T test by Greg Clyde Divino

The t-test is a statistical test whether two sample means or proportions are equal. The t-test (or student's t-test) gives an indication of the separateness of two sets of measurements, and is thus used to check whether two sets of measures are essentially different (and usually that an experimental effect has been demonstrated). The typical way of doing this is with the null hypothesis that means of the two sets of measures are equal. The t-test was developed by “Student”, whose actual name was Williams Sealy Gossett, but the company he was working for, Guiness Brewery, wouldn’t let him publish under his own name.

It is used when there is random assignment and only two sets of measurement to compare.

There are two main types of t-test:

• Independent-measures t-test: when samples are not matched.
• Matched-pair t-test: When samples appear in pairs

Independent mean test is used when we have two different groups of subjects, one group performing one condition in the experiment, and the other group performing the other condition.

In both cases, we have one independent variable, with two levels. We have one dependent variable.

The matched pair t-test or dependent means test is used when the same subjects participate in both conditions of the experiment.

The value of t may be calculated using packages such as SPSS. The actual calculation for two groups is:

t = experimental effect / variability

  = difference between group means /
     standard error of difference between group means

  =  Dg / SEg

Dg = AVERAGE(Xt) - AVERAGE(Xc)

where Xt are the measures in the treatment group and Xc are the measures in the control group. Note that any minus sign is removed, so that 't' remains positive.

SEg = SQRT( VAR(Xt)/nt + VAR(Xc)/nc)

where n is the number of people in the group and VAR(X) is the variance of X.

VAR(X) = SUM((X-AVERAGE(X))2)/(n-1)

The t-value will be positive if the first mean is larger than the second and negative if it is smaller. Once you compute the t-value you have to look it up in a table of significance to test whether the ratio is large enough to say that the difference between the groups is not likely to have been a chance finding. To test the significance, you need to set a risk level (called the alpha level). In most social research, the "rule of thumb" is to set the alpha level at .05. This means that five times out of a hundred you would find a statistically significant difference between the means even if there was none (i.e., by "chance"). You also need to determine the degrees of freedom (df) for the test. In the t-test, the degrees of freedom is the sum of the persons in both groups minus 2. Given the alpha level, the df, and the t-value, you can look the t-value up in a standard table of significance (available as an appendix in the back of most statistics texts) to determine whether the t-value is large enough to be significant. If it is, you can conclude that the difference between the means for the two groups is different (even given the variability). Fortunately, statistical computer programs routinely print the significance test results and save you the trouble of looking them up in a table.