Assumptions

Most $t$-test statistics have the form $t=\frac{Z}{s}$, where $Z$ and $s$ are functions of the data. Typically, $Z$ is designed to be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas $s$ is a scaling parameter that allows the distribution of $t$ to be determined.

As an example, in the one-sample $t$-test:

$Z=\dfrac{\bar{X}}{(\hat{\sigma}/\sqrt{n})}$

where $\bar { X }$ is the sample mean of the data, $n$ is the sample size, and $\hat { \sigma }$ is the population standard deviation of the data; $s$ in the one-sample $t$-test is $\hat { \sigma } /\sqrt { n }$, where $\hat { \sigma }$ is the sample standard deviation.

The assumptions underlying a $t$-test are that:

• $Z$ follows a standard normal distribution under the null hypothesis.
• $s^2$ follows a $\chi^2$ distribution with $p$ degrees of freedom under the null hypothesis, where $p$ is a positive constant.
• $Z$ and $s$ are independent.

In a specific type of $t$-test, these conditions are consequences of the population being studied, and of the way in which the data are sampled. For example, in the $t$-test comparing the means of two independent samples, the following assumptions should be met:

• Each of the two populations being compared should follow a normal distribution. This can be tested using a normality test, or it can be assessed graphically using a normal quantile plot.
• If using Student's original definition of the $t$-test, the two populations being compared should have the same variance (testable using the $F$-test or assessable graphically using a Q-Q plot). If the sample sizes in the two groups being compared are equal, Student's original $t$-test is highly robust to the presence of unequal variances. Welch's $t$-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.
• The data used to carry out the test should be sampled independently from the two populations being compared. This is, in general, not testable from the data, but if the data are known to be dependently sampled (i.e., if they were sampled in clusters), then the classical $t$-tests discussed here may give misleading results.