t-test Mathematically

The task of the t-test is to determine whether the difference between the two samples' means is significant. What should we take into consideration to perform it?

• Obviously, we should consider the difference between the means itself;

• As shown in the image below, the variance matters too;

• Also, the size of each sample should be taken into consideration.

To account for the difference between the means, we simply calculate that difference:

The situation becomes more complex when it comes to variance. The t-test assumes that the variance is equal for both samples. We will delve deeper into this in the t-test assumptions chapter. To estimate the variance from two samples, the pooled variance formula is applied:

And to account for the size, we need sample sizes:

Let's put it all together into t statistic.

You may have noticed that sample sizes are not used in the most intuitive manner. However, this approach ensures that t follows the t-distribution, as we'll explore in the next chapter.