The paired t-test (see Student's t-test) is useful for looking at differences in two variables. The data must be paired e.g. looking at pre and post-diet weight. The variables being averaged must also be numerical and adequately normal.

If you are looking at independent/unpaired data e.g. height between two countries, you need to use the independent t-test.

If your data is not normally distributed, or there are outliers, you may be better using the Wilcoxon Signed Ranks Test.

The p value tells you how likely the difference observed would occur if sampling from a population in which there is no actual difference. A small p value tells you that it would be rare to observe such a difference if there is no actual difference between the variables. From this we might reject the null hypothesis in favour of the alternative hypothesis - namely, that there is a difference.

In the example below (based on false data for illustration only), the p value is very low. In which case we can reject the hypothesis that there is no difference in weight pre and post-diet. Of course, whether the difference is of practical significance is another matter entirely.

Be aware that there is a certain sensitivity about terminology around this area. According to a widespread convention, we shouldn't conclude that there is a relationship, only that we reject the null hypothesis (see Hypothesis testing). We might go so far as to reject the null hypothesis in favour of the alternative hypothesis. See Statistical hypothesis testing.

A video is available showing how to do paired t-tests using SOFA Statistics: https://www.youtube.com/watch?v=DiAYu0aM9Zw

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