How to Perform a Paired t-Test by Hand

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

The following step-by-step example shows how to perform a paired samples t-test to determine if the population means are equal between the following two groups:

Step 1: Calculate the Test Statistic

The test statistic of a paired t-test is calculated as:

t = xdiff / (sdiff/√n)


  • xdiffsample mean of the differences
  • s: sample standard deviation of the differences
  • n: sample size (i.e. number of pairs)

We will calculate the mean of the differences between the two groups and the standard deviation of the differences between the two groups:

Thus, our test statistic can be calculated as:

  • t = xdiff / (sdiff/√n)
  • t = 1.75 / (1.422/√12)
  • t = 4.26

Step 2: Calculate the Critical Value

Next, we need to find the critical value to compare our test statistic to.

For this example, we’ll use a two-tailed test with α = .05 and df = n-1 degrees of freedom.

According to the t-Distribution table, the critical value that corresponds to these values is 2.201:

Step 3: Reject or Fail to Reject the Null Hypothesis

Our paired samples t-test uses the following null and alternative hypothesis:

  • H0: μ1 = μ2 (the two population means are equal)
  • HA: μ1 ≠ μ2 (the two population means are not equal)

Since the absolute value of our test statistic (4.26) is greater than the critical value found in the t-table (2.201), we reject the null hypothesis.

This means we have sufficient evidence to say that the mean between the two groups is not equal.

Bonus: Feel free to use the Paired Samples t-test Calculator to confirm your results.

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