An important probability distribution used instead of the normal distribution when the standard deviation is estimated from data. Discovered by W. S. Gosset (‘Student’) in 1908, the t distribution gives wider confidence intervals than the normal distribution because of the uncertainty in the estimate of the standard deviation (see measures of variation). The probability values depend on an integer f, the number of degrees of freedom, which is the number associated with the estimate of the standard deviation. Tables of the t distribution are widely available, but algorithms for direct computation are relatively lengthy.

The most common applications are:

1. (a) testing differences between means of two samples;

2. (b) testing differences from zero of estimated parameters in regression analysis and experimental design;

3. (c) evaluation of confidence intervals for means and other estimated quantities.