# Why a T-statistic needs the data to follow a normal distribution

I was looking at this notebook, and I am puzzled by this statement:

When we talk about normality what we mean is that the data should look like a normal distribution. This is important because several statistic tests rely on this (e.g. t-statistics).

I don't understand why a T-statistic needs the data to follow a normal distribution.

Indeed, Wikipedia says the same thing:

Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population

However, I don't understand why this assumption is necessary.

Nothing from its formula indicates to me that the data has to follow a normal distribution: I looked a bit on its definition but I don't understand why the condition is necessary.

The information you require is in the "Characterization" section of the Wiki page. A $t$-distribution with degrees of freedom $\nu$ may be defined as the distribution of the random variable $T$ such that $$T = \dfrac{Z}{\sqrt{V/\nu}} \,,$$ where $Z$ is a standard normal distribution random variable and $V$ is a $\chi^2$ random variable with degrees of freedom $\nu$. In addition, $Z$ and $V$ must be independent. So given any $Z$ and $V$ that follow the above definition, you can then arrive at a random variable that has a $t$-distribution.

Now, suppose $X_1, X_2, \dots, X_n$ is distributed according to a distribution $F$. Let $F$ have mean $\mu$ and variance $\sigma^2$. Let $\bar{X}$ be the sample mean and $S^2$ be the sample variance. We will then look at the formulae:

$$\dfrac{\bar{X} - \mu}{S/\sqrt{n}} = \dfrac{\frac{\bar{X} - \mu}{\sigma/\sqrt{n}}}{\sqrt{\frac{(n-1)S^2}{(n-1)\sigma^2}}} \,.$$

If, $F$ denotes the normal distribution, then $\bar{X} \sim N(\mu, \sigma^2/n)$, and thus $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)$. In addition, $\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}$ by Cochran's Theorem. Finally, by an application of Basu's theorem, $\bar{X}$ and $S^2$ are independent. This then implies that the resulting statistic has a $t$-distribution with $n-1$ degrees of freedom.

If the original data distribution $F$ was not normal, then, the exact distribution of the numerator and denominator will not be standard normal and $\chi^2$, respectively, and thus the resulting statistics will not have a $t$-distribution.

• I've always found it quite interesting how much mathematical technology go into these foundational results in mathematical statistics. – Matthew Drury Dec 20 '17 at 16:04
• Good post. However, we don't need to invoke those big theorems to prove the independence between $\bar{X}$ and $S$, as well as the $\chi^2$ distribution. See the first answer of this post. – Zhanxiong Dec 20 '17 at 21:40

I think there may be some confusion between the statistic and its formula, versus the distribution and its formula. You can apply the t-statistic formula to any dataset and get a "t-statistic", but this statistic will not be distributed according to the student-t distribution unless the data came from a normal distribution (or at least, will not be guaranteed to be; my guess is that non-normal distributions won't produce a student-t distribution when the t-statistic formula is applied, but I'm not certain of that). The reason for this is simply that the distribution of the t-statistic is calculated from the distribution of the data that generated it, so if you have a different underlying distribution, then you're not guaranteed to have the same distribution for derived statistics.