# Statistics Dictionary

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### Matched-Pairs t-Test

A matched-pairs t-test is used to test whether there is a significant mean difference between two sets of paired data.

Here is how to use the test.

• Define paired differences. Define a new variable d, based on the difference between paired values from two data sets.

d = x1 - x2

where x1 is the value of variable x in the first data set, and x2 is the value of variable from the second data set that is paired with x1.

• Define hypotheses. The table below shows three sets of null and alternative hypotheses. Each makes a statement about how the true difference in population values μd is related to some hypothesized value M. (In the table, the symbol ≠ means " not equal to ".)

Set Null hypothesis Alternative hypothesis Number of tails
1 μd = M μd ≠ M 2
2 μd > M μd < M 1
3 μd < M μd > M 1

• Specify significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.

• Find degrees of freedom. The degrees of freedom (DF) is:

DF = n - 1

where n is the number of paired observations.

• Compute test statistic. The test statistic is a t statistic (t) defined by the following equation.

t = (d - M ) / [ sd /sqrt(n) ]

where d is the sample mean difference between paired observations, M is the hypothesized mean difference (from the null hypothesis), and sd is the standard deviation of the differences (i.e., of the d values).

• Compute P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the t statistic, having the degrees of freedom computed above.

• Evaluate null hypothesis. The evaluation involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.

The matched-pairs t-test can be used with large or small samples.