# Conduct and Interpret a One-Sample T-Test

**What is the One-Sample T-Test?**

The one-sample t-test is a member of the t-test family. All the tests in the t-test family compare differences in mean scores of continuous-level (interval or ratio), normally distributed data. The 1-sample t-test does compare the mean of a single sample. Unlike the other tests, the independent and dependent sample t-test it works with only one mean score.

The independent sample t-test compares one mean of a distinct group to the mean of another group from the same sample. It would examine the question, “*Are old people smaller than the rest of the population*?” The dependent sample t-test compares before/after measurements, like for example, “*Do pupils’ grades improve after they receive tutoring*?”

So if only a single mean is calculated from the sample what does the 1-sample t-test compare the mean with? The 1-sample t-test compares the mean score found in an observed sample to a hypothetically assumed value. Typically the hypothetically assumed value is the population mean or some other theoretically derived value.

There are some typical applications of the 1-sample t-test: 1) testing a sample a against a pre-defined value, 2) testing a sample against an expected value, 3) testing a sample against common sense or expectations, and 4) testing the results of a replicated experiment against the original study.

First, the hypothetical mean score can be a generally assumed or pre-defined value. For example, a researcher wants to disprove that the average age of retiring is 65. The researcher would draw a representative sample of people entering retirement and collecting their ages when they did so. The 1-sample t-test compares the mean score obtained in the sample (e.g., 63) to the hypothetical test value of 65. The t-test analyzes whether the difference we find in our sample is just due to random effects of chance or if our sample mean differs systematically from the hypothesized value.

Secondly, the hypothetical mean score also can be some derived expected value. For instance, consider the example that the researcher observes a coin toss and notices that it is not completely random. The researcher would measure multiple coin tosses, assign one side of the coin a 0 and the flip side a 1. The researcher would then conduct a 1-sample t-test to establish whether the mean of the coin tosses is really 0.5 as expected by the laws of chance.

Thirdly, the 1-sample t-test can also be used to test for the difference against a commonly established and well known mean value. For instance a researcher might suspect that the village she was born in is more intelligent than the rest of the country. She therefore collects IQ scores in her home village and uses the 1-sample t-test to test whether the observed IQ score differs from the defined mean value of 100 in the population.

Lastly, the 1-sample t-test can be used to compare the results of a replicated experiment or research analysis. In such a case the hypothesized value would be the previously reported mean score. The new sample can be checked against this mean value. However, if the standard deviation of the first measurement is known a proper 2-sample t-test can be conducted, because the pooled standard deviation can be calculated if the standard deviations and mean scores of both samples are known.

Although the 1-sample t-test is mathematically the twin brother of the independent variable t-test, the interpretation is somewhat different. The 1-sample t-test checks whether the mean score in a sample is a certain value, the independent sample t-test checks whether an estimated coefficient is different from zero.

*The One-Sample T-Test in SPSS*

The 1-sample t-test does compare the mean of a single sample. Unlike the independent and dependent sample t-test, the 1-sample t-test works with only one mean score. The 1-sample t-test compares the mean score found in an observed sample to a hypothetically assumed value. Typically the hypothetically assumed value is the population mean or some other theoretically derived value.

The statement we will examine for the 1-sample t-test is as follows: *The average age in our student sample is 9½ years. *

Before we actually conduct the 1-sample t-test, our first step is to check the distribution for normality. This is best done with a Q-Q Plot. We simply add the variable we want to test (age) to the box and confirm that the test distribution is set to *Normal*. This will create the diagram you see below. The output shows that small values and large values somewhat deviate from normality. As a check we can run a K-S Test to tests the null hypothesis that the variable is normally distributed. We find that the K-S Test is not significant thus we cannot reject H_{0} and we might assume that the variable age is normally distributed.

Let's move on to the 1 sample t-test, which can be found in *Analyze/Compare Means/One-Sample T-Test…*

The 1-sample t-test dialog box is fairly simple. We add the test variable age to the list of *Test Variables* and enter the *Test Value*. In our case the hypothetical test value is 9.5. The dialog *Options…* gives us the setting how to manage missing values and also the opportunity to specify the width of the confidence interval used for testing.