# Correct normalty preconditions for the t-Test

Lately, I have had several data sets at hand that I want to use to study the effect of a treatment. The data is in the form of:

$$\begin{array} {|c|c|c|c|} \hline \text{Participant} \quad & \text{value before} \quad & \text{value after} \quad & \text{difference} \quad \\ \hline 1 & 64 & 62 &2 \\ 2 & 58 & 53 & 5 \\ \vdots & & & \\ \hline \end{array}$$

I assume the paired t-test to be a good test to decide if the difference in values before and after the treatment are significant. However, I'm unsure about the pre-conditions of this test.
Half of the sources I find state that I need to check for normality of the distribution of values before and after the test (this is given in my sample).
The other half states that I need to check the normality of the difference of the values (this is not given in my sample).

Which (if not both?) of the checks is truly necessary and why?

• I think you can use MathJax t create arrays which serve the same function as your table. This is explained in one of the answers to the post you link to. – mdewey Dec 21 '18 at 15:19
• It is the differences which are relevant (the test is often referred to as the one-sample t-test which gives you a clue). Having said that moderate departures from normality ar eunlikely to affect much. – mdewey Dec 21 '18 at 15:20
• thanks @mdewey, didn't scroll down long enough... The long headers where a bit tricky, but it looks nice now. Also thanks for the answer. – nirolo Dec 21 '18 at 16:46

## 1 Answer

If you are doing a paired t-test, you should look for normality in the differences. A paired t-test uses the differences to compute the test statistic, so the differences are all that matter, not the original data. See Bland's "Introduction to Medical Statistics" for more.

• Thanks @demetri-pananos, that book helped. I think I understand now why the differences are of interest and why there is the assumption that the difference is normally distributed. One last thing: I can test for this normality assumption by testing my sample differences because if the difference follows a normal distribution in the population, then I would also assume that my sample should follow a normal distribution, correct? – nirolo Dec 21 '18 at 18:15
• @nirolo Yea, sure – Demetri Pananos Dec 21 '18 at 19:22
• In reality the Gauss-Markov theorem specifies that the model errors are normally distributed, but this would imply normality of the differences. – StatsStudent Dec 22 '18 at 3:48