Sampling and Statistics

When the population standard deviation () is unknown and the sample size is less than 30 (n < 30), the distribution of the test statistic can not be guaranteed to be normal. In fact, the test statistic can be said to conform to what is called a t distribution.

The t distribution is similar to the standard normal distribution in that it is symmetrically distributed around a mean value. But where the t distribution varies from the standard normal, is that its standard deviation is determined by what is known as the number of degrees of freedom.

Degrees of freedom are calculated from the size of the sample. They are a measure of the amount of information from the sample data that has been used up. Every time a statistic is calculated from a sample, one degree of freedom is used up.

When you have a very large sample drawn from your data set, the difference between t values and Z values is miniscule. But as the size of your sample falls, the t distribution takes on a standard deviation increasingly greater than 1. In other words, this means that the t distribution when n < 30, is more spread out than the standard normal distribution.

To show this, have a look at the explanation of the normal distribution curve to confirm that 95% of all Z values lie between 1.96 and -1.96.

It is the case that in a t distribution with 20 degrees of freedom, 95 % of all t values will lie between 2.09 and -2.09. But with only 5 degrees of freedom, 95 % of all t values will lie between 2.57 and -2.57.

There is an illustration of Significance Tests in the 'Crunching' section of TimeWeb.

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