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Karl Pearson: Everybody Knows His Correlation Coefficient, but Not How "Close" the Binomial Distribution is to a Normal Distribution

Karl Pearson was born 150 years ago on March 27, 1857; he died April 27, 1936. He made important contributions to statistics; he is usually remembered for two path-breaking achievements: his "product-moment" estimate of the correlation coefficient (dating from 1896), and the chi-square test (introduced in 1900). But, on the 150th anniversary of his birth, there is a small striking discovery he made in 1895 that is virtually unknown today, yet well worth knowing.

Everybody knows that the binomial distribution is "like" a normal distribution if the number of independent trials (n) is large and the probability of success (p) on a single trial is not too near 0 or 1. Everybody knows this because Abraham De Moivre proved it to be true in 1733.

Because we are always reminded that this is an approximation, there is a nagging doubt as to how close the binomial and normal distributions really are. Pearson discovered something quite remarkable: There is a case where the agreement is much, much closer than anyone would have expected. In fact, if one particular definition of "agreement" is adopted, and if p = 1/2, the "agreement" is actually exact for all n (even for n =1), providing one minor fudge factor is allowed ...

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