The T Test
The t test decides whether the means of groups differ more than random chance would explain. It is the workhorse when comparing averages with realistic sample sizes.
Why not a z test
A z test assumes you know the true population standard deviation. In practice you only have a sample estimate, which adds uncertainty. The t test uses the t distribution, a bell curve with heavier tails that widen for small samples, properly accounting for that extra uncertainty.
Common forms
- A one sample t test checks whether a sample mean differs from a known target.
- A two sample t test compares the means of two independent groups.
- A paired t test compares two measurements on the same subjects, like before and after.
Degrees of freedom
The shape of the t distribution depends on the degrees of freedom, roughly the sample size minus the number of estimated means. As degrees of freedom grow, the t distribution approaches the normal, which is why large samples let a z test substitute.
Reading the result
The test produces a t statistic and a p value. A small p value says the observed difference in means is unlikely under the null of equal means.
Key idea
The t test compares means using a sample based standard error and the heavier tailed t distribution, making it valid for small samples with unknown variance.