Critical values of Student's t distribution

The derivation of the t distribution was first published by W. S. Gosset in 1908. Gosset was employed by an Irish brewery that did not allow its employees to publish the results of their research. Gosset instead published his work using the pseudonym ‘Student.’ Consequently, the t distribution is often referred to as Student's t distribution.

The t distribution is commonly used for calculating confidence intervals. It is bell-shaped and symmetric around a mean of zero, similar to the Gaussian (normal) distribution. The t distribution has a higher variance than a Gaussian distribution, making it appear flatter and more spread out. As the number of degrees of freedom in the t distribution becomes very large, it is well approximated by a Gaussian distribution with a mean of 0 and a variance of 1, that is, the so-called ‘standard normal distribution.’ Consequently, the last line in the following table shows the critical values for a standard normal distribution.

The values shown in the following table are the critical values of the t distribution. Each ta;n value in the table is the value on the x axis for which there is an area of a to the left of ta;n, as shown in Figure C.I, where n is the number of degrees of freedom. For example, say you want to find the t value necessary to compute a 95% confidence interval with eight degrees of freedom. The corresponding significance level is α = 1 − 0.95 = 0.05.