Paired t-Test

Given two paired sets $X_i$ and $Y_i$ of $n$ measured values, the paired $t$-test determines if they differ from each other in a significant way. Let

$$\hat X_i = (X_i-\bar X_i)$$

$$\hat Y_i = (Y_i-\bar Y_i),$$

then define $t$ by

$$t=(\bar X-\bar Y)\sqrt{n(n-1)\over \sum_{i=1}^n(\hat X_i-\hat Y_i)^2}\,.$$

This statistic has $n-1$ Degrees of Freedom.

A table of Student's t-Distribution confidence interval can be used to determine the significance level at which two distributions differ.

See also Fisher Sign Test, Hypothesis Testing, Student's t-Distribution, Wilcoxon Signed Rank Test

References

Goulden, C. H. Methods of Statistical Analysis, 2nd ed. New York: Wiley, pp. 50-55, 1956.