Home Â» How to Perform a Paired t-Test by Hand

# 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 = xdiffÂ / (sdiff/âˆšn)

where:

• xdiff:Â sample mean of the differences
• s:Â sample standard deviation of the differences
• n:Â 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 = xdiffÂ / (sdiff/âˆšn)
• t = 1.75 / (1.422/âˆš12)
• t =Â 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, weâ€™ll use a two-tailed test with Î± = .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: Î¼1 = Î¼2 (the two population means are equal)
• HA: Î¼1 â‰  Î¼2 (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.