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Statistical Inference: A Short Course by Michael J. Panik

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Appendix 11.A Runs Test for Two Independent Samples

The single-sample runs test, presented in Appendix 10. A, can also be used to compare the identity of two population distributions given that we have two independent random samples of sizes n1 and n2, respectively. Let us assume that the underlying population characteristic that these samples represented can be described by a variable that follows a continuous distribution. Hence the following test applies to interval- or ratio- scale data.

Let us assign a letter code to each of the n = n1 + n2 observations in these samples. For instance, any observation in sample 1 can be marked with the letter a, and any observation in sample 2 can be marked with the letter b. Next, let us rank all n observations according to the magnitude of their scores, with an a placed below each observation belonging to sample 1 and a b placed beneath each observation belonging to sample 2. We thus have an ordered sequence of a's and b's so that we can now conduct a test for the randomness of this arrangement.

We now consider the runs or clusterings of the a's and b's. If the two samples have been drawn from identical populations, then we should expect to see many runs since the n observations from the two samples should be completely intermingled when placed in numerical order. But if the two populations are not identical (e.g., they differ with respect to location or central tendency), then we shall expect fewer runs in the ordered arrangement.

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