Table of Contents
What Is the T-Statistic?
The t-statistic is a ratio that measures how many standard errors the sample mean is away from a hypothesized population mean. It follows Student's t-distribution (developed by William Sealy Gosset under the pseudonym "Student" in 1908) and is used when the population standard deviation is unknown and estimated from the sample.
The t-statistic is the foundation of t-tests, which are among the most commonly used statistical tests. The larger the absolute value of t, the stronger the evidence against the null hypothesis. The t-distribution approaches the normal distribution as the sample size increases.
Formula
Critical T-Values (Two-Tailed)
| df | α=0.10 | α=0.05 | α=0.01 |
|---|---|---|---|
| 10 | 1.812 | 2.228 | 3.169 |
| 20 | 1.725 | 2.086 | 2.845 |
| 30 | 1.697 | 2.042 | 2.750 |
| 120 | 1.658 | 1.980 | 2.617 |
Frequently Asked Questions
When should I use t-test vs z-test?
Use a t-test when the population standard deviation is unknown (which is almost always the case in practice) and you are estimating it from the sample. Use a z-test when the population standard deviation is known or when the sample size is very large (n > 30), though even then the t-test gives identical results asymptotically.
What does degrees of freedom mean?
Degrees of freedom (df = n - 1 for a one-sample t-test) represent the number of independent pieces of information available to estimate the population variance. More degrees of freedom make the t-distribution closer to the normal distribution, leading to narrower critical regions.