T scores allow a person to calculate various statistics within a specific sample. For example, a t score can be used to calculate whether the estimate of a sample mean should be rejected or not. The t score can also be used to test various hypotheses about samples, such as whether a person's gender is relevant to SAT scores. Two calculations need to be made to conduct t tests: the critical t score, and the actual t score.

## Critical T Score - Definition

The critical t score is the threshold used to determine whether a hypothesis should be rejected or not. The value of the critical t score depends on the degrees of freedom in the regression, and level of significance the test is being conducted at. It also depends on whether a one- or two-tail test is being conducted.

## T Score Distribution Table

To find the critical t score, a copy of a t distribution table is required. The table is set up so that the degrees of freedom are listed in ascending order down the left side of the table, and the levels of significance are listed in descending order across the top of the table. The levels of significance are listed with the percent in each tail for a two-tail test on top, and the percent in the tail for a one-tail test on the bottom.

## Finding the Critical T Score

To locate a specific critical t score, first locate the degrees of freedom closest to the one for your regression, then follow that row to the level of significance you are using. For example, if the degrees of freedom are 3, and you are conducting a two-tail test at a 5 percent level of significance, your critical t score would be 3.182.

## Sample T Score Formula

Once the critical t score is determined, you will need to find the t score for your information to determine whether or not to reject your hypothesis. The formula for the t score is the sample mean minus the population mean, all over the sample standard deviation divided by the square root of the number of observations. The sample mean, sample standard deviation and number of observations are all available in the data from your sample. The population mean is given by your hypothesis.

## T Score Example

For example, given a sample mean of 20, an estimated population mean of 22, a sample standard deviation of 3 and a total of 4 observations, you would have a t score of 2.667. Given the critical t score from above of 3.182, your hypothesis would fail to be rejected, because the t score is less than the critical t score.

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#### References

- Essentials of Economics; Damodar N. Gujarati; 2006