Steve Simon


*Dear Professor Mean

A t-test covers a wide range of tests. It appears when you are testing whether the mean for a given group has exceeded a certain standard. It appears when you compare the means of two different groups. It also appears in linear regression models.

When you take a statistic and dividie by its estiamted variation, which is often known as the standard error

Most computer software will provide a p-value to accompany the t-test. A p-value makes the use of t percentile tables unnecessary.

Short explanation

A simple interpretation of the t-test is that it measures how many standard errors our statistical estimate is from a hypothesized value. A large positive t-test implies that our estimate is quite a bit larger than the hypothesized value. A large negative t-test implies that our estimate is quite a bit smaller than our hypothesized value.

More details

The behavior of the t-test depends greatly on how good our standard error is. If we have a very precise estimate of the variation in our statistic

We can quantify how good our standard error is by the degrees of freedom. The degrees of freedom is related to how much data we have and how many things we are trying to estimate with that data.

Here’s an example of how a t-test would behave if it had 25 degrees of freedom. Notice that it looks quite a bit like a standard normal distribution.


It looks and behaves quite a bit like a standard normal distibution. Here’s a t-distribution with 2 degrees of freedom.


It has the same bell-hspaed curve



We have a sample of 30 informational pamphlets. We record the reading level of each pamphlet and notice that the average level is 9.8. We want an average reading level for all the pamphlets that we produce to be at an 8th grade level. It looks like our sample of pamphlets is writen at a level 1.8 years higher than we want. But could a deviation of that size be due to sampling error?

We can use a one-sample t-test in SPSS to check. Select ANALYZE | COMPARE MEANS | ONE-SAMPLE T TEST from the menu. The dialog box appears in Figure 1. Select the variable that you want to test

SPSS reports a mean difference of 1.8 and a standard error of 0.53. If you divide the mean difference by the standard error

We could also look at the p-value. Since the p-value is so small (.002), the deviation that we see is unlikely to arise just by sampling error.


The t-test is a general test that involves dividing a test statistic by its standard error. The value is then compared to the t-distribution. The t-distribution looks a lot like a normal distribution

Further reading

Just about any introductory Statistics book will talk about the t-distribution. See

–> SurfSTAT Australia. Annette Dobson

–> **Fundamentals of Biostatistics

You can find an earlier version of this page on my original website.