t-test for Between Groups and Related Groups

Some times you have to decide if two groups are different. You have calculated mean values and standard deviations for the groups assuming you have measurement data. For example:

• Do men and women do differently on a test of spatial ability?
• Does one drug treat headache pain more effectively than another?
• Do College Juniors sleep more than College Seniors?

The problem is that if you calculate two sample means and they are physically different, there are two possible reasons for the difference:

• Each group comes from a different population and the sample means represent two different population means. When this happens, you reject the Null Hypothesis.

• The groups comes from the same population and the means vary by chance. You just happened to pick two groups with means that are far apart. The groups aren't really different. When this happens you fail to reject the Null Hypothesis - see the graphic on the bottom of the page.

The way to decide which is the case if you have two groups is with the t-test. You will compare the two means and get an estimate of the probability that the means are different by chance. How does the t-test work?

First, you will calculate the means and standard deviations for each group. A computer might actually do all of this for you. Next:

• You will use the t-test formula to compute the t-value. The formula you use will depend on if you have equal numbers of subjects, the standard deviations are not different or if you are testing the same people twice or using matched groups.
• The t-score is like a z-score and tells you if the difference is far away from the center of a distribution. In this case, it is the sampling distribution of differences.
• If the t-value is big enough (you look it up in a t-table), you will reject the Null Hypothesis and say you have a difference. If it is not big enough you will say that you have not found a difference and fail to reject the Null Hypothesis.
  

How does the t-ratio do this?:

• The t-test sets up a sampling distribution of differences with a mean of zero.

• The t-test calculates if your difference between the means (m1 - m2) is far in the tails of the sampling distribution or far away from zero.

• Thus, the difference may be big to be chance. If your groups were really the same, then the difference should be close to zero.
• The t-score is made a relative score by dividing the difference between the means by the standard error of the difference, which is like the standard deviation in a score. See your book for the standard error formula.
• That's how we know if your difference is rare by chance as we can calculate the areas in the tales (see the z-score workshop). See the following graphic: