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Seven (plus or minus two) Clusters, A Monte Carlo Study Larry Hoyle, Policy Research Institute, The University of Kansas

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1972 Kansas Statistical Abstract

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Shading by Overprinting

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Shading by Line Spacing

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Line Shading Detail

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What did they have in common? Neither method is continuous So both methods required grouping or classes Fixed number of combinations Characters on a fixed grid Integer number of lines in the polygon Lines are relatively coarse

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How to Group for Shading Equal Intervals Equal numbers (quantiles) By clusters Dont group (unclassed)

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Population Density 7 Equal Intervals 100 counties fall into the bottom class

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Population Density - Equal Numbers 15 counties in each class - a very different picture

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Population Density - Cluster Means Group around the 7 values that best represent the data

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Population Density - Unclassed No classes, just shade in proportion to value

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Clustering Tries for Best grouping Each member of cluster can be represented by the mean of the group

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Proc Fastclus You specify the number of clusters Minimizes cluster sum of squared distance (e.g. minimum within cluster variance) inspired by: k-means (MacQueen) leader algorithm (Hartigan)

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Example clustering - data

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4 clusters y cluster data. x 0102030405060708090 R-squared=.9912

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4 clusters data Correlation.9956 R-squared=.9912

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3 clusters y cluster data. x 0102030405060708090 R-squared=.9609

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How many clusters is enough?

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Plot R-squared by number of clusters Sample of 300 observations, Uniform distribution, 11 cluster analyses

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What happens if there really arent any clusters? Lets try 500 samples

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Uniform, 300 obs. per sample 500 samples, 11 clusterings each

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Uniform, 1000 obs. per sample 500 samples, 11 clusterings each

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Normal, 300 obs. per sample 500 samples, 11 clusterings each

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Normal, 1000 obs. per sample 500 samples, 11 clusterings each

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Exponential, 300 obs. per sample 500 samples, 11 clusterings each

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Distribution of worst sample

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Exponential, 1000 obs. per sample 500 samples, 11 clusterings each

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So Whats with 7 2?

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Uniform, 7 2 500 samples, 11 clusterings each

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Normal, 7 2 500 samples, 11 clusterings each

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Exponential, 7 2 500 samples, 11 clusterings each

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Minimum R squared by sample size and distribution At least 95% of the variance for all

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Histograms Equal intervals Number of observations in each interval

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Needle Plot of Cluster Means

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Bar chart needs more bars

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The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Information Processing George Miller, The Psychological Review 1956, vol.63 pp. 81-97

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Limits on Categories for Absolute Judgments Pitch 6 Loudness 5 Visual position 9 Size of a square 5 Hue 8 Name the colors in this slide

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And finally, what about the magical number seven? George A. Miller

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Miller Quote 1 seven wonders of the world seven seas seven deadly sins seven daughters of Atlas in the Pleiades seven ages of man seven levels of hell seven primary colors seven notes of the musical scale seven days of the week What about the

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Miller Quote 2 seven-point rating scale seven categories for absolute judgment seven objects in the span of attention seven digits in the span of immediate memory What about the

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Perhaps there is something deep and profound behind all these sevens, something just calling out for us to discover it. Miller Quote 3

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Miller - close But I suspect that it is only a pernicious, Pythagorean coincidence.

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Coincidence or Natures Parsimony? Does our capacity match whats needed for 95% of the variance? 95%? Hmmmm. confidence intervals an A 19 fingers and toes 970,000 web pages Larry Hoyle Policy Research Institute University of Kansas LarryHoyle@ku.edu