Student t Distribution

The population standard deviation is rarely known in many day-to-day examples of statistics. When the sample sizes are large, it is easy to estimate the population standard deviation using a confidence interval, which provides results close enough to the original value. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.

The Student t distribution was developed by William S. Goset (1876–1937) of the Guinness brewery in Dublin, Ireland, to estimate the population standard deviation when the sample sizes were small. The name for this distribution comes from the pen name "Student" used by Gosset.

The Student t distribution is used whenever s is used to estimate σ. If a simple random sample of size n is drawn from an approximately normally distributed with mean μ and unknown population standard deviation σ and t scores are calculated, the t scores follow the Student t distribution with n – 1 degrees of freedom. The t score is interpreted similarly to the z score. It measures how far a value is from its mean μ. For each sample size n, there is a different Student t distribution.

The t score or statistic is given as follows:

Properties of the Student  t distribution:

1. The graph for the Student t distribution is similar to the standard normal curve.
2. The mean for the Student t distribution is zero, and the distribution is symmetric about zero.
3. The Student t distribution has more probability in its tails than the standard normal distribution because the spread of the t-distribution is greater than that of the standard normal. So the Student t distribution curve is thicker in the tails and shorter in the center than the graph of the standard normal distribution.
4. The exact shape of the Student t distribution depends on the degrees of freedom. As the degrees of freedom increase, the graph of the Student t distribution becomes more like the graph of the standard normal distribution.
5. The underlying population of individual observations is assumed to be normally distributed with an unknown population mean μ and unknown population standard deviation σ.

This text is adapted from Section 8.2, A Single Population Mean using the Student's t distribution, Introductory Statistics, Openstax,