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We have discussed the sampling distribution of the sample mean when the population standard deviation, σ, is known.  However, in practice, we rarely know the population standard deviation.  In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size can cause inaccuracies in the confidence interval.

Student’s t Distribution

William S. Gosset (1876–1937) of the Guinness brewery in Dublin, Ireland ran into this problem. His experiments with hops and barley produced very few samples. Just replacing σ with s did not always produce accurate results when he tried to use existing inference techniques. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size.  This is because s is a more reliable estimate of σ as samples get bigger.   This problem led him to “discover” what is called Student's t-distribution. The name “Student’s T distribution” comes from the fact that Gosset wrote under the pen name “Student.”

Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and used the Student’s t-distribution only for sample sizes of at most 30. I our current age of technology, the accepted practice now is to simply use the Student’s t-distribution whenever s is used as an estimate for σ.

In summary, if you draw a simple random sample of size n from a population that has an approximately normal distribution with mean μ and unknown population standard deviation σ and calculate the t-score t = [latex]\frac{\overline{x}–\mu }{\left(\frac{s}{\sqrt{n}}\right)}[/latex], then the t-scores follow a Student’s t-distribution with n – 1 degrees of freedom. The t-score has the same interpretation as the z-score. It measures how far [latex]\overline{x}[/latex] is from its mean μ. For each sample size n, there is a different Student’s t-distribution.

The following images compare the Z (Standard Normal) and t (Student’s T).  What differences do you notice?  

Degrees of Freedom

The degrees of freedom, n – 1, come from the calculation of the sample standard deviation s. Remember when we calculated a sample standard deviation we divided the sum of the squared deviations by n − 1, but we used n deviations \(\left(x–\overline{x}\right)\) to calculate s. Because the sum of the deviations is zero, we can find the last deviation once we know the other n – 1 deviations. The other n – 1 deviations can change or vary freely. We call the number n – 1 the degrees of freedom (df).

For example, if we have a sample of size n = 20 items, then we calculate the degrees of freedom as df = n – 1 = 20 – 1 = 19 and we write the distribution as T ~ t₁₉.

The following image shows what happens to the t distribution as you change the degrees of freedom.  What happens as the df increases? What happens once n becomes around 30 and how does that relate to what you already know about the CLT?  

Finding T Distribution Probabilities

A probability table for the Student’s t-distribution can also be used. The table gives t-scores that correspond to the confidence level (column) and degrees of freedom (row).  When using a t-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.  Notice that most t tables gives t-scores given the degrees of freedom and the right-tailed probability.

You’ll find that the t table adequate for finding critical values, but is very limited when trying to find p-values. Calculators and computers can easily calculate any Student’s t-probabilities.