T-test (created 1999-04-18)

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Dear Professor Mean, How do you analyze a t-test. I have a t-test value, and I know that I have to compare it to a t-distribution. I'm not sure how to do that.

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, the result is a t-test. You would compare this t-test to percentiles from a t distribution. Tables of these percentiles are found in the back of most statistical text books.

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, then the t-test has a distribution that is very close to a standard normal distribution. If we have an imprecise estimate of the variation in our statistic, then the t-test will be more variable than a standard normal distribution.

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.

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

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It has the same bell-hspaed curve, but notice that the tails of the distribution don't touch the axis, even at plus or minus 4. That means that extreme values are more likely with this t-distribution than the previous one.

Usually, you compare your t-test with a value from a t-percentile table. Extreme values of the t-test (for example, a value larger than the 95th percentile) indicate that your statistic is more extreme than you would expect from sampling error. If you get a t-test from computer software like SPSS, then you will usually get a p-value with your t-test. If your p-value is small, that implies that your t-test is more extreme than most percentiles from a t-distribution.


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, and insert 8 into the TEST VALUE field. Then click on the OK button. SPSS produces two tables of statistics (see Figure 2).

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, you get a t-test value of 3.380, which SPSS shows in table 2. SPSS also informs us that the degrees of freedom is 29 (which for the one-sample t-test is always one less than the smaple size). Looking in any standard textbook, we would find that the 95th percentile of a t-distribution with 29 degrees of freedom is 1.699. Since our t-test is much larger than 1.699, we would conclude that a deviation as large as 1.8 years is unlikely to arise just by sampling 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, but it tends to be more spread out, especially if the degrees of freedom are small.

Further reading

Just about any introductory Statistics book will talk about the t-distribution. See, for example, chapter 7 of Rosner's book. There are web pages that will calculate probabilities and percentiles of the t distribution, such as SurfSTAT (click on the TABLES button on the main page).

--> SurfSTAT Australia. Annette Dobson, Anne Young, Bob Gibson, and others. (Accessed May 15, 2002). www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html

--> Fundamentals of Biostatistics, Third Edition. Rosner B. Belmont CA: Duxbury Press (1990). ISBN: 0-534-91973-1.