Student's t curve is a family of curves indexed by a parameter called the degrees of freedom, which can take the values 1, 2, … Student's t curve is used to approximate some probability histograms. Consider a population of numbers that are nearly normally distributed and have population mean is μ. Consider drawing a random sample of size n with replacement from the population, and computing the sample mean M and the sample standard deviation S. Define the random variable

T = (M − μ)/(S/n^½).

If the sample size n is large, the probability histogram of T can be approximated accurately by the normal curve. However, for small and intermediate values of n, Student's t curve with n − 1 degrees of freedom gives a better approximation. That is,

P(a < T < b) is approximately the area under Student's T curve with n − 1 degrees of freedom, from a to b.

Student's t curve can be used to test hypotheses about the population mean and construct confidence intervals for the population mean, when the population distribution is known to be nearly normally distributed. This page contains a tool that shows Student's t curve and lets you find the area under parts of the curve.