Tuesday, June 10, 2008

Cumulative Distribution Tabs

The image above shows a cumulative distribution function that can be seen and understood with relative ease. This is a side view of the pages of the paperback version of the Oxford Advanced Learners’ Dictionary. Small colored squares for each letter are shown on the edge of each page. These colored squares act as tabs running from the top of the page for letters early in the alphabet to the bottom of the page for those letters that come later. These printed alphabetical marks on the edges of the pages help speed the look up of words and their definitions. This is really one of the first search engines.

When the dictionary is placed on its side, these colored tabs produce a cumulative distribution function for words from the English language. We have a visual and understandable image of a cumulative distribution function. We can quantify this by noting, for example, that 93 pages of this dictionary are devoted to the letter "A", so 93 pages have tabs colored to act as guide tabs to words starting with "A". The last page number of each letter’s tab indicates the number of pages devoted to words that begin with letters occurring, in alphabetical order, before that tab. Let G(x) denote the page number of this last page for each letter x in the set {A,B,C,…,Z}. These are the cumulative counts of pages for words beginning with each letter in the English alphabet. The maximum of G(x), call it M, is just the number of the last page of the dictionary for the letter "Z". Define F(x) to be G(x)/M. Then F(x) represents the cumulative relative frequency of letters of the alphabet. This is the alphabet’s cdf. Learn more from my paper "A Photographic View of Cumulative Distribution Functions" in the Journal of Statistics Education.