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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

$\displaystyle \hat X_i$ $\textstyle =$ $\displaystyle (X_i-\bar X_i)$  
$\displaystyle \hat Y_i$ $\textstyle =$ $\displaystyle (Y_i-\bar Y_i),$  

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


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

© 1996-9 Eric W. Weisstein