Student’s T-Distribution: Definition, Formula and Examples

Student’s t-distribution is a probability distribution used to estimate population parameters from small samples when the population standard deviation is unknown. It is bell-shaped and symmetric like the normal distribution but has heavier tails. It was published in 1908 by William Sealy Gosset under the pen name ‘Student’ while he worked at the Guinness brewery in Dublin.

The t-statistic is calculated as t = (x̄ − μ) / (s / √n), where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation and n is the sample size. The term s / √n is the standard error of the mean. You then read the probability or critical value for that t-value from a t-table using df = n − 1 degrees of freedom.

Both are symmetric, bell-shaped curves centred on zero, but the t-distribution is shorter and wider with heavier tails, giving more probability to extreme values. The t-distribution depends on degrees of freedom, while the normal distribution has a fixed shape. The t is used for small samples with an unknown population standard deviation; the normal is used for large samples with a known standard deviation.

Use a t-test when the population standard deviation is unknown and the sample is small (typically fewer than 30 observations). Use a z-test when the population standard deviation is known, or when the sample is large enough that the sample standard deviation reliably estimates it. Because the t-distribution approaches the normal as the sample grows, the two give almost identical results for large samples.

The t-distribution underpins one-sample, two-sample and paired t-tests, the construction of confidence intervals for a mean from a small sample, and significance testing of regression coefficients. It is widely used wherever conclusions about a population mean must be drawn from a small sample with an unknown standard deviation.