Student’s t-test is a statistical test used to determine whether two normally distributed samples are drawn from populations with the same mean, or whether one normally distributed sample is drawn froma population with a given reference mean.
The basic idea of Student’s t-test is to measure the difference in means in terms of the standard error. This is the t statistic. If the standard deviation of the populations were known in advance, the distribution of the t statstic would be normal. But because the standard deviation, and thus the standard error, is itself estimated from the samples, the distribution of the t statistic is not quite normal. The precise distribution of t was calculated by William Gosset, using the pen name Student, in 1908. The departure of the distribution of t from normality is particularly pronounced for small samples; for large samples the distribution of t tends to the normal distribution.
Three variations of Student’s t-test are in common use:
The one-sample t-test compares the the mean of a sample to a reference mean.
The two-sample t-test compares the means of two samples.
The paired t-test compares the means of two samples with paired entries. Each pair is assumed to share some variance, so that the pairing reduces the variance in the sample.