By Matthew Schieltz – Updated Aug 30, 2022

When you collect data or run an experiment, you often need to determine whether a change in one variable is linked to a change in another. T‑tests are the standard statistical tools for testing whether the difference between two groups is significant, beyond what might be expected by random chance.

Independent Samples

Step 1

Create a summary‑statistics table for each group. Calculate and record the sum, sample size (n), and mean. Label each row as sum, n, and mean.

Step 2

Compute the degrees of freedom for each group: df = n – 1. Write this value beside the corresponding summary statistics.

Step 3

Determine the variance and standard deviation for each group and add these to the table.

Step 4

Sum the degrees of freedom from both groups and record this as df‑total.

Step 5

Calculate the pooled variance:

1. Multiply each group’s df by its variance.
2. Add the two products.
3. Divide the sum by df‑total.

Write the result as pooled variance.

Step 6

Compute the standard error of the difference:

1. Divide the pooled variance by each group’s n.
2. Add the two quotients.
3. Take the square root of the sum.

Label this value standard error of the difference.

Step 7

Find the t‑value:

1. Subtract the smaller mean from the larger mean.
2. Divide this difference by the standard error of the difference.

Record the result as t‑obtained or t‑value.

Dependent Samples

Step 1

For each paired observation, subtract the second score from the first and place the result in a column titled Difference. Sum all differences to obtain D.

Step 2

Square each difference, store in a column D‑squared, and sum these to get ΣD².

Step 3

Compute the divisor:

1. Multiply the number of pairs (n) by ΣD².
2. Subtract D² from this product.
3. Divide the result by (n – 1).
4. Take the square root of the quotient.

Label the final value divisor.

Step 4

Divide D by the divisor to obtain the t‑value for the paired‑samples t‑test.

TL;DR

Compare the calculated t‑value with the critical t‑value from a t‑distribution table. If the absolute t‑value exceeds the critical value, reject the null hypothesis; otherwise, do not reject it.

For further reading, see Wikipedia – T‑test.