**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 t**ake 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.

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, 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.

**Example**

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.

**Summary**

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.