The t-test assesses whether the means of two groups are statistically different from each other. This analysis is appropriate whenever you want to compare the means of two groups.
Figure 1 shows the distributions for the treated (blue) and control (green) groups in a study. The figure indicates where the control and treatment group means are located. The question the t-test addresses is whether the means are statistically different.
What does it mean to say that the averages for two groups are statistically different? Consider the three situations shown in Figure 2. The first thing to notice about the three situations is that the difference between the means is the same in all three. But the three situations don't look the same -- The top example shows a case with moderate variability of scores within each group. The second situation shows the high variability case. The third shows the case with low variability. Clearly, we would conclude that the two groups appear most different or distinct in the bottom or low-variability case. Why? Because there is relatively little overlap between the two bell-shaped curves. In the high variability case, the group difference appears least striking because the two bell-shaped distributions overlap so much.
This leads us to a very important conclusion: when we are looking at the differences between scores for two groups, we have to judge the difference between their means relative to the spread or variability of their scores. The t-test does just this.
Statistical Analysis of the t-test
The formula for the t-test is a ratio. The top part of the ratio is just the difference between the two means or averages. The bottom part is a measure of the variability or dispersion of the scores. This formula is essentially another example of the signal-to-noise metaphor in research: the difference between the means is the signal that, in this case, we think our program or treatment introduced into the data; the bottom part of the formula is a measure of variability that is essentially noise that may make it harder to see the group difference. Figure 3 shows the formula for the t-test and how the numerator and denominator are related to the distributions.
The top part of the formula the difference between the means. The bottom part is called the standard error (SE) of the difference. To compute it, we take the variance for each group and divide it by the number of people in that group. We add these two values and then take their square root. The specific formula is given in Figure 4:
The variance is of course simply the square of the standard deviation.
The final formula for the t-test is shown in Figure 5:
The t-value will be positive if the first mean is larger than the second and negative if it is smaller. Once we compute the t-value we 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, we 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 we would find a statistically significant difference between the means even if there was no such difference (i.e., "chance" occurance). We also need to determine the degrees of freedom (df) for the test. In the t-test, the df is the sum of the persons in both groups minus 2. Given the alpha level, the df, and the t-value, we can look the t-value up in a standard table of significance (given below) to determine whether the t-value is large enough to be significant. If it is, we can conclude that the difference between the means for the two groups is different (even given the variability).
- From http://www.socialresearchmethods.net/kb/index.php
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