*n*drawn from a Normal population with mean and standard deviation . Let denote the sample mean and

*s*, the sample standard deviation. Then the quantity

has a

*t*distribution with

*n*-1 degrees of freedom.

The

*t*density curves are symmetric and bell-shaped like the normal distribution and have their peak at 0. However, the spread is more than that of the standard normal distribution. This is due to the fact that in formula (1), the denominator is

*s*rather than . Since

*s*is a random quantity varying with various samples, the variability in

*t*is more, resulting in a larger spread.

The larger the degrees of freedom, the closer the *t*-density is to the normal density. This reflects the fact that the standard deviation *s* approaches for large sample size *n*.

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