitude, are in accordance with Google Earth Pro (and 1870's/1890's Ordnance Survey overlays, where applicable). ***

Conventional wisdom dictates that the 'Macnaghten-Five' victims of 'Jack the Ripper' (i.e. those, which comprise the supposed 'Canon': Nichols, Chapman, Stride, Eddowes and Kelly) were all murdered within an area of 'one square-mile'. As the measured area of the 'Convex Hull' would indicate; this assessment is actually too conservative. But, as the 'Convex Hull' in this particular case is an irregular polygon, depicting the 'tightest possible fit'; it would be somewhat inappropriate to base the 'size' of the 'Ripper's killing field' on its measured area.

As demonstrated later in this project; the 'Ripper's killing field' can be justly defined by its as yet undetermined Murder-Site Mean-Center (i.e. 'Mean-Center'; as opposed to 'Median-Center'), along with its corresponding 'Circle of Greatest Single Deviation' (0.72 Square-Miles) or 'Ellipse of Greatest Single Deviation' (0.53 Square-Miles). Using a 'happy medium' of 0.63 Square-Miles, it can be rightly asserted that the 'Macnaghten-Five' were murdered within an area of approximately 5/8 of a square-mile; that being substantially less than the convention of 'one square-mile'.

*** As previously stated; my efforts to isolate the murder-site 'center of minimum distance', at this juncture, amount to 'experimentation'. An underlying assumption, which as yet I have been unable to validate, is the notion that the 'Center of Minimum Distance' possesses all of the qualities of the 'Median-Center' (i.e. it is itself the 'Median-Center', for some given orientation of the 'field' under consideration). ***

It should be obvious that the multitude of points that each bears the distinction of being a murder-site 'median-center' (for some given orientation of the 'killing field'), lies well within the 'Convex Hull'. It should also be obvious that certain portions of the 'Convex Hull' itself could not 'play host' to the murder-site median-center.

Cases in Point:

- The triangle having the Kelly, Chapman and Nichols murder-sites as its apexes (easily visualized by constructing a green line-segment between the Kelly and Nichols murder-sites)

- The triangle having the Chapman, Nichols and Stride murder-sites as its apexes (easily visualized by constructing a green line-segment between the Chapman and Stride murder-sites)

- The triangle having the Nichols, Stride and Eddowes murder-sites as its apexes (easily visualized by constructing a green line-segment between the Nichols and Eddowes murder-sites)

- The triangle having the Stride, Eddowes and Kelly murder-sites as its apexes (easily visualized by constructing a yellow line-segment between the Stride and Kelly murder-sites) *

* An 'adjustment' will be depicted in Figure 3, to accommodate the fact that this triangle includes a fourth murder-site (i.e. Tabram), and is therefore not necessarily exclusive of any points of 'median-center'

- The triangle having the Eddowes, Kelly and Chapman murder-sites as its apexes (easily visualized by constructing a green line-segment between the Eddowes and Chapman murder-sites)

The result of this iteration is the formation of an irregular pentagram; having an irregular pentagon (white; i.e. the 'Convex Hull') as its external 'foundation', and another irregular pentagon (green/yellow) as its internal 'foundation'. The isolation of the murder-site 'center of minimum distance' effected thus far, confines its location to the pentagram's internal 'foundation' (i.e. its internal pentagon). *

This, the first iteration of the 'process of isolation', has served to isolate not only the murder-site 'center of minimum distance', but all points of murder-site 'median-center'. * This will not be inherent however, in subsequent iterations.

* Pending the 'adjustment' to be depicted in Figure 3.

Of Note: The extent of the pentagram's irregular shape or 'skew', should serve to depict the 'Median's' aforementioned immunity to 'outlier-effect'. The location of any point of 'median-center' can be affected by the relative directions to each of the outlying murder-sites, but not by the corresponding distances. If for example, the Nichols murder-site were shifted one thousand miles along the existing azimuth from the murder-site 'center of minimum distance' thereto; the location of that 'median' would not change.


Figure 3: Isolating the Murder-Site 'Center of Minimum Distance' (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Figure 3 depicts the 'adjustment' to accommodate the prohibited inclusion of a fourth murder-site (i.e. Tabram) within the 'Stride-Eddowes-Kelly triangle of elimination'; as seen in the first iteration of the process of 'murder-site median-center isolation'. This 'adjustment' is merely a slight shift of the 'yellow' side of the pentagram's internal pentagon; to the extent that it passes directly through the Tabram murder-site. The result is a 'synthetic' internal pentagon (five 'green' sides), which should include all points of murder-site 'median-center'.

The first iteration of the process of 'murder-site median-center isolation' is now completed.

The second iteration is begun with the construction of a hexagon (white), having the following points as its six apexes:

- The median of the pentagonal base of the 'Chapman' triangle (i.e. the 'Chapman' point of the pentagram

- The median of the pentagonal base of the 'Nichols' triangle (i.e. the 'Nichols' point of the pentagram

- The median of the pentagonal base* of the 'Stride' triangle (i.e. the 'Stride' point of the pentagram

- The Tabram murder-site

- The median of the pentagonal base of the 'Eddowes' triangle (i.e. the 'Eddowes' point of the pentagram

- The median of the pentagonal base* of the 'Kelly' triangle (i.e. the 'Kelly' point of the pentagram

* In accordance with the earlier 'adjustment'

The utilization of a hexagon brings a sixth murder-site (i.e. Tabram) into the 'equation'; while the orientation of that hexagon facilitates a continued 'focus' on the relative directions to each of the six sites.

As there would not appear to be any justification for believing that all points of murder-site median-center are necessarily contained within the newly constructed hexagon; this process of 'murder-site median-center isolation' has now crossed a certain 'threshold', to become specifically a process of 'murder-site 'center of minimum distance' isolation'.

The second iteration of the process is continued with the construction of a hexagram (green); by the same reasoning, and in the same manner as that seen in the construction of the pentagram.

The process has now found its 'rhythm', and can continue ad infinitum; using precisely the same iterative steps:

- Construct External Hexagon (white); Generate Hexagram (green), having Internal Hexagon (green)

- Construct External Hexagon (white); Generate Hexagram (green), having Internal Hexagon (green)

- Construct External Hexagon (white); Generate Hexagram (green), having Internal Hexagon (green)

- …


Figure 4: Isolating the Murder-Site 'Center of Minimum Distance' (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

The third iteration of the process of 'murder-site 'center of minimum distance' isolation':

- Construct External Hexagon (white); Generate Hexagram (green), having Internal Hexagon (green)

It would appear that subsequent iterations should be 'blessed' with the formation of external hexagons (white) of increasingly regular shape; thus leading to the generation of hexagrams (green) of increasingly regular shape; having themselves, internal hexagons (green) of increasingly regular shape. If indeed this is the case; then a gradual convergence should occur between the stationary murder-site 'center of minimum distance' and the successive points of internal-hexagon 'centroid'. *

In other words: As successive iterations generate internal hexagons (green) of increasingly regular shape; a point is reached, at which the internal-hexagon 'centroid' provides a reasonable estimate of the murder-site 'center of minimum distance'.

* A polygonal 'Centroid' is the 'center of mass' / 'center of gravity' of a convex polygon, in which 'mass' (hypothetical; in this particular case) is evenly distributed; either throughout the figure or at each of its points of apex. If 'mass' is assumed specifically to be evenly distributed at the polygonal points of apex, then the 'Centroid' can be described as being the 'Mean-Center' of that distribution of points. This will be discussed in more detail, as the project progresses.

The 'centroid' (green dot) of the third iteration's internal hexagon (green) is easily pinpointed:

- Begin at any apex

- Move one half of the distance toward any of the five remaining apexes

- From that point; move one third of the distance toward any of the four remaining apexes

- From that point; move one fourth of the distance toward any of the three remaining apexes

- From that point; move one fifth of the distance toward any of the two remaining apexes

- From that point; move one sixth of the distance toward the lone remaining apex

- Arrive at the 'centroid' (green dot)

*** Albeit through 'experimentation', and perhaps an insufficient degree of iteration; I believe that we have attained a worthwhile estimate of the murder-site 'center of minimum distance'. ***

Interior of Common Lodging House, 18 Thrawl Street, Parish of Christ Church Spitalfields, County of Middlesex
Longitude: 0° 4' 19.06" West
Latitude: 51° 31' 3.23" North

18 Thrawl Street, Parish of Christ Church Spitalfields: The common lodging house, where Polly Nichols is known to have resided during the weeks prior to her murder (excepting the final week, during which time her whereabouts were unknown; but presumed by Emily Holland to have been 56 Flower & Dean Street, Parish of Christ Church Spitalfields).


Figure 5: Three 'Medians' (i.e. Two Points of Murder-Site Median-Center, and One Estimation of the Murder-Site 'Center of Minimum Distance') (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

The isopleths (red, orange and yellow color-shadings) represent the 'central tendency' of varying degrees of likelihood, with regard to the location of the murder-site 'center of minimum distance', within the internal hexagon of the third iteration.

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As the 'Murder-Site Median-Center' is not a unique statistic (i.e. as there is a distinct murder-site median-center for every conceivable orientation of the murder-site 'field'); and as the 'Center of Minimum Distance' is in fact quite elusive (i.e. as the lone instance of a murder-site median-center possessing all of the qualities of a true 'median' is forever 'incognito'); the utilization of the 'Median' for the establishment of a "particular place" to serve as the focal point of the murder 'locale', would seem to be somewhat impractical.

And as its immunity to 'outlier-effect' renders the 'Median' of little use as a descriptive statistic beyond the establishment of a "particular place"; its utilization for the establishment of a "limited portion of space" to serve as a specified degree of murder-site vicinity, would seem to be rather untenable.

Another widely used measure of 'central tendency', which unlike the 'Median' minimizes the aggregate 'squared' distance to a set of observations, is the 'Mean'; more commonly referred to as the 'Average'.

Consider again, the following set of six numbers: 12, 36, 33, 3, 11 and 55.

The arithmetic mean or 'average' is easily calculated by adding the six quantities, and dividing the sum (150) by the number of quantities within the set (6): Arithmetic Mean or 'Average Quantity' = 25.

Again; this measure of 'central tendency' (i.e. the 'Mean') minimizes the aggregate 'squared' distance to a set of observations; such that in this particular case, the aggregate 'squared' distance from '25' to each of the six components of the set, is the smallest attainable.

It should also be noted that the 'Mean', unlike the 'Median', is affected by outliers. If for example, the sixth observed quantity in the above set, were '1,055', rather than just '55'; the mean or 'average quantity' of the set would be '191.67', instead of just '25'.

This sensitivity to 'outlier-effect' actually renders the 'Mean' (i.e. the 'Mean-Center', in the case of a two-dimensional 'field' of murder-sites) of tremendous value as a descriptive statistic beyond the establishment of a "particular place" to serve as the focal point of the murder 'locale'. In other words: The 'Murder-Site Mean-Center' can in fact, lead to the establishment of a "limited portion of space" to serve as a specified degree of vicinity, on the basis of murder-site dispersion around its location.

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While there is a distinct manifestation of the 'Median-Center' for every conceivable orientation of a two-dimensional 'field' of observations; there is only one 'Mean-Center'. And whereas the unique manifestation of the 'Median-Center', which minimizes the aggregate distance to the set of observations (i.e. the 'Center of Minimum Distance') is nearly impossible to pinpoint; the unique statistic, which minimizes the aggregate 'squared' distance to the set of observations (i.e. the 'Mean-Center') is actually quite simple to pinpoint.

Reference: The murder-site longitudinal and latitudinal coordinates, given above.

By converting each of these coordinates to decimal form, the mean or 'average' longitudinal and latitudinal coordinates can be easily calculated.

Martha Tabram
Longitude: 0.07179167° West
Latitude: 51.51683333° North

Polly Nichols
Longitude: 0.06042500° West
Latitude: 51.52003889° North

Annie Chapman
Longitude: 0.07261111° West
Latitude: 51.52046389° North

Elizabeth Stride
Longitude: 0.06559444° West
Latitude: 51.51373333° North

Catherine Eddowes
Longitude: 0.07807222° West
Latitude: 51.51370833° North

Mary Jane Kelly
Longitude: 0.07513056° West
Latitude: 51.51865833° North

Mean or 'Average' Longitudinal and Latitudinal Coordinates
Mean Longitude: 0.07060417° West
Mean Latitude: 51.51723935° North

The point, at which the mean longitude and mean latitude intersect (having the two respective means as its own longitudinal and latitudinal coordinates) is the mean-center of the six murder-sites, i.e. the 'average murder-site'.

Murder-Site Mean-Center
Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North

Before discussing the murder-site mean-center and its relevance as the "particular place" or focal point of the murder 'locale'; some other more 'direct' methods of pinpointing its location shall be considered.


Figure 6: Pinpointing the Murder-Site Mean-Center (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

In the above figure, the location of the murder-site mean-center or 'average murder-site' is determined by dividing the set of six murder-sites into three subsets (each consisting of two murder-sites: Kelly & Chapman; Stride & Nichols; Eddowes & Tabram); finding the mean-center of each of the three subsets; and then pinpointing the overall mean-center of the three points of subset mean-center.

Step 1: Kelly and Chapman Subset. Measure the straight-line distance, by which the two sites are separated: 290.82 yards. Move one half of this distance, from the Kelly murder-site directly toward the Chapman murder-site: 145.41 yards. Arrive at the mean-center of the two sites: Point '1'.

Step 2: Stride and Nichols Subset. Measure the straight-line distance, by which the two sites are separated: 860.66 yards. Move one half of this distance, from the Stride murder-site directly toward the Nichols murder-site: 430.33 yards. Arrive at the mean-center of the two sites: Point '2'.

Step 3: Eddowes and Tabram Subset. Measure the straight-line distance, by which the two sites are separated: 609.32 yards. Move one half of this distance, from the Eddowes murder-site directly toward the Tabram murder-site: 304.66 yards. Arrive at the mean-center of the two sites: Point '3'.

The three points of subset mean-center, Points '1', '2' and '3', form the vertices of a triangle, which is color-shaded red. The overall mean-center of the three points of subset mean-center coincides therefore, with the 'centroid' of the triangle.

As Previously Stated: A polygonal 'Centroid' is the 'center of mass' / 'center of gravity' of a convex polygon, in which 'mass' (hypothetical; in this particular case) is evenly distributed; either throughout the figure or at each of its points of apex. If 'mass' is assumed specifically to be evenly distributed at the polygonal points of apex, then the 'Centroid' can be described as being the 'Mean-Center' of that distribution of points.

Step 4: Red Color-Shaded Triangle. Measure the straight-line distance, by which Vertices '3' and '1' are separated: 527.65 yards. Move one half of this distance, from Vertex '3' directly toward Vertex '1': 263.83 yards. Arrive at the mean-center of the two vertices: Point '4'.

Step 5: Red Color-Shaded Triangle. Measure the straight-line distance, by which Point '4' and Vertex '2' are separated: 866.30 yards. Move one third of this distance, from Point '4' directly toward Vertex '2': 288.77 yards. Arrive at the 'centroid' / 'center of mass' / 'center of gravity' of the triangle; i.e. the mean-center of its three vertices; that being the overall mean-center of the three points of subset mean-center: Point '5'.

Point '5' is of course, the mean-center of the overall set of six murder-sites. And indeed, it is precisely the same 'average murder-site' that was determined above, by combining the mean longitudinal and latitudinal coordinates of the overall set.

Mean or 'Average' Longitudinal and Latitudinal Coordinates
Mean Longitude: 0.07060417° West
Mean Latitude: 51.51723935° North

Murder-Site Mean-Center: Point '5' Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North

It should be noted that the selection of murder-site subsets (e.g. Kelly & Chapman; Stride & Nichols; Eddowes & Tabram) and order of triangle vertex-factorization is unimportant. Any chosen grouping of three pairs of murder-sites will render three points of subset mean-center, which will invariably coincide with the vertices of a triangle (excepting the highly improbable scenario, in which the three points of subset mean-center form a straight line). The triangle's 'centroid' (i.e. the mean-center of the overall set of six murder-sites) is then easily determined by moving one half of the distance from any of the three vertices, directly toward either of the other two; and then moving one third of the distance from that point (i.e. the mean-center of the two chosen vertices), directly toward the lone remaining vertex.


Figure 7: Pinpointing the Murder-Site Mean-Center (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Three distinct sequences of measurement (Red; Blue; Yellow); which lead to precisely the same murder-site mean-center.

In conducting the three sequences of measurement; one should perceive the six murder-sites under consideration, as being situated (i.e. stationary) in an open plane of 'free space' (i.e. a plane, in which there is a complete lack of urban, rural and geologic topography, as well as any other physical features that could pose any form of impediment or semblance of physical obstruction). One should also perceive the set of six murder-sites as being a system of six physical 'particles'; in which each 'particle' is of equivalent mass, and is therefore exerting an equivalent degree of 'gravitational pull'.

If the site of an impending subsequent murder (i.e. a seventh 'particle', having mass equivalent to that of the other six) is brought into the 'equation' as a random variable (i.e. as a freely floating 'particle'); then its location will be determined by the interaction of its own 'gravitational pull' with that of the other six murder-sites, whose locations have already been established. In other words: If an impending subsequent murder site is tossed into the open plane of the murder-site 'field', the interaction of its own 'gravitational pull' with that of the six stationary murder-sites will draw the seventh murder-site to a point, at which there is a maximization of total gravitational force (i.e. an equilibrium of the seven individual forces of 'gravitational pull').

The concept is more easily understood if the 'gravitational pull' of the impending subsequent murder-site is stricken from the 'equation'. As the effect of the 'gravitational pull' of this seventh 'particle' is actually 'redundant' to that of each of the other six 'particles', its inclusion in the overall 'equation' is conceptually unnecessary. *

* As the 'gravitational pull' exerted by the impending subsequent murder-site interacts with that of any one of the established murder-sites (e.g. the Chapman murder-site); the resultant degree of gravitational force is directly proportional to the mass of each of the two sites (i.e. the 'product' of the masses of the two murder-sites); and inversely proportional to the distance from each of the two sites to the other (i.e. the 'square' of the distance between the two murder-sites). It would therefore be technically impossible to strike the 'gravitational pull' of the impending subsequent murder-site from the 'equation'. But again; as its effect is essentially a 'redundancy', in this particular case, its inclusion in the overall 'equation' is conceptually unnecessary.

One can now perceive the impending subsequent murder-site as a freely floating random variable, whose destiny lies in the location of the point of equilibrium or 'balance', in the competing efforts of each of the six stationary murder-sites to 'reign-in' the seventh.

Put simply: One can now perceive the impending subsequent murder-site as being caught in the middle of a six-way tug-of-war, in which each of the six stationary murder-sites is competing for its physical companionship. This as yet freely floating murder-site will come to a complete stand-still and remain forever stationary at the point of maximum aggregate tug-of-war (i.e. gravitational) force: The Murder-Site 'Center of Mass'. *

* In this particular case; the Murder-Site 'Center of Mass'; Murder-Site 'Center of Gravity'; and Murder-Site 'Mean-Center' are all one and the same. But, on the basis of the concept just described; "Murder-Site 'Center of Mass'" is the appropriate label.

Imagine now, that an impending subsequent murder-site is tossed into the plane of the murder-site 'field', with a set of step-by-step 'directions' to help guide it to its destination. Imagine also, that the force of 'gravitational pull' of each respective stationary murder-site is not enacted until the impending subsequent murder-site initiates its respective 'Step' in this set of 'directions'.

These hypothetical 'directions', of course, are the following sequence of measurement:

Red Sequence of Measurement: Order of Perceived Level of 'Acceptance' as Victim of 'Jack the Ripper'

Step '0': Red Sequence of Measurement. Locate the Chapman murder-site. As this site accounts for all of the total 'gravitational pull' being exerted thus far; move the entire distance, from any point in the plane, directly toward the Chapman murder-site. Arrive at the Chapman murder-site.

Step 1: Red Sequence of Measurement. Measure the straight-line distance, by which the Chapman and Nichols murder-sites are separated: 925.52 yards. As each of the two sites accounts for one half of the total 'gravitational pull' being exerted thus far; move one half of this distance, from the Chapman murder-site directly toward the Nichols murder-site: 462.76 yards. Arrive at the mean-center of the two murder-sites (Chapman and Nichols): Point '1'.

Step 2: Red Sequence of Measurement. Measure the straight-line distance, by which Point '1' and the Eddowes murder-site are separated: 1,183.31 yards. As each of the three sites (Chapman, Nichols and Eddowes) accounts for one third of the total 'gravitational pull' being exerted thus far; move one third of this distance, from Point '1' directly toward the Eddowes murder-site: 394.44 yards. Arrive at the mean-center of the three murder-sites (Chapman, Nichols and Eddowes): Point '2'.

Step 3: Red Sequence of Measurement. Measure the straight-line distance, by which Point '2' and the Kelly murder-site are separated: 367.98 yards. As each of the four sites (Chapman, Nichols, Eddowes and Kelly) accounts for one fourth of the total 'gravitational pull' being exerted thus far; move one fourth of this distance, from Point '2' directly toward the Kelly murder-site: 92.00 yards. Arrive at the mean-center of the four murder-sites (Chapman, Nichols, Eddowes and Kelly): Point '3'.

Step 4: Red Sequence of Measurement. Measure the straight-line distance, by which Point '3' and the Stride murder-site are separated: 708.24 yards. As each of the five sites (Chapman, Nichols, Eddowes, Kelly and Stride) accounts for one fifth of the total 'gravitational pull' being exerted thus far; move one fifth of this distance, from Point '3' directly toward the Stride murder-site: 141.65 yards. Arrive at the mean-center of the five murder-sites (Chapman, Nichols, Eddowes, Kelly and Stride): Point '4'.

Step 5: Red Sequence of Measurement. Measure the straight-line distance, by which Point '4' and the Tabram murder-site are separated: 123.18 yards. As each of the six sites (Chapman, Nichols, Eddowes, Kelly, Stride and Tabram) accounts for one sixth of the total 'gravitational pull' being exerted by the overall set; move one sixth of this distance, from Point '4' directly toward the Tabram murder-site: 20.53 yards. Arrive at the mean-center of the overall set of six murder-sites: Point '5'.

Murder-Site Mean-Center (i.e. Murder-Site 'Center of Mass'): Point '5'
Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex*
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North

* Specifically: 2.23 Yards ~North of the Northeast Corner of the Building, which now Occupies this Location

Put Simply (Steps 1–5):

- From the Chapman murder-site, move one half of the distance toward the Nichols murder-site
- From that point ('1'), move one third of the distance toward the Eddowes murder-site
- From that point ('2'), move one fourth of the distance toward the Kelly murder-site
- From that point ('3'), move one fifth of the distance toward the Stride murder-site
- From that point ('4'), move one sixth of the distance toward the Tabram murder-site
- Arrive at the mean-center (i.e. 'center of mass') of the overall set of six murder-sites: '5'

So, if the following conditions were to apply:

- The six murder-sites under consideration were situated (i.e. stationary) in an open plane of 'free space' (i.e. a plane, in which there was a complete lack of urban, rural and geologic topography, as well as any other physical features that could pose any form of impediment or semblance of physical obstruction) …

- The set of six murder-sites under consideration was a system of six physical 'particles'; in which each 'particle' was of equivalent mass, and would therefore exert an equivalent degree of 'gravitational pull' …

- Apart from the equivalent degree of 'gravitational pull' that each of the six murder-sites would exert; and the corresponding degree of aggregate gravitational force that each of the six murder-sites would 'feel'; the open plane of 'free space' and all things contained therein, was under no internal or external influences of any kind …

And:

- An impending subsequent murder-site (i.e. a freely floating random variable) of mass equivalent to that of each of the six stationary murder-sites was tossed into the plane …

Then:

- The impending subsequent murder-site would come to a complete stand-still and remain forever stationary at the murder-site mean-center (i.e. the murder-site 'center of mass').

Put Simply:

- If the six murder-sites under consideration were placed in a 'vacuum' of a purely 'physical' context; then any correlated subsequent murder that was to occur, would do so precisely at the murder-site mean-center.

Murder-Site Mean-Center (i.e. Murder-Site 'Center of Mass')
Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North

So why is the concept of 'Center of Mass' tantamount to that of 'Mean-Center'?

Consider again, the following set of six numbers: 12, 36, 33, 3, 11 and 55.

The arithmetic mean or 'average' is easily calculated by adding the six quantities, and dividing the sum (150) by the number of quantities within the set (6): Arithmetic Mean or 'Average Quantity' = 25.

But, the arithmetic mean or 'average quantity' in this instance, is just as easily determined by perceiving it as being a 'center of mass'; i.e. the point on the mathematical number-line, at which an added variable quantity would become stationary, given an equal degree of 'gravitational pull' from each of the six quantities within the set.

Step 1: Number Line Measurement. Measure the number-line distance, by which '12' and '36' are separated: 24 units. As each of the two quantities accounts for one half of the total 'gravitational pull' being exerted thus far; move one half of this distance, from '12' directly toward '36': 12 units. Arrive at the arithmetic mean or 'average' of the two quantities ('12' and '36'): '24'.

Step 2: Number Line Measurement. Measure the number-line distance, by which '24' and '33' are separated: 9 units. As each of the three quantities ('12', '36' and '33') accounts for one third of the total 'gravitational pull' being exerted thus far; move one third of this distance, from '24' directly toward '33': 3 units. Arrive at the arithmetic mean or 'average' of the three quantities ('12', '36' and '33'): '27'.

Step 3: Number Line Measurement. Measure the number-line distance, by which '27' and '3' are separated: 24 units. As each of the four quantities ('12', '36', '33' and '3') accounts for one fourth of the total 'gravitational pull' being exerted thus far; move one fourth of this distance, from '27' directly toward '3': 6 units. Arrive at the arithmetic mean or 'average' of the four quantities ('12', '36', '33' and '3'): '21'.

Step 4: Number Line Measurement. Measure the number-line distance, by which '21' and '11' are separated: 10 units. As each of the five quantities ('12', '36', '33', '3' and '11') accounts for one fifth of the total 'gravitational pull' being exerted thus far; move one fifth of this distance, from '21' directly toward '11': 2 units. Arrive at the arithmetic mean or 'average' of the five quantities ('12', '36', '33', '3' and '11'): '19'.

Step 5: Number Line Measurement. Measure the number-line distance, by which '19' and '55' are separated: 36 units. As each of the six quantities ('12', '36', '33', '3', '11' and '55') accounts for one sixth of the total 'gravitational pull' being exerted thus far; move one sixth of this distance, from '19' directly toward '55': 6 units. Arrive at the arithmetic mean or 'average' of the six quantities ('12', '36', '33', '3', '11' and '55'): '25'.

Put Simply (Steps 1–5):

- From '12', move one half of the distance toward '36'
- From that point ('24'), move one third of the distance toward '33'
- From that point ('27'), move one fourth of the distance toward '3'
- From that point ('21'), move one fifth of the distance toward '11'
- From that point ('19'), move one sixth of the distance toward '55'
- Arrive at the arithmetic mean or 'average': '25'

Remember too; that the 'Mean' minimizes the aggregate 'squared' distance to a set of observations; such that in this particular case, the aggregate 'squared' distance from '25' to each of the six components of the set, is the smallest attainable. Accordingly; the 'Mean-Center' minimizes the aggregate 'squared' distance to a set of observations; such that in the case of a two-dimensional 'field' of murder-sites, the aggregate 'squared' distance from the murder-site mean-center to each of the murder-sites in the 'field', is the smallest attainable. Well; this in fact, is the catalyst in the theoretical 'physical vacuum', in which any correlated subsequent murder that was to occur, would do so precisely at the murder-site mean-center. This is due to the fact, that total gravitational force is maximized at the point, at which aggregate 'squared' distance from the impending subsequent murder-site to each of the six stationary murder-sites, is the smallest attainable.

The significance of the Murder-Site 'Mean-Center', therefore, lies in the capacity that it fills as the 'Average Murder-Site' of the overall set: That of 'Expected Murder-Site'.

Once Again: If the six murder-sites under consideration were placed in a 'vacuum' of a purely 'physical' context; then any correlated subsequent murder that was to occur, would do so precisely at the murder-site mean-center.

What is now being considered, however, is a 'vacuum' more-or-less, of a purely 'mathematical' (i.e. 'statistical') context. In this case, there is no 'guarantee' that the impending subsequent murder will occur precisely at the murder-site mean-center; rather there is simply an 'expectation' that it will do so.

If the set of six numbers (12, 36, 33, 3, 11 and 55) used in the above example, were a sample of observed 'outcomes' (i.e. the outcomes of a series of experimentation 'trials'), the 'expected outcome' of an impending subsequent 'trial' would be the arithmetic mean or 'average' of the six quantities: 25. Likewise, if the set of six murder-sites under consideration were considered to be a 'sample', in as much as there was not only a possibility of subsequent correlated murders, but perhaps a degree of perceived likelihood; the 'expected murder-site' of any such subsequent murder would be the mean-center or 'average murder-site' of the overall set.

At this point, one should assume the perspective of those who were actually investigating this series of six murders, in the closing weeks of the autumn of 1888: That the series of seemingly correlated atrocities showed no signs of abating; so there was indeed a distinct possibility, if not a likelihood, of subsequent murders. From this point of view, one could more easily conceptualize the overall set of six murder-sites as constituting a mere 'sample'; and one would then be attuned to the meaning of any forthcoming references to this "murder-site 'sample'".

Of Particular Note:

- Murder-Site 'Sample': The overall set of six murder-sites under consideration.

- Murder-Site 'Population': The overall set of six murder-sites under consideration (i.e. the Murder-Site 'Sample'), 'plus' the hypothetical set of any correlated subsequent murder-sites that would come under consideration.

So, to reiterate the above assertion, regarding the significance of the murder-site mean-center: It lies in the capacity that is filled, or the role that is played by the 'average murder-site' of the 'sample' set: That of 'Expected Murder-Site'. The mean-center of the murder-site 'sample' is, in other words: The point, on which a hypothetical impending subsequent murder would be most likely to occur.*

* i.e.: The point, on which the 'next' murder is most likely to occur.

This facet of the 'average murder-site' must not be misconstrued. Indeed, as the 'expected murder-site'; it is the single most-probable point within the plane of the murder site 'sample', for the location of the 'next' murder. But, even in light of its superior 'probability density'; as a single point within a plane, the probability of it 'playing host' to the impending subsequent murder is effectively 'zero'. In fact, in the case of a circle having the 'expected murder-site' as its center, and a radius of one yard (Area: 3.14 Square-Yards), i.e. sufficient space for the containment of a sprawled corpse; the probability of the 'next' murder occurring within would be less than one eighth of one percent. This seemingly low probability, however, would be greater than that for any other circle of like size, anywhere within the plane of the murder-site 'sample'.*

* Note: The probabilities of various areas 'playing host' to the hypothetical impending subsequent murder will be covered later, in great detail.

As the single most-probable point for the location of the 'next' murder, the murder-site mean-center is clearly the most meaningful "particular place" or focal point, for the establishment of a murder 'locale'. So, having determined the 'average murder-site'; and having considered the significance of its role as the 'expected murder-site'; a brief examination of the site itself, and the geography of its immediate surroundings, may be worthwhile.

But first; the approach to pinpointing the murder-site mean-center, depicted in Figure 7, will be further evaluated through consideration of the 'Blue' and 'Yellow' sequences of measurement.

Blue Sequence of Measurement: Order of Murder Chronology.

Step 1: Blue Sequence of Measurement. Measure the straight-line distance, by which the Tabram and Nichols murder-sites are separated: 945.83 yards. As each of the two sites accounts for one half of the total 'gravitational pull' being exerted thus far; move one half of this distance, from the Tabram murder-site directly toward the Nichols murder-site: 472.92 yards. Arrive at the mean-center of the two murder-sites (Tabram and Nichols): Point '1'.

Step 2: Blue Sequence of Measurement. Measure the straight-line distance, by which Point '1' and the Chapman murder-site are separated: 551.28 yards. As each of the three sites (Tabram, Nichols and Chapman) accounts for one third of the total 'gravitational pull' being exerted thus far; move one third of this distance, from Point '1' directly toward the Chapman murder-site: 183.76 yards. Arrive at the mean-center of the three murder-sites (Tabram, Nichols and Chapman): Point '2'.

Step 3: Blue Sequence of Measurement. Measure the straight-line distance, by which Point '2' and the Stride murder-site are separated: 684.50 yards. As each of the four sites (Tabram, Nichols, Chapman and Stride) accounts for one fourth of the total 'gravitational pull' being exerted thus far; move one fourth of this distance, from Point '2' directly toward the Stride murder-site: 171.13 yards. Arrive at the mean-center of the four murder-sites (Tabram, Nichols, Chapman and Stride): Point '3'.

Step 4: Blue Sequence of Measurement. Measure the straight-line distance, by which Point '3' and the Eddowes murder-site are separated: 934.67 yards. As each of the five sites (Tabram, Nichols, Chapman, Stride and Eddowes) accounts for one fifth of the total 'gravitational pull' being exerted thus far; move one fifth of this distance, from Point '3' directly toward the Eddowes murder-site: 186.93 yards. Arrive at the mean-center of the five murder-sites (Tabram, Nichols, Chapman, Stride and Eddowes): Point '4'.

Step 5: Blue Sequence of Measurement. Measure the straight-line distance, by which Point '4' and the Kelly murder-site are separated: 460.97 yards. As each of the six sites (Tabram, Nichols, Chapman, Stride, Eddowes and Kelly) accounts for one sixth of the total 'gravitational pull' being exerted by the overall set; move one sixth of this distance, from Point '4' directly toward the Kelly murder-site: 76.83 yards. Arrive at the mean-center of the overall set of six murder-sites: Point '5'.

Put simply (Steps 1–5):

- From the Tabram murder-site, move one half of the distance toward the Nichols murder-site
- From that point ('1'), move one third of the distance toward the Chapman murder-site
- From that point ('2'), move one fourth of the distance toward the Stride murder-site
- From that point ('3'), move one fifth of the distance toward the Eddowes murder-site
- From that point ('4'), move one sixth of the distance toward the Kelly murder-site
- Arrive at the mean-center of the overall set of six murder-sites: '5'

This sequence of measurements, in which the points of mean-center of incrementally larger murder-site subsets converge toward the mean-center of the overall set, has led to precisely the same 'average murder-site' as that determined earlier, through other methodology:

Murder-Site Mean-Center: Point '5'
Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North

As the basis of sequential order in this case, is the chronology of the six murders under consideration, the incremental murder-site subsets in two specific instances, have points of mean-center that would have been of particular interest to those investigating these crimes, in the weeks leading up to the so-called 'Double Event'; and again, in the weeks prior to the murder of Mary Jane Kelly.

The mean-center or 'average murder-site' of the 'Tabram, Nichols and Chapman' subset (Point '2'), would have played the role of 'expected murder-site' (i.e. the single most-probable point for the location of the impending subsequent murder), from the time of discovery of the body of Annie Chapman (8 September), until the discovery of the body of Elizabeth Stride (30 September). As the murder of Annie Chapman was likely to have been the catalyst that spawned the perception throughout London's Metropolitan Police Force, that a 'series' of murders was underway; an awareness of this 'expected murder-site' and an understanding of its significance might have proved quite useful to those charged with investigating these atrocities.

Subset Mean-Center (Tabram, Nichols and Chapman): Point '2'
Rear Portion of Building, North Side of Chicksand Street, Hamlet of Mile End New Town, County of Middlesex
Longitude: 0° 4' 5.79" West
Latitude: 51° 31' 8.80" North

Obviously, the 'next' murder did not take place on what had been the single most-probable point for the location of its occurrence: The 'expected murder-site'. But the joint establishment of this "particular place" as a focal point, and a corresponding "limited portion of space" as a specified measure of vicinity (i.e. the establishment of a murder 'locale'); might have given Scotland Yard a better feeling for the likelihood of varying degrees of deviation from its 'expectations' (e.g. the degree of deviation that was realized upon the discovery of Elizabeth Stride's body). *

* Note: The probabilities or "likelihood" of varying degrees of deviation from the 'expected murder-site' will be covered later, in great detail.

As the mean-center or 'average murder-site' of the 'Tabram, Nichols, Chapman and Stride' subset (Point '3'), would have filled the capacity of 'expected murder-site' for less than one hour (~1:00AM - ~1:45AM) on the morning of 30 September 1888; it would never have been of any significance to those involved in the investigations of these crimes. However, the mean-center or 'average murder-site' of the 'Tabram, Nichols, Chapman, Stride and Eddowes' subset (Point '4'), would have played the role of 'expected murder-site' from the morning of the 'Double Event' (30 September), until the discovery of the body of Mary Jane Kelly (9 November).

Subset Mean-Center (Tabram, Nichols, Chapman, Stride and Eddowes): Point '4'
Rear Portion of Building, off West Side of Green Dragon Place, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 10.92" West
Latitude: 51° 31' 1.04" North

Yellow Sequence of Measurement: Order of Proximity to 'Known' Murder-Site Mean-Center.

Step 1: Yellow Sequence of Measurement. Measure the straight-line distance, by which the Tabram and Kelly murder-sites are separated: 336.65 yards. As each of the two sites accounts for one half of the total 'gravitational pull' being exerted thus far; move one half of this distance, from the Tabram murder-site directly toward the Kelly murder-site: 168.33 yards. Arrive at the mean-center of the two murder-sites (Tabram and Kelly): Point '1'.

Step 2: Yellow Sequence of Measurement. Measure the straight-line distance, by which Point '1' and the Chapman murder-site are separated: 336.43 yards. As each of the three sites (Tabram, Kelly and Chapman) accounts for one third of the total 'gravitational pull' being exerted thus far; move one third of this distance, from Point '1' directly toward the Chapman murder-site: 112.14 yards. Arrive at the mean-center of the three murder-sites (Tabram, Kelly and Chapman): Point '2'.

Step 3: Yellow Sequence of Measurement. Measure the straight-line distance, by which Point '2' and the Stride murder-site are separated: 829.48 yards. As each of the four sites (Tabram, Kelly, Chapman and Stride) accounts for one fourth of the total 'gravitational pull' being exerted thus far; move one fourth of this distance, from Point '2' directly toward the Stride murder-site: 207.37 yards. Arrive at the mean-center of the four murder-sites (Tabram, Kelly, Chapman and Stride): Point '3'.

Step 4: Yellow Sequence of Measurement. Measure the straight-line distance, by which Point '3' and the Eddowes murder-site are separated: 684.94 yards. As each of the five sites (Tabram, Kelly, Chapman, Stride and Eddowes) accounts for one fifth of the total 'gravitational pull' being exerted thus far; move one fifth of this distance, from Point '3' directly toward the Eddowes murder-site: 136.99 yards. Arrive at the mean-center of the five murder-sites (Tabram, Kelly, Chapman, Stride and Eddowes): Point '4'.

Step 5: Yellow Sequence of Measurement. Measure the straight-line distance, by which Point '4' and the Nichols murder-site are separated: 1,012.20 yards. As each of the six sites (Tabram, Kelly, Chapman, Stride, Eddowes and Nichols) accounts for one sixth of the total 'gravitational pull' being exerted by the overall set; move one sixth of this distance, from Point '4' directly toward the Nichols murder-site: 168.70 yards. Arrive at the mean-center of the overall set of six murder-sites: Point '5'.

Put simply (Steps 1–5):

- From the Tabram murder-site, move one half of the distance toward the Kelly murder-site
- From that point ('1'), move one third of the distance toward the Chapman murder-site
- From that point ('2'), move one fourth of the distance toward the Stride murder-site
- From that point ('3'), move one fifth of the distance toward the Eddowes murder-site
- From that point ('4'), move one sixth of the distance toward the Nichols murder-site
- Arrive at the mean-center of the overall set of six murder-sites: '5'

Once Again: A sequence of measurements, in which the points of mean-center of incrementally larger murder-site subsets converge toward the mean-center of the overall set, has led to precisely the same 'average murder-site' as that determined earlier, through other methodology:

Murder-Site Mean-Center: Point '5'
Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North


Figure 8: Three 'Medians' and One 'Mean' (i.e. Two Points of Murder-Site Median-Center, One Estimation of the Murder-Site 'Center of Minimum Distance', and the Murder-Site Mean-Center) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Of Note: Having established the "particular place" or focal point, for the parameterization of a murder 'locale'; it will henceforth be referred to as the Murder-Site 'Epicenter'. Of course, the term is neither the most conceptually applicable ('Average Murder-Site' / 'Expected Murder-Site'), nor the most technically accurate (Murder-Site 'Center of Mass' (physical perspective) / Murder-Site 'Center of Gravity' (physical perspective, as yet not discussed) / Murder-Site 'Mean-Center' (statistical perspective)). However, it provides the most 'fluid' manner of reference to the point in question; and alleviates the need for constant 'crisscross' clarification of the 'perspective' being considered. Its 'meaning' therefore, in the given context, should be fully understood.

------------

It is often the case that the 'expected outcome' of a given set of observations is itself impractical, or perhaps impossible. Consider for instance, some of the subset points of 'epicenter' that were encountered in each of the three sequences of measurement, depicted in Figure 7: "Rear Portion of a 'Chemical Works' Facility …"; "Rear Portion of a Building …"; etc … The murder-sites of Martha Tabram and Mary Jane Kelly notwithstanding; urban topography can place significant limitations on the practicality, and even the possibility of a particular point 'playing host' to the impending subsequent murder. It is therefore quite remarkable that the murder-site 'epicenter', in this case, not only affords both physical possibility and 'Ripperesque' practicality; but actually coincides with a prominent feature in the landscape of the 'Whitechapel Murders': The very spot, on which many surmise that Emma Smith was confronted by her alleged assailants.


Figure 9: Murder-Site Mean-Center (i.e. Murder-Site 'Epicenter') (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Most accounts of the circumstances, in which Emma Smith was allegedly assaulted by a group of ruffians, on the morning of 3 April 1888, include references to 'Osborn Street' and/or the vicinity of a 'cocoa factory', with regard to the location of the attack. The references are generally vague and somewhat difficult to comprehend, as the thoroughfare 'Osborn Street' became 'Brick Lane' as it progressed northward through its intersection with Wentworth Street (west) / Old Montague Street (east), before passing the east side of Taylor Brothers' Chocolate & Mustard Factory. As the northwestern extremity of Osborn Street, i.e. the southwest corner of its junction with Wentworth Street, was the point at which it was most closely 'connected' to the 'cocoa factory', it has perhaps been deemed to have been the most likely venue for the assault.

However, a very specific reference to the location of the attack was included in a report filed by Inspector Edmund Reid, Metropolitan Police Force, H Division (date unknown): "The offence had been committed on the pathway opposite No. 10 Brick Lane". Ironically, this assertion was contradictory to one made earlier in the same report: "She had been assaulted and robbed in Osborne (sic.) Street". But the specificity of the "opposite No. 10 Brick Lane" reference should not be ignored; especially in light of the distinct possibility that all 'primary' references to 'Osborn Street' were actually intended to describe the point, at which Smith first encountered the alleged group of men, who then followed her north into Brick Lane.

It would seem unlikely therefore, that the murder-site 'epicenter' actually coincides with the spot that Emma Smith identified as being the location, in which she was confronted by her alleged assailants. In fact, if Reid's 'Brick Lane' reference is assumed to be accurate, then the murder-site 'epicenter' lies approximately thirty eight yards southeast of the spot, on which Smith claimed she was attacked.

"the pathway opposite No. 10 Brick Lane"
Northeastern Exterior of Taylor Brothers' Chocolate & Mustard Factory, West Side of Brick Lane, Parish of Christ Church Spitalfields, County of Middlesex
Longitude: 0° 4' 15.15" West
Latitude: 51° 31' 3.02" North

In any case, the murder-site 'epicenter' is in remarkably close proximity to the spot, on which the 'Whitechapel Murders' saga purportedly began. If nothing else; the murder of Emma Smith set the 'stage' for the six murders that followed, within the 'Whitechapel' series. But, the purported location of Smith's encounter with her alleged assailants notwithstanding; the intersection of Wentworth Street / Old Montague Street and Osborn Street / Brick Lane is nonetheless a prominent feature in the landscape of the 'Whitechapel Murders'.

This crossroads of two major thoroughfares (four 'named' streets) was a pivotal point in the boundary that separated the Civil Parishes of Christ Church Spitalfields and St. Mary Whitechapel. The boundary ran easterly along Wentworth Street, from Middlesex Street to Brick Lane; and then northerly along Brick Lane to a point just beyond Chicksand Street; and then easterly again, to its termination as a 'T'-shaped junction with the boundary of The Hamlet of Mile End New Town. As such, the northwestern quadrant of the intersection was situated within the Parish of Christ Church Spitalfields; whereas the remaining three quadrants lay within the Parish of St. Mary Whitechapel. Hence a subtle, but very significant difference between the aforementioned distinctions of the murder-site 'epicenter' and the 'more likely' purported location of Emma Smith's encounter with her alleged assailants.

Murder-Site Mean-Center (i.e. Murder-Site 'Epicenter')
Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex

"the pathway opposite No. 10 Brick Lane"
Northeastern Exterior of Taylor Brothers' Chocolate & Mustard Factory, West Side of Brick Lane, Parish of Christ Church Spitalfields, County of Middlesex

Having established a "particular place" (i.e. the murder-site epicenter) to serve as the focal point of the murder 'locale'; it is now meaningful to consider a "limited portion of space" that will serve as a specified degree of murder vicinity, e.g. 'immediate vicinity', 'general vicinity' or 'broad vicinity'. As the location of the "particular place" is based entirely on the relative locations of the six murder-sites under consideration; the parameters, which define the "limited portion of space" will be based entirely on the relative distribution of the six murder-sites, around the epicenter, i.e. the extent to which the locations of the six murder-sites deviate from the location of their mean-center.

As the locations of the six murder-sites and their epicenter have been expressed in terms of longitudinal and latitudinal coordinates; the deviations of each of those sites from that epicenter, can be expressed accordingly.


Figure 10: Longitudinal Deviation / Latitudinal Deviation / Absolute Deviation from Murder-Site Epicenter (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

If each longitudinal deviation were depicted as a horizontal line-segment 'a', extending from the murder-site epicenter to the corresponding point of longitudinal deviation; and its corresponding latitudinal deviation were depicted as a vertical line-segment 'b', extending from that point to the corresponding murder-site; then the depiction of the corresponding absolute deviation as the line-segment 'c', extending from the murder-site epicenter to the corresponding murder-site, would join 'a' and 'b' in forming a right triangle 'abc', having 'c' as its hypotenuse (i.e. the side of a right triangle, which lies opposite to its right angle (90°)).

According to the Pythagorean Theorem (a benchmark of Euclidean Geometry): The length of the hypotenuse of a right triangle is equal to the 'square-root' of the sum of the 'squared' lengths of the other two sides.



… where 'a' and 'b' are the 'legs' of the right angle, i.e. "the other two sides"; and 'c' is the side opposite the right angle, i.e. the "hypotenuse".

Therefore; the 'square' of the longitudinal deviation, plus the 'square' of the latitudinal deviation, equals the 'square' of the absolute deviation

Or; the 'square-root' of [the 'square' of the longitudinal deviation, plus the 'square' of the latitudinal deviation], equals the absolute deviation.

For ease of conceptualization; the longitudinal and latitudinal deviations can be expressed in 'yards', on the basis that at the murder-site epicenter:

- One 'Second' of Longitude = 21.07 Yards
- One 'Second' of Latitude = 33.76 Yards

Murder-Site Epicenter
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North

Martha Tabram
Longitude: 0° 4' 18.45" West
Latitude: 51° 31' 0.60" North

Longitudinal Deviation from Murder-Site Epicenter: 0° 0' 4.27500000" = 90.05 Yards
Latitudinal Deviation from Murder-Site Epicenter: 0° 0' 1.46166667" = 49.35 Yards

Absolute Deviation from Murder-Site Epicenter:



Polly Nichols
Longitude: 0° 3' 37.53" West
Latitude: 51° 31' 12.14" North

Longitudinal Deviation from Murder-Site Epicenter: 0° 0' 36.64500000" = 771.93 Yards
Latitudinal Deviation from Murder-Site Epicenter: 0° 0' 10.07833333" = 340.24 Yards

Absolute Deviation from Murder-Site Epicenter:



Annie Chapman
Longitude: 0° 4' 21.40" West
Latitude: 51° 31' 13.67 North

Longitudinal Deviation from Murder-Site Epicenter: 0° 0' 7.22500000" = 152.19 Yards
Latitudinal Deviation from Murder-Site Epicenter: 0° 0' 11.60833333" = 391.90 Yards

Absolute Deviation from Murder-Site Epicenter:



Elizabeth Stride
Longitude: 0° 3' 56.14" West
Latitude: 51° 30' 49.44 North

Longitudinal Deviation from Murder-Site Epicenter: 0° 0' 18.03500000" = 379.91 Yards
Latitudinal Deviation from Murder-Site Epicenter: 0° 0' 12.62166667" = 426.11 Yards

Absolute Deviation from Murder-Site Epicenter:



Catherine Eddowes
Longitude: 0° 4' 41.06" West
Latitude: 51° 30' 49.35" North

Longitudinal Deviation from Murder-Site Epicenter: 0° 0' 26.88500000" = 566.33 Yards
Latitudinal Deviation from Murder-Site Epicenter: 0° 0' 12.71166667" = 429.15 Yards

Absolute Deviation from Murder-Site Epicenter:



Mary Jane Kelly
Longitude: 0° 4' 30.47" West
Latitude: 51° 31' 7.17" North

Longitudinal Deviation from Murder-Site Epicenter: 0° 0' 16.29500000" = 343.25 Yards
Latitudinal Deviation from Murder-Site Epicenter: 0° 0' 5.10833333" = 172.46 Yards

Absolute Deviation from Murder-Site Epicenter:



The calculated absolute deviation of each of the six murder-sites, from the murder-site epicenter, is remarkably similar to its corresponding measured absolute deviation (i.e. its measured straight-line deviation from the murder-site epicenter); such that any dissimilarities are probably attributable to 'rounding-error'.

Absolute Deviations from Murder-Site Epicenter:

Martha Tabram
102.69 Yards (calculated)
102.67 Yards (measured)

Polly Nichols
843.59 Yards (calculated)
843.50 Yards (measured)

Annie Chapman
420.41 Yards (calculated)
420.38 Yards (measured)

Elizabeth Stride
570.87 Yards (calculated)
570.85 Yards (measured)

Catherine Eddowes
710.56 Yards (calculated)
710.69 Yards (measured)

Mary Jane Kelly
384.14 Yards (calculated)
384.13 Yards (measured)


Figure 11: Absolute Deviations from Murder-Site Epicenter (Straight-Line Distance) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Martha Tabram: 102.67 Yards
Polly Nichols: 843.50 Yards
Annie Chapman: 420.38 Yards
Elizabeth Stride: 570.85 Yards
Catherine Eddowes: 710.69 Yards
Mary Jane Kelly: 384.13 Yards

Mean Absolute Deviation (i.e. 'Average Deviation'): 505.37 Yards


Figure 12: Absolute Deviations from Murder-Site Epicenter (Circular Perspective) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Martha Tabram: 102.67 Yards
Mary Jane Kelly: 384.13 Yards
Annie Chapman: 420.38 Yards
Mean Absolute Deviation (i.e. 'Average Deviation') (Green): 505.37 Yards
Elizabeth Stride: 570.85 Yards
Catherine Eddowes: 710.69 Yards
Polly Nichols: 843.50 Yards

------------

Even in light of the suggested 'perception' that the six murder-sites under consideration are situated in a plane, in which there is a complete lack of urban, rural and geologic topography; the fact that various methods of deviation measurement (i.e. Straight-Line Distance, Manhattan-Grid Distance, Network Distance) are effectively different means to the same end, may be difficult to comprehend.

- 'Straight-Line Distance' Measurement; i.e. Measurement of Absolute Deviation from Murder-Site Epicenter

- 'Manhattan-Grid Distance' Measurement; i.e. Measurement of Longitudinal plus Latitudinal Deviation from Murder-Site Epicenter

- 'Network Distance' Measurement; i.e. Measurement of Cumulative Deviation, Along Randomly Selected Network of Practical Routes, from Murder-Site Epicenter


Figure 13: Deviations from Murder-Site Epicenter (Network Distance Measurement) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

White: Randomly Selected Networks of Practical Routes, from the Murder-Site Epicenter, to Each of the Six Murder-Sites Under Consideration.

Martha Tabram: 135.58 Yards
- South Side of Wentworth Street, Parish of St. Mary Whitechapel: 98.65 Yards
- West Side of George Yard, Parish of St. Mary Whitechapel: 31.27 Yards
- Main Corridor of George Yard Buildings, George Yard, Parish of St. Mary Whitechapel: 5.66 Yards

Polly Nichols: 1,067.38 Yards
- West Side of Osborn Street, Parish of St. Mary Whitechapel: 149.69 Yards
- North Side of Whitechapel High Street / Whitechapel Road, Parish of St. Mary Whitechapel: 613.57 Yards
- East Side of Baker's Row, Parish of St. Mary Whitechapel: 64.72 Yards
- South Side of White's Row / Buck's Row, Parish of St. Mary Whitechapel: 239.40 Yards

Annie Chapman: 460.73 Yards
- West Side of Brick Lane, Parish of Christ Church Spitalfields: 395.82 Yards
- North Side of Hanbury Street, Parish of Christ Church Spitalfields: 48.91 Yards
- Corridor of 29 Hanbury Street, Parish of Christ Church Spitalfields: 16.00 Yards

Elizabeth Stride: 676.47 Yards
- West Side of Osborn Street, Parish of St. Mary Whitechapel: 149.69 Yards
- West Side of Church Lane, Parish of St. Mary Whitechapel: 156.85 Yards
- South Side of Commercial Road, Parish of St. Mary Whitechapel / Parish of St. George in the East: 243.44 Yards
- West Side of Berner Street, Parish of St. George in the East: 122.43 Yards
- Dutfield's Yard, Berner Street, Parish of St. George in the East: 4.06 Yards

Catherine Eddowes: 978.27 Yards
- South Side of Wentworth Street, Parish of St. Mary Whitechapel: 359.45 Yards
- East Side of Goulston Street, Parish of St. Mary Whitechapel: 61.21 Yards
- South Side of New Goulston Street, Parish of St. Mary Whitechapel: 85.28 Yards
- East Side of Middlesex Street, Parish of St. Mary Whitechapel: 37.38 Yards
- North Side of Ellison Street, Parish of St. Botolph without Aldgate: 64.22 Yards
- North Side of New Street, Parish of St. Botolph without Aldgate: 63.96 Yards
- West Side of Gravel Lane, Parish of St. Botolph without Aldgate: 109.58 Yards
- South Side of Houndsditch, Parish of St. Botolph without Aldgate: 37.03 Yards
- East Side of Duke Street, Parish of St. Botolph without Aldgate / Parish of St. James: 67.38 Yards
- St. James's Place / St. James's Passage, Parish of St. James: 63.81 Yards
- Mitre Square, Parish of St. James: 28.97 Yards

Mary Jane Kelly: 497.20 Yards
- South Side of Wentworth Street, Parish of St. Mary Whitechapel: 98.65 Yards
- South Side / Center / North Side of Wentworth Street, Parish of St. Mary Whitechapel / Parish of Christ Church Spitalfields: 23.87 Yards
- West Side of George Street, Parish of Christ Church Spitalfields: 118.01 Yards
- North Side of Flower & Dean Street, Parish of Christ Church Spitalfields: 106.08 Yards
- East Side / Center of Commercial Street, Parish of Christ Church Spitalfields: 102.49 Yards
- North Side of Dorset Street, Parish of Christ Church Spitalfields: 38.13 Yards
- Miller's Court, Dorset Street, Parish of Christ Church Spitalfields: 9.97 Yards

Mean Network Distance (i.e. 'Average Network Distance'): 635.94 Yards


Figure 14: Deviations from Murder-Site Epicenter (Network Distance Measurement / Points of Mean Network Distance (Yellow)) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Yellow Dots: Points of Mean Network Distance (i.e. Mean Cumulative Deviation), Along Each of the Randomly Selected Networks of Practical Routes, from the Murder-Site Epicenter.

Mean Network Distance (Yellow Dots): 635.94 Yards

… i.e. a 25.84% increase from the Mean Absolute Deviation or 'Mean Straight-Line Distance' (505.37 Yards)

In order to 'accommodate' (i.e. sufficiently depict) the extent of the mean network distance; the following randomly selected 'networks' of practical routes, from the murder-site epicenter, are expanded accordingly:

Martha Tabram:
- West Side of George Yard, Parish of St. Mary Whitechapel: 149.77 Yards
- North Side / Center of Whitechapel High Street, Parish of St. Mary Whitechapel: 148.57 Yards
- South Side of Whitechapel High Street, Parish of St. Mary Whitechapel: 156.04 Yards
- East Side of Mansell Street, Parish of St. Mary Whitechapel: 51.64 Yards

Annie Chapman:
- North Side of Hanbury Street, Parish of Christ Church Spitalfields: 52.85 Yards
- East Side of John Street / Wilkes Street, Parish of Christ Church Spitalfields: 138.36 Yards

Mary Jane Kelly:
- North Side of Dorset Street, Parish of Christ Church Spitalfields: 65.84 Yards
- Little Paternoster Row, Parish of Christ Church Spitalfields: 62.37 Yards
- South Side of Brushfield Street, Parish of Christ Church Spitalfields: 20.50 Yards


Figure 15: Deviations from Murder-Site Epicenter (Network Distance Measurement / Points of Mean Network Distance / Measurement of Straight-Line Distances to Points of Mean Network Distance) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

One should now imagine that the objective is to cast a circular 'net' upon the murder-site epicenter; having sufficient radius to 'capture' the mean or 'average' extent of cumulative deviation, along each of the selected 'networks'. In order to determine 'sufficient radius', in this instance; the straight-line distances to each of the points of mean network distance must be taken into account.

Straight-Line Distances to Points of Mean Network Distance:

Martha Tabram: 453.13 Yards
Polly Nichols: 488.97 Yards
Annie Chapman: 541.37 Yards
Elizabeth Stride: 544.53 Yards
Catherine Eddowes: 492.75 Yards
Mary Jane Kelly: 475.64 Yards

The 'sufficient radius' is then determined by calculating the mean straight-line distance to the points of mean network distance.

Mean Straight-Line Distance to Points of Mean Network Distance: 499.40 Yards


Figure 16: Deviations from Murder-Site Epicenter (Points of Mean Network Distance / Measurement of Straight-Line Distances to Points of Mean Network Distance / Circular Perspective of Mean Straight-Line Distance) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Mean Straight-Line Distance to Points of Mean Network Distance (Yellow Circle): 499.40 Yards

Hence, a circular 'net' cast upon the murder-site epicenter; having sufficient radius (i.e. 499.40 Yards); will 'capture' the mean or 'average' extent of cumulative deviation, along each of the selected 'networks' of practical routes.


Figure 17: Deviations from Murder-Site Epicenter (Points of Mean Network Distance / Circular Perspective of Mean Straight-Line Distance to Points of Mean Network Distance (Yellow) / Circular Perspective of Mean Absolute Deviation (Green)) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Mean Straight-Line Distance to Points of Mean Network Distance (Yellow Circle): 499.40 Yards
Mean Absolute Deviation (i.e. Mean Straight-Line Distance to Murder-Sites) (Green Circle): 505.37 Yards

So again: A circular 'net' cast upon the murder-site epicenter; having a radius of 499.40 Yards; will 'capture' the mean or 'average' extent of cumulative deviation, along each of the selected 'networks' of practical routes.

While: A circular 'net' cast upon the murder-site epicenter; having a radius of 505.37 Yards; will 'capture' the mean or 'average' extent of absolute deviation, along each of the perceived vectors extending from the murder-site epicenter, to its respective murder-site.

In this particular instance (i.e. having specifically used this set of randomly selected networks of practical routes, as a means of estimating a 'general-case' Mean Network Distance); a disparity of 25.84% between the results of two distinct methods of measurement, has produced a disparity of just 1.18% between the results of the corresponding estimations of point-pattern dispersion.

In other words: Two distinctly different methods of deviation measurement (i.e. Straight-Line Distance and Network Distance) each concluded that a circular 'net' cast upon the murder-site epicenter, and having a radius of approximately five hundred yards; would succeed in 'capturing' those murder-sites, which lie within the average extent of deviation from the 'mean'.

Put simply: The two methods of deviation measurement (i.e. Straight-Line Distance and Network Distance), in this particular instance, are effectively different means to the same end.

------------

Again; Of Particular Note:

- Murder-Site 'Sample': The overall set of six murder-sites under consideration.

- Murder-Site 'Population': The overall set of six murder-sites under consideration (i.e. the Murder-Site 'Sample'), 'plus' the hypothetical set of any correlated subsequent murder-sites that would come under consideration.

In this particular instance; the mean absolute deviation of the overall set of six murder-sites under consideration (i.e. the murder-site 'sample') is a viable measure of the 'central tendency' of the 'sample' deviations:

- It is greater than three of the six 'sample' deviations; and less than the remaining three:

(Tabram, Kelly and Chapman Deviations) < Mean Absolute Deviation < (Stride, Eddowes and Nichols Deviations), i. e. …

(102.67, 384.13 and 420.38) < 505.37 < (570.85, 710.69 and 843.50)

- It lies in relatively close proximity to the mid-point of the 'sample' deviation range:
(102.67 Yards –to- 843.50 Yards): 473.09 Yards

- It lies in relatively close proximity to the 'sample' deviation median; i.e. the mid-point of the range between the two most 'central' deviations:
[(570.85 – 420.38) / 2] + 420.38 = 495.62 Yards

In other words: The mean absolute deviation, in this instance, would appear to be a meaningful 'average' of the 'sample' deviations. If for example, the above figure (Figure 17) merely depicted the murder-site epicenter (green dot) and mean absolute deviation (green circle); a reasonable portrayal of the murder-site distribution would be apparent.

Put simply: The murder-site epicenter and mean absolute deviation, in the case of this particular 'sample', are very useful 'descriptive statistics'; and their respective depictions (green dot and green circle) themselves, provide a worthwhile 'description' of the murder-site distribution.


Figure 18: Outlier Effect (Rose Mylett Murder-Site) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Point of Clarification: The Murder-Site 'Epicenter' is specifically the Mean-Center of the Murder-Site 'Sample'; but it is merely an estimate of the Mean-Center of the Murder-Site 'Population'.

This is due to the fact that the sites of any correlated subsequent murders would invariably 'weigh-in' to the determination of a displaced 'subsequent' Mean-Center; unless of course, all such subsequent murders were to occur precisely on the 'Sample' Mean-Center.

Consider now; the subsequent murder in the 'Whitechapel' series:

Rose Mylett (20 December 1888) (Yellow Dot: East)
Interior of Clarke's Yard, Poplar High Street, Parish of All Saints Poplar, County of Middlesex
Longitude: 0° 0' 49.25" West
Latitude: 51° 30' 31.56" North

Absolute Deviation from ('Sample') Murder-Site Epicenter: 4,437.84 Yards (White Circle)

The Mylett murder-site was in fact, almost as distantly removed from the 'sample' murder-site epicenter, as was the site of the discovery of the Whitehall Torso.

Whitehall Torso (Discovered: 2 October 1888) (Yellow Dot: West)
Interior of Basement Vault, Western Wing, New Scotland Yard (Construction in Process)
Near Northeast Corner of the Intersection of Derby Street and Canon Row, Parish of St. Margaret, Liberty of the City of Westminster, County of Middlesex
Longitude: 0° 7' 29.59" West
Latitude: 51° 30' 8.04" North

Its inclusion in the 'Whitechapel Murders' case-file notwithstanding; the murder of Rose Mylett has never been widely 'embraced' as a likely component of the 'tally' compiled by 'Jack the Ripper'. As such; it has been unable to 'pass muster' as the 'impending subsequent murder', that contemporary investigators should have anticipated during the final weeks of the Autumn of 1888. And accordingly; it does not serve to augment the murder-site 'sample', in this instance, as its inclusion would be 'ill-conceived', to say the least.


Figure 19: Outlier Effect (Rose Mylett Murder-Site) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

It is quite enlightening nonetheless, to examine the affect that factorization of the Mylett murder-site would have on the location of the murder-site epicenter.

'Sample' Murder-Site Epicenter (Green Dot: Center)
Southwest Corner of the Intersection of Wentworth Street and Osborn Street, Parish of St. Mary Whitechapel, County of Middlesex
Longitude: 0° 4' 14.18" West
Latitude: 51° 31' 2.06" North

Displaced 'Subsequent' Murder-Site Epicenter (i.e. Given Factorization of the Mylett Murder-Site) (Green Dot: Off-Center/East)
Southern Exterior of Dwelling, Northwest Corner of the Intersection of Fordham Street and New Road, Hamlet of Mile End Old Town, County of Middlesex
Longitude: 0° 3' 44.90" West
Latitude: 51° 30' 57.71" North

And it is just as enlightening, to consider the affect that factorization of the Mylett murder-site would have on mean absolute deviation (i.e. 'average deviation'), relative to the existing 'sample' murder-site epicenter.

Martha Tabram: 102.67 Yards
Mary Jane Kelly: 384.13 Yards
Annie Chapman: 420.38 Yards
'Sample' Mean Absolute Deviation (i.e. 'Average Deviation') (Green Circle): 505.37 Yards
Elizabeth Stride: 570.85 Yards
Catherine Eddowes: 710.69 Yards
Polly Nichols: 843.50 Yards

'Subsequent' Mean Absolute Deviation (i.e. 'Average Deviation'; Given Factorization of the Mylett Murder-Site) (Yellow Circle): 1,067.15 Yards

In this particular instance; the 'subsequent' mean absolute deviation of the overall set of seven murder-sites under consideration (i.e. the murder-site 'sample' 'plus' the Mylett murder-site) is not a viable measure of the 'central tendency' of the seven deviations:

- It is greater than six of the seven deviations; and less than only one:
(Tabram, Kelly, Chapman, Stride, Eddowes and Nichols Deviations) < Mean Absolute Deviation < (Mylett Deviation), i. e. …

(102.67, 384.13, 420.38, 570.85, 710.69 and 843.50) < 1,067.15 < (4,437.84)

- It does not lie in close proximity to the mid-point of the deviation range:
(102.67 Yards –to- 4,437.84 Yards): 2,270.26 Yards

- It does not lie in close proximity to the deviation median; i.e. the most 'central' deviation (Stride): 570.85 Yards

For comparison:

Murder-Site 'Sample'
- Deviation Mean: 505.37 Yards
- (Deviation Range: 102.67 Yards –to- 843.50 Yards)
- Mid-Point of Deviation Range: 473.09 Yards
- Deviation Median: 495.62 Yards

In this instance, the three measures of central tendency (mean, 'mid-range' and median) lie within a sub-range of just 32.29 Yards (473.09 Yards –to- 505.37 Yards), or 4.36% of the overall deviation range.

Murder-Site 'Sample' + Mylett
- Deviation Mean: 1,067.15 Yards
- (Deviation Range: 102.67 Yards –to- 4,437.84 Yards)
- Mid-Point of Deviation Range: 2,270.26 Yards
- Deviation Median: 570.85 Yards

But, in this instance, the three measures of central tendency (mean, 'mid-range' and median) lie within a sub-range of 1,699.41 Yards (570.85 Yards –to- 2,270.26 Yards); or 39.20% of the overall deviation range.

It must be understood that the Mylett murder-site, in this instance, is 'weighing-in' against a 'sample' of six murder-sites, of which it is not a part. Its affect therefore, on mean absolute deviation (i.e. 'average deviation') from the existing epicenter of the murder-site 'sample', bears the scent of a 'red herring'.

However, if one takes into account the two-fold purpose of this particular 'assessment';

- A comparison of two instances of Mean Absolute Deviation; in which one instance entails a viable measure of 'central tendency', while the other instance does not.

- A depiction of the extent, to which the location of the Mylett murder-site deviated markedly from the 'expectation' that should have prevailed, in the final weeks of the Autumn of 1888. *

* A depiction, which in its own right, makes a strong case for the murder of Rose Mylett having not been the 'impending subsequent murder', that contemporary investigators should have anticipated (i.e. a strong case for the murder of Rose Mylett having not been a correlation of the 'sample' set).

… then the affect of the Mylett murder-site, on mean absolute deviation (i.e. 'average deviation') from the existing epicenter of the murder-site 'sample', bears more pertinence.

------------

"… the mean absolute deviation of the overall set of six murder-sites under consideration (i.e. the murder-site 'sample') is a viable measure of the 'central tendency' of the 'sample' deviations …"

"… the 'subsequent' mean absolute deviation of the overall set of seven murder-sites under consideration (i.e. the murder-site 'sample' 'plus' the Mylett murder-site) is not a viable measure of the 'central tendency' of the seven deviations …"

In the former instance; the 'sample' deviation mean lies in relatively close proximity to the 'sample' deviation median; thus contributing to its viability as a measure of 'central tendency'.

But in the latter instance; the deviation mean does not lie in relatively close proximity to the deviation median; thus contributing to its lack of viability as a measure of 'central tendency'.

Hence the implication; that perhaps the deviations of the murder-site 'sample' are suggestive of a 'normally' distributed murder-site 'population'. And of course, the corresponding implication; that the deviations of the 'forced' murder-site set (i.e. the murder-site 'sample' 'plus' the Mylett murder-site), are not.

A 'Normal' Distribution is a set of observed data, in which the 'observations' tend to symmetrically congregate around a centrally located mean (i.e. 'average observation'). It can be depicted graphically, by way of a 'Bell-Curve', in which the degree of density of the 'observations' (i.e. the height of the 'Bell') is highest at the mean, and lowest at the outer extremities. Because of the symmetry of its 'central tendency'; the 'Normal' Distribution 'enjoys' a mean and median that are both one and the same.

It is often the case that the deviations of a 'sample' set of observations (e.g. the heights of twenty four adult males, living in the Parish of Christ Church Spitalfields, in late November 1888) will be suggestive of a 'population' set (e.g. the heights of all adult males, living in the Parish of Christ Church Spitalfields, in late November 1888) that is normally distributed.

"… suggestive of a 'population' set that is normally distributed."

Of course, in dealing with a 'sample' set of just six murder-sites, in which correlation between no more than three of the associated murders would seem a certainty; it would be very difficult to draw the conclusion that the 'population' set is normally distributed.

On the basis of subjective reasoning, however, the inference could be drawn; that the 'population' set is likely to be normally distributed.

*** My research has given me the distinct impression that Victorian London's East End did not have any semblance of a monopoly, where poverty, vice and criminal behavior within the metropolis were concerned. This of course, is contrary to today's 'conventional wisdom', as well as that of 1888. Charles Booth, himself, was quite surprised by the amount of poverty that his research team uncovered in areas such as Greenwich, Bermondsey, Southwark, Holborn and Clerkenwell. In fact, Booth eventually concluded that the Southwark Parishes of Christ Church, St. Saviour and St. George the Martyr were the most impoverished in the whole of the metropolis.

Considerable wealth and abject poverty both tended to be concentrated in various enclaves, throughout the four 'quarters' (i.e. North, East, South, and West) of London's metropolis in 1888. There being two notable exceptions: Certain parts of the West End, which were too large to be considered mere 'enclaves', enjoyed considerable wealth; while nowhere in the East End was such wealth at all prevalent. The only characteristic of the East End, which truly differentiated it from the other 'quarters' of the metropolis, was just that: An apparent lack of any enclaves of considerable wealth. Indeed, 1888's East End constituted a vast landscape, in which 'blue-collar' society overwhelmingly prevailed. But, it was burdened with just slightly more than its fair share of enclaves of abject poverty, vice and criminal elements.

Three of these enclaves were situated in very close proximity to the murder-site epicenter:

'Great Pearl Street'
- Great Pearl Street, Parish of Christ Church Spitalfields
- Little Pearl Street, Parish of Christ Church Spitalfields
- various adjoining courts

'Dorset Street'
- Dorset Street, Parish of Christ Church Spitalfields
- Little Paternoster Row, Parish of Christ Church Spitalfields
- various adjoining courts

'Flower & Dean Street'
- Fashion Street, Parish of Christ Church Spitalfields (specifically; its adjoining courts)
- Flower & Dean Street, Parish of Christ Church Spitalfields (excepting its southwestern quarter)
- George Street, Parish of Christ Church Spitalfields (eastern side)
- Thrawl Street, Parish of Christ Church Spitalfields (eastern half)
- Wentworth Street, Parish of Christ Church Spitalfields / Parish of St. Mary Whitechapel (between George Street and Brick Lane)
- George Yard, Parish of St. Mary Whitechapel (northeastern quarter)
- various adjoining courts

I am inclined to believe that two of these enclaves, namely 'Dorset Street' and 'Flower & Dean Street', were most unusual in that they were apparently home to an extraordinarily large concentration of middle-aged, alcoholic, totally destitute and completely vulnerable 'dollymops' (i.e. 'casual' prostitutes).

I have little doubt that had these murders continued indefinitely, the epicenter of their locations would have gradually moved into increasingly closer proximity to the epicenter of the two enclaves mentioned above (i.e. 'Dorset Street' and 'Flower & Dean Street'); regardless of the location(s) of any sort of 'base', from which the perpetrator(s) might have operated.

In other words: I am of the opinion that the distribution of murder-sites, in this particular instance, is mostly a function of the tightly clustered locations of the victims' residences, and the correspondingly confined dispersion of their presumed 'activity spaces'.

Put simply: I tend to believe that each of the victims died in areas, to which they were drawn by totally random circumstances; and that these areas were those, in which they likely went about their normal routines (e.g. begging, scavenging, pick-pocketing, soliciting, hawking, etc …). ***

It would seem reasonable therefore (i.e. on the basis of the above subjective reasoning), to assume that the Murder-Site 'Population', in this particular instance, likely constitutes a 'Normal' Distribution. Specifically: A distribution, in which the murder-sites would tend to symmetrically congregate around the epicenter of the 'Dorset Street' and 'Flower & Dean Street' 'rookeries'; one, in which the degree of murder-site density would be highest at the Mean-Center (i.e. the epicenter of the aforementioned 'rookeries'); and one, in which the symmetry of its 'central tendency' would dictate that the Mean-Center and 'Center of Minimum Distance' both be one and the same.

Again; it would seem reasonable therefore (i.e. on the basis of the above subjective reasoning), to assume that the Murder-Site 'Population', in this particular instance, likely constitutes a 'Normal' Distribution.

And Once Again; Of Particular Note:

- Murder-Site 'Sample': The overall set of six murder-sites under consideration.

- Murder-Site 'Population': The overall set of six murder-sites under consideration (i.e. the Murder-Site 'Sample'), 'plus' the hypothetical set of any correlated subsequent murder-sites that would come under consideration.

*** With the conclusion that it would seem reasonable to assume that the murder-site 'population' likely constitutes a normal distribution; the utilization of the concept of 'Standard Deviation', is clearly the next step. Specifically: The 'next step' toward the parameterization of a "limited portion of space" to serve as a specified degree of vicinity (e.g. 'immediate vicinity', 'general vicinity' or 'broad vicinity'), for the establishment of a murder 'locale'.

I am concluding this portion of the informal presentation with seven aerial images, which utilize the concept of 'Standard Deviation' accordingly. I have included some applicable statistics that should serve to clarify (at least to some degree) the purposes that these images fulfill.

An explanation of the concept of 'Standard Deviation', along with a review of these seven images, will be forthcoming in the next portion of the informal presentation. This will hopefully materialize within four-to-five weeks. ***


Figure 20: Mean Absolute Deviation / Standard Deviation (Circular) / 'Sample' Standard Deviation (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Standard Deviation:





'Sample' Standard Deviation:







… a graphic depiction of the Distribution Density Function (i.e. the 'Probability Density Function') ('one-tailed') for six data points, i.e. five 'degrees of freedom' …



… a graphic depiction of the Distribution Accumulation Function (i.e. the 'Cumulative Distribution Function') for six data points, i.e. five 'degrees of freedom' …

------------


Figure 21: Cumulative Probability Distribution (0.00 - 1.00 Standard Deviations) (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red: 0.00 - 1.00 Standard Deviations
- Radius: 612.74 Yards
- Area: 0.38 Square-Miles
- 'Expected' Distribution Accumulation: 63.68% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 63.68% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 63.68%.

Entire Image (Assumed Circle): 0.00 - 2.00 Standard Deviations
- Radius (Assumed Circle): 1,225.48 Yards
- Area (Assumed Circle): 1.52 Square-Miles
- Area (Square): 1.94 Square-Miles
- 'Expected' Distribution Accumulation (Assumed Circle): 89.80% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 89.80% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 89.80%.


Figure 22: Cumulative Probability Distribution (0.00 - 3.00 Standard Deviations) (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red: 0.00 - 1.00 Standard Deviations
- Radius: 612.74 Yards
- Area: 0.38 Square-Miles
- 'Expected' Distribution Accumulation: 63.68% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 63.68% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 63.68%.

Red/Orange: 0.00 - 2.00 Standard Deviations
- Radius: 1,225.48 Yards
- Area: 1.52 Square-Miles
- 'Expected' Distribution Accumulation: 89.80% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 89.80% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 89.80%.

Red/Orange/Yellow: 0.00 - 3.00 Standard Deviations
- Radius: 1,838.22 Yards
- Area: 3.43 Square-Miles
- 'Expected' Distribution Accumulation: 97.00% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 97.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 97.00%.

Entire Image (Assumed Circle): 0.00 - 4.00 Standard Deviations
- Radius (Assumed Circle): 2,450.96 Yards
- Area (Assumed Circle): 6.09 Square-Miles
- Area (Square): 7.76 Square-Miles
- 'Expected' Distribution Accumulation (Assumed Circle): 98.96% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 98.96% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 98.96%.


Figure 23: Cumulative Probability Distribution (0.00 - 5.00 Standard Deviations) (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red: 0.00 - 1.00 Standard Deviations
- Radius: 612.74 Yards
- Area: 0.38 Square-Miles
- 'Expected' Distribution Accumulation: 63.68% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 63.68% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 63.68%.

Red/Orange: 0.00 - 2.00 Standard Deviations
- Radius: 1,225.48 Yards
- Area: 1.52 Square-Miles
- 'Expected' Distribution Accumulation: 89.80% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 89.80% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 89.80%.

Red/Orange/Yellow: 0.00 - 3.00 Standard Deviations
- Radius: 1,838.22 Yards
- Area: 3.43 Square-Miles
- 'Expected' Distribution Accumulation: 97.00% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 97.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 97.00%.

Red/Orange/Yellow/Green: 0.00 - 4.00 Standard Deviations
- Radius: 2,450.96 Yards
- Area: 6.09 Square-Miles
- 'Expected' Distribution Accumulation: 98.96% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 98.96% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 98.96%.

Red/Orange/Yellow/Green/Aqua: 0.00 - 5.00 Standard Deviations
- Radius: 3,063.71 Yards
- Area: 9.52 Square-Miles
- 'Expected' Distribution Accumulation: 99.58% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 99.58% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 99.58%.

Entire Image (Assumed Circle): 0.00 - 6.00 Standard Deviations
- Radius (Assumed Circle): 3,676.45 Yards
- Area (Assumed Circle): 13.71 Square-Miles
- Area (Square): 17.45 Square-Miles
- 'Expected' Distribution Accumulation (Assumed Circle): 99.82% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 99.82% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 99.82%.


Figure 24: Cumulative Probability Distribution (0.00% - 50.00%) (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red: 0.00% - 50.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 0.73 Standard Deviations
- Radius: 445.28 Yards
- Area: 0.20 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 50.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 50.00%.

Entire Image (Assumed Circle): 0.00% - 75.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 1.30 Standard Deviations
- Radius (Assumed Circle): 797.18 Yards
- Area (Assumed Circle): 0.64 Square-Miles
- Area (Square): 0.82 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 75.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 75.00%.


Figure 25: Cumulative Probability Distribution (0.00% - 90.00%) (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red: 0.00% - 50.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 0.73 Standard Deviations
- Radius: 445.28 Yards
- Area: 0.20 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 50.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 50.00%.

Red/Orange: 0.00% - 60.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 0.92 Standard Deviations
- Radius: 563.42 Yards
- Area: 0.32 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 60.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 60.00%.

Red/Orange/Yellow: 0.00% - 70.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 1.16 Standard Deviations
- Radius: 708.33 Yards
- Area: 0.51 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 70.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 70.00%.

Red/Orange/Yellow/Green: 0.00% - 80.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 1.48 Standard Deviations
- Radius: 904.41 Yards
- Area: 0.83 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 80.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 80.00%.

Red/Orange/Yellow/Green/Aqua: 0.00% - 90.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 2.02 Standard Deviations
- Radius: 1,234.67 Yards
- Area: 1.55 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 90.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 90.00%.

Entire Image (Assumed Circle): 0.00% - 95.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 2.57 Standard Deviations
- Radius (Assumed Circle): 1,575.36 Yards
- Area (Assumed Circle): 2.52 Square-Miles
- Area (Square): 3.20 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 95.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 95.00%.


Figure 26: Cumulative Probability Distribution (0.00% - 99.50%) (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red: 0.00% - 50.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 0.73 Standard Deviations
- Radius: 445.28 Yards
- Area: 0.20 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 50.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 50.00%.

Red/Orange: 0.00% - 60.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 0.92 Standard Deviations
- Radius: 563.42 Yards
- Area: 0.32 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 60.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 60.00%.

Red/Orange/Yellow: 0.00% - 70.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 1.16 Standard Deviations
- Radius: 708.33 Yards
- Area: 0.51 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 70.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 70.00%.

Red/Orange/Yellow/Green: 0.00% - 80.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 1.48 Standard Deviations
- Radius: 904.41 Yards
- Area: 0.83 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 80.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 80.00%.

Red/Orange/Yellow/Green/Aqua: 0.00% - 90.00% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 2.02 Standard Deviations
- Radius: 1,234.67 Yards
- Area: 1.55 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 90.00% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 90.00%.

Red/Orange/Yellow/Green/Aqua/Blue: 0.00% - 99.50% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 4.77 Standard Deviations
- Radius: 2,924.61 Yards
- Area: 8.67 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 99.50% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 99.50%.

Entire Image (Assumed Circle): 0.00% - 99.75% Stipulated 'Expectation' of Distribution Accumulation *
- 0.00 - 5.60 Standard Deviations
- Radius (Assumed Circle): 3,433.80 Yards
- Area (Assumed Circle): 11.96 Square-Miles
- Area (Square): 15.23 Square-Miles
* Were these murders to have continued ad infinitum; the 'expectation' would be that 99.75% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 99.75%.


Figure 27: Cumulative Probability Distribution (Greatest Deviation: Polly Nichols) (Circular) (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red: Greatest 'Sample' Deviation (Polly Nichols) 0.00 - 1.38 Standard Deviations
- Radius: 843.50 Yards
- Area: 0.72 Square-Miles
- 'Expected' Distribution Accumulation: 77.30% *
* Were these murders to have continued ad infinitum; the 'expectation' would be that 77.30% would have occurred within the specified circular area. This can be loosely interpreted, to mean that the 'probability' of the impending subsequent murder occurring within this circular area; would have been 77.30%.

[Dave O]

Hi Colin,

You write "I am inclined to believe that two of these enclaves, namely 'Dorset Street' and 'Flower & Dean Street', were most unusual in that they were apparently home to an extraordinarily large concentration of middle-aged, alcoholic, totally destitute and completely vulnerable 'dollymops' (i.e. 'casual' prostitutes)."

Indeed, and going by Wynne Baxter's summation at the Alice Mackenzie inquest, not only was the size of this concentration or parish-within-a-parish unusual, it was unique!

It must now be patent to the whole world that in Spitalfields there is a class of persons who, I think, cannot be found in such numbers, not only in any other part of this metropolis, but in any other metropolis; and the question arises, should this state of affairs continue to exist? I do not say it is for you to decide. The matter is one for a higher power than ourselves to suggest a remedy. But it certainly appears to me there are two ways in which the matter ought to be attacked. In the first place, it ought to be attacked physically. Many of the houses in the neighbourhood are unfit for habitation. They want clearing away and fresh ones built. Those are physical alterations which, I maintain, require to be carried out there. Beyond this there is the moral question. Here we get a population of the same character, and not varied, as in a moderately-sized town or village. Here there is a population of 20,000 of the same character, not one of whom is capable of elevating the other. Of course there is an opinion among the police that it is a proper thing that this seething mass should be kept together rather than be distributed all over the metropolis. Every effort ought to be made to elevate this class. I am constantly struck by the fact that all the efforts of charitable and religious bodies here are comparatively unavailing. It is true a great deal has been done of late years, especially to assist the moral development of the East-end, but it is perfectly inadequate to meet the necessities of the case. If no other advantage comes from these mysterious murders, they will probably wake up the Church and others to the fact that it is the duty of every parish in the West to have a mission and localize work in the East-end, otherwise it will be impossible to stop these awful cases of crime. Here is a parish of 21,000 persons with only one church in it. There are not only cases of murder here, but many of starvation. I hope at least these cases will open the eyes of those who are charitable to the necessity of doing their duty by trying to elevate the lower classes.

Cheers,
Dave

[richardnunweek]

Hi Colin,
Remarkable work, athough a bit hard to fathom out this early in the morning, could you express a more simplified version, so that we can all comprehend its obvious merit.
Regards Richard.

[Sam Flynn]

My research has given me the distinct impression that Victorian London's East End did not have any semblance of a monopoly, where poverty, vice and criminal behavior within the metropolis were concerned. This of course, is contrary to today's 'conventional wisdom', as well as that of 1888. Charles Booth, himself, was quite surprised by the amount of poverty that his research team uncovered in areas such as Greenwich, Bermondsey, Southwark, Holborn and Clerkenwell.

Some very important points, there - and valid ones, too - which have equally important implications.

[Scott Nelson]

I'm reminded of John Michell's books, The View Over Atlantis and City of Revelation. Quite remarkable. Now all you need to do is portion out the diameter of the moon and the circumference of the Great Pyramid.

[John Bennett]

This is a titanic effort and a lot of work has evidently gone into it. However (with no disrespect to Colin) I am struggling with it. What is it actually attempting to achieve? I gather it's nothing to do with 'profiling', ie finding where the murderer probably lived. But I also see that there is an effort to 'predict' a potential subsequent crime location. Is this what it's about?

Mathematics has never been my strong point and perhaps I'm being a tad thick here, but I'm just wondering if someone can post a brief overview of the concept at work here.

Septic Blue? Anyone?

[Septic Blue]

This is a titanic effort and a lot of work has evidently gone into it. However (with no disrespect to Colin) I am struggling with it. What is it actually attempting to achieve? I gather it's nothing to do with 'profiling', ie finding where the murderer probably lived. But I also see that there is an effort to 'predict' a potential subsequent crime location. Is this what it's about?

Mathematics has never been my strong point and perhaps I'm being a tad thick here, but I'm just wondering if someone can post a brief overview of the concept at work here.

Septic Blue? Anyone?

How near is 'near'?

How far is 'far'?

How local is 'local'?

For that matter; what exactly constitutes 'local'?

Can a murder 'locale' be defined in terms of varying degrees of vicinity: 'Immediate Vicinity'; 'General Vicinity'; 'Broad Vicinity'?

[ATTACH]6164[/ATTACH]
Figure 3: Immediate Vicinity; General Vicinity; Broad Vicinity (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Can those varying degrees of vicinity be based not just on the 'observed' dispersion of established murder-sites; but also on the 'expected' dispersion of impending subsequent murder-sites?

Perhaps they can be! Perhaps they should be!

Are the terms 'local' and 'East-Ender' interchangeable?

Does our focus on the East End perhaps amount to 'tunnel-vision'?

[ATTACH]6165[/ATTACH]
Figure 4: Ancient Parochial Establishment of East London (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Part of an effort to define that, which was 'local' in the context of political geography:

- Bright Yellow: St. John at Hackney
- Orange: St. Leonard Shoreditch
- Red: St. Dunstan Stepney
- Green: Bromley St. Leonard
- Purple: City of London
- Gold: The Liberty of His/Her Majesty's Tower of London
- Blue: Precinct of the Priory of St. Mary without Bishopsgate (St. Mary Spital)
- Green: Precinct of the Priory of the Holy Trinity, Aldgate (County of Middlesex)

The Ancient Parish of St. Dunstan Stepney (Red) was divided, between the years 1329 - 1817, into the following Civil Parishes:

- St. Matthew Bethnal Green
- Christ Church Spitalfields
- The Hamlet of Mile End New Town
- St. Mary Whitechapel
- St. John of Wapping
- St. George in the East
- St. Paul Shadwell
- The Hamlet of Mile End Old Town
- The Hamlet of Ratcliff
- St. Anne Limehouse
- St. Mary Stratford Bow
- All Saints Poplar

The Precinct of the Priory of St. Mary without Bishopsgate (St. Mary Spital) (Blue) was originally part of St. Botolph without Bishopsgate (excepting the northernmost 'tip', which was part of St. Leonard Shoreditch). By the end of the seventeenth century, it had become The Liberty of Norton Folgate (north) and The Old Artillery Ground (south).

The green areas, which constituted the Middlesex portions of the Precinct of the Priory of the Holy Trinity, Aldgate were originally part of St. Botolph without Aldgate. The smaller area (north) eventually became Holy Trinity (Minories), while the larger area (south) became The Precinct of St. Katharine (west), and once again St. Botolph without Aldgate (east).

...


Figure 5: Murder-Site 'Vicinity' in the Context of a Larger 'East End' (Click to View in flickr)
Underlying Aerial Imagery: Copyright Google Earth, 2007
Overlying Plots, Labels and Color-Shadings: Copyright Colin C. Roberts, 2009

Red Outline and Portions of White-Dotted 'Compensation' Outline: An estimation of the area most likely to have been considered the 'East End', in 1888

The Parliamentary Borough of Shoreditch
- The Parish of St. Leonard Shoreditch

The Parliamentary Borough of Bethnal Green
- The Parish of St. Matthew Bethnal Green

The Parliamentary Borough of Tower Hamlets
- The Liberty of Norton Folgate
- The Old Artillery Ground
- The Parish of Christ Church Spitalfields
- The Hamlet of Mile End New Town
- The Parish of Holy Trinity (Minories)
- The Parish of St. Mary Whitechapel (Middlesex portion)
- The Liberty of Her Majesty's Tower of London
--- The Liberty of the Tower
--- The Precinct of Old Tower Without
--- The Tower
- The Precinct of St. Katharine
- The Parish of St. Botolph without Aldgate (Middlesex portion)
- The Parish of St. John of Wapping
- The Parish of St. George in the East
- The Parish of St. Paul Shadwell
- The Hamlet of Mile End Old Town
- The Hamlet of Ratcliff
- The Parish of St. Anne Limehouse
- The Parish of St. Mary Stratford Bow
- The Parish of Bromley St. Leonard
- The Parish of All Saints Poplar

My analysis would suggest that a sizable portion of 1888's 'East End' was not 'local' to this series of murders; and that a sizable portion of the area, which was 'local' to this series of murders was not in the 'East End'.

Wasn't the Parish of St. Luke just as 'local' as the Parish of St. Anne Limehouse?

It's exclusion from the East End notwithstanding; wasn't the Parish of St. Luke actually more 'local' to this particular set of six murder-sites, than was the Parish of St. Anne Limehouse?

Was the Parish of St. John at Hackney 'local'? What about the Parish of St. John Horselydown?

Should Horselydown have been considered more 'local' than Hackney?

Were Druitt's chambers in The Temple actually more 'local' than Bury's residence in the Parish of Bromley St. Leonard?

Should those who actually lived in The City be discounted; even though Charles Booth estimated, in 1889, that 85% were of 'blue-collar' society, and that 31% were actually 'poor'?

Can this analysis lead to the development of a very simple geographic-profile 'model'? Yes, it can; and it will!

But; will this geographic-profile 'model' provide us with anything more than an enhanced perspective? No! Of course not!

[Sam Flynn]

Should those who actually lived in The City be discounted; even though Charles Booth estimated, in 1889, that 85% were of 'blue-collar' society, and that 31% were actually 'poor'?

...85% and 31% of how many, though?

[John Bennett]

Thanks Colin,

It wasn't brief, but it is a bit clearer. Is it about what is defined as 'local'?

...85% and 31% of how many, though?

Erm. I know my maths isn't great, but doesn't that make 116%? Presumably there must be some sort of crossover? Perhaps?

Boy, do I sound dumb!

[Sam Flynn]

Erm. I know my maths isn't great, but doesn't that make 116%? Presumably there must be some sort of crossover?

Indeed, John - think of an overlapping Venn Diagram.

[John Bennett]

Ah! Venn diagrams. I understood those. Cheers Gareth.

[richardnunweek]

Hi ,
I thought odds compiling was confusing[ my occupation] but to get ones head round such detailed work, with the explanations shown, tends to make [ at least me] one dizzy.
To me, it is blatently obvious that the killer proberly resided very close to the killing zone, and if the opportunety arose again to kill, would have struck within that safety blanket.
With respect we proberly are all barking up the wrong tree, when [a] attempting to identify the killer. and[B] attempting to pinpoint his location, and although all these attempts are praiseworthy where does it lead us?
Regards Richard.

[Septic Blue]

Should those who actually lived in The City be discounted; even though Charles Booth estimated, in 1889, that 85% were of 'blue-collar' society, and that 31% were actually 'poor'?

...85% and 31% of how many, though?

From Charles Booth's first of three surveys: "Labour and Life of the People: London" (two volumes + appendix), Williams & Norgate, 1889-1891

The City of London:
- Estimated Population; City of London: 42,561
- Estimated Percentage; Class 'A': 1%
- Estimated Percentage; Class 'B': 6%
- Estimated Percentage; Classes 'C' & 'D': 24%
- (Total Percentage; Below 'Line of Poverty': 31%)
- Estimated Percentage; Classes 'E' & 'F': 54%
- Estimated Percentage; Classes 'G' & 'H': 15%

As compared to the Whitechapel Registration District, for which greater detail is provided:
- Estimated Population; Whitechapel Registration District *: 73,518
- Estimated Percentage; Class 'A': 3.3%
- Estimated Percentage; Class 'B': 8.9%
- Estimated Percentage; Class 'C': 10.7%
- Estimated Percentage; Class 'D': 16.3%
- (Total Percentage; Below 'Line of Poverty': 39.2%)
- Estimated Percentage; Class 'E': 43.3%
- Estimated Percentage; Class 'F': 11.3%
- Estimated Percentage; Class 'G': 4.4%
- Estimated Percentage; Class 'H': 1.8%

* Whitechapel Registration District:
- The Liberty of Norton Folgate
- The Old Artillery Ground
- The Parish of Christ Church Spitalfields
- The Hamlet of Mile End New Town
- The Parish of Holy Trinity (Minories)
- The Parish of St. Mary Whitechapel (Middlesex portion)
- The Liberty of Her Majesty's Tower of London
--- The Liberty of the Tower
--- The Precinct of Old Tower Without
--- The Tower
- The Precinct of St. Katharine
- The Parish of St. Botolph without Aldgate (Middlesex portion)

[Sam Flynn]

Thanks, Colin.

[DarkPassenger]

Could someone please sum up the OP for someone who's maths and statistics knowledge could be written on the back of a postage stamp with a fat crayon?