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What is a 'T-Test'

A t-test is an analysis framework used to determine the difference between two sample means from two normally distributed populations with unknown variances. A t-test is an analysis of two populations means through the use of statistical examination; analysts commonly use a t-test with two samples with small sample sizes, testing the difference between the samples when they do not know the variances of two normal distributions.

A t-test looks at the t-statistic, the t-distribution and degrees of freedom to determine the probability of difference between populations; the test statistic in the test is the t-statistic. To conduct a test with three or more variables, one must use an analysis of variance.

BREAKING DOWN 'T-Test'

A form of hypothesis testing, the t-test is just one of many tests used for this purpose. Statisticians must use tests other than the t-test to examine more variables and tests with larger sample sizes. For a large sample size, statisticians use a z-test. Other testing options include the chi-square test and the f-test.

Statistical Analysis of the T-Test

The formula used to calculate the test is a ratio: The top portion of the ratio is the easiest portion to calculate and understand as it is the difference between the means or averages of the two samples. The lower half of the ratio is a measurement of the dispersion, or variability, of the scores. The bottom part of this ratio is the standard error of the difference. To compute this part of the ratio, one must determine the variance for each sample and divide by the number of individuals comprising the sample or group. These two values are then added together, and a square root is taken of the result.

Example of a T-Test

For example, consider that an analyst wants to study the amount that Pennsylvanians and Californians spend on clothing per month. It would not be practical to record the spending habits of every individual or family in both states. Thus, a sample of spending habits is taken from a selected group of individuals from each state. The group may be of any small to moderate size — for this example, assume that the sample group is 200 individuals.

The average amount for Pennsylvanians comes out to $500, and the average amount for Californians is $1,000. The t-test questions whether the difference between the groups represents a true difference between people in Pennsylvania and people in California or if it is likely a meaningless statistical difference. In this example, if all Pennsylvanians spent $500 per month on clothing and all Californians spent $1,000 per month, it is highly unlikely that 200 randomly selected individuals all spent that exact amount, respective to state. Thus, if an analyst or statistician yielded the results listed in the example above, it is safe to conclude that the difference between sample groups indicates a significant difference between the populations of each state.

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