How to Calculate a T-Statistic

How to Calculate a T-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 the 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 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

      1- Subtract the population mean from the sample mean: x-bar - μ.
      2- Divide s by the square root of n, the number of units in the sample: s ÷ √(n).
      3- 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]).

      The t score formula is:

      If you have only one item in your sample, the square root in the denominator becomes √1. This means the formula becomes:
      t score formula 2
      In simple terms, the larger the t score, the larger the difference is between the groups you are testing. It’s influenced by many factors including:
      • How many items are in your sample.
      • The means of your sample.
      • The mean of the population from which your sample is drawn.
        • The standard deviation of your sample.

        Example of the T Score Formula


        Sample question:
        A law school claims it’s graduates earn an average of $300 per hour. A sample of 15 graduates is selected and found to have a mean salary of $280 with a sample standard deviation of $50. Assuming the school’s claim is true, what is the probability that the mean salary of graduates will be no more than $280?
        Step 1: Plug the information into the formula and solve:
        x̄ = sample mean = 280
        μ0 = population mean = 300
        s = sample standard deviation = 50
        n = sample size = 15
        t = (280 – 300)/ (50/√15) = -20 / 12.909945 = -1.549.

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