# Sign Test Dialog

This dialog is activated by selecting the Paired Sample Sign Test... command from the Statistics -> Nonparametric Tests -> menu. It can be used in order to test for consistent differences between pairs of observations. The Wikipedia article on the sign test makes for excellent reading on this topic.

For two paired samples, `Xi` and `Yi`, the test statistics (`S`) is the number of pairs for which Xi < Yi. QtiPlot also computes the reduced sample size (`Nr`), which is the number of non-tied paires (Xi ≠ Yi) and a probability `p`.

For a lower tail test, the p-value is the probability of observing a value ≤ S, if S ≤ Nr/2, or of observing a value < S, if S > Nr/2 and is calculated using the cumulative distribution function of the binomial distribution: binocdf(S, Nr, 0.5). For an upper tailed test, the probability is calculated as the complement of the lower tail probability: 1-p. For a two-tailed test, the probability is calculated as the double of the lower of these two values: 2*min(p, 1-p). The null hypothesis is rejected if the probability is lower than the value of the significance level.

If Nr > 25, QtiPlot also computes a Z-score, using the following formula: Z = (S-mean-0.5*sign(S-mean))/sqrt(variance), where 0.5 is a continuity correction factor, the mean of the binomial distribution with probability p is Nr*p and its variance is Nr*p*(1-p). In our case positive and negative differences are equally probable for the paired samples, therefore p is 0.5.

If there are missing values in the two source data columns, the whole pair will be excluded from the analysis.