## Overlapping Normal Distributions. John M. Linacre

Are men taller than women? Are women more patient than men? We answer such questions by comparing putatively normal distributions. A typical analysis computes a t-test of differences between sample means. When samples are large, any difference between means will be declared "statistically significant". Before drawing substantive conclusions from a "significant" result, however, it is wise to discover how much the distributions overlap, contradicting any finding of clear "difference".  Consider two normal distributions, N1 and N2, with the same sample sizes but different means (M1 and M2) and standard deviations (SD1 and SD2). Number the samples so that N1 has the smaller standard deviation, SD1. To discover the expected percentage by which each sample distribution overlaps the other, consult the nomogram. For the x-coordinate, compute a standardized absolute distance between the means, |M2- M1|/SD1. For the y-coordinate, compute the ratio of the standard deviations, SD2/SD1. Interpolate by eye between contours to estimate the percent, p%, of each distribution that is in common with the other. The unique amount of each distribution is then 100-p%.

If M1=2.4, SD1=1.6, and M2=3.2, SD2=2.0, then |M2-M1|/SD1=0.5, SD2/SD1=1.25, and, by reference to the nomogram, about 80% of each distribution overlaps the other.

These results were computed using numerical integration of the areas under the normal curves.

Overlapping normal distributions. Linacre JM. … Rasch Measurement Transactions, 1996, 10:1 p.487

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