Student’s t-distribution is a perfect example of where the text books miss an opportunity to explain a statistical device in terms of why it was needed. With large samples we can reliably estimate the variance parameter of a normally distributed data but this not possible when it comes to small samples. Gosset showed that when estimating the mean of a normally distributed population the ratio of the sample mean to the variance has the same distribution no matter what the values of the population mean and variance. Not only is this amazing (I would never have assumed this) but it is also extremely useful: a hypothesis test can be carried out on the ratio, thus obviating the need for a sample estimate of the variance.
The point is missed, and the understanding foregone, when statistics is taught using the format: “This is what you do when…” instead of “This is why you do…”
Even better if the “why” is answered with “You need to do … because as you can see that … doesn’t work in this situation.”