T-statistics are used in the calculation of small-sample statistics (that is, where a sample size, n, is less than or equal to 30), and take the place of the z-statistic. A t-statistic is necessary because the population standard deviation, defined as the measure of variability in a population, is not known for a small sample. T-statistics, on the other hand, allow for the use of the sample standard deviation, or s, which measures a specific sample's variation, and is more applicable to smaller-sized samples.

## Finding the Values

Find the sample mean, x-bar. This is calculated by adding all of the values in the sample and dividing by the number of units in this summation, n. In certain cases, this value will be given to you by default.

Find the population mean, μ (the Greek letter mu). You can calculate this value by adding all of the values in the observed population and then dividing by number of units in this summation, n. This value is often given by default.

Calculate the sample standard deviation, s. Do this by taking the square root of the variance, if it is given. If not, find the variance: Take a value in the sample, subtract it from the sample mean, and square the difference. Do this for each value, and then add all the values together. Divide this total value by the number of units in the calculation minus 1, or n-1. After you find the variance, take the square root of it.

## Calculate the T-statistic

Subtract the population mean from the sample mean: x-bar - μ.

Divide s by the square root of n, the number of units in the sample: s ÷ √(n).

Take the value you got from subtracting μ from x-bar and divide it by the value you got from dividing s by the square root of n: (x-bar - μ) ÷ (s ÷ √[n]).

References

About the Author

Robert Frankel began writing professionally in 2010. He has written for "The Daily of the University of Washington" as a film critic and cultural analyst. Frankel is pursuing a Bachelor of Science in economics at the University of Washington, Seattle, and is a cadet in the Air Force ROTC detachment on campus.