t Distribution

David M. Lane

Prerequisites

Normal Distribution, Areas Under Normal Distributions, Degrees of Freedom, Confidence Interval for the Mean

Learning Objectives
1. State the difference between the shape of the t and normal distribution
2. State how the difference between the shape of the t and normal distribution is affected by the degrees of freedom
3. Use a t table to find the value of t to use in a confidence interval
4. Use the t calculator to find the value of t to use in a confidence interval

The t distribution is very similar to the normal distribution when the estimate of variance is based on many degrees of freedom but has relatively more scores in its tails when there are fewer degrees of freedom. The normal distribution has relatively more scores in the center of the distribution and the t distribution has relatively more in the tails. The t distribution is therefore leptokurtic

Since the t distribution is leptokurtic, the percentage of the distribution within 1.96 standard deviations of the mean is less than the 95% for the normal distribution. Table 1 shows the number of standard deviations from the mean required to contain 95% and 99% of the area of the t distribution for various numbers of degrees of freedom. These are the values of t that you use in a confidence interval. The corresponding values for the normal distribution are 1.96 and 2.58 respectively. Notice that with few degrees of freedom, the values of t are much higher than the corresponding values for a normal distribution and that the difference decreases as the degrees of freedom increase. The values in Table 1 can be obtained from the "Find t for a confidence interval" calculator.

Table 1. Abbreviated t table.
 df 0.95 0.99 2 4.303 9.925 3 3.182 5.841 4 2.776 4.604 5 2.571 4.032 8 2.306 3.355 10 2.228 3.169 20 2.086 2.845 50 2.009 2.678 100 1.984 2.626