Independent Samples t-Test

Compare means of two independent groups

Independent Samples t-Test#

The independent samples t-test (also: two-sample t-test) tests whether the means of two independent groups differ significantly.

When to Use#

Use the t-test when you want to:

Compare two independent groups
The dependent variable is metric (continuous)
The data in both groups are approximately normally distributed

Assumptions#

Independence of observations
Metric scale of the dependent variable
Normal distribution in both groups (Shapiro-Wilk test)
Homogeneity of variance (Levene's test) – if violated: Welch's t-test

Formula#

The test statistic is calculated as:

t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}

where $s_p$ is the pooled standard deviation:

s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}

Example#

Practical Example: Drug Effectiveness

A researcher wants to know if a new drug lowers blood pressure. They randomly assign 40 patients to two groups:

Group 1 (n=20): Receives the drug
Group 2 (n=20): Receives a placebo

After 4 weeks, blood pressure is measured. The t-test compares the mean blood pressure values of both groups.

Effect Size#

Cohen's d as a measure of effect size:

d = \frac{\bar{X}_1 - \bar{X}_2}{s_p}

Effect Size	Cohen's d
Small	0.2
Medium	0.5
Large	0.8

Further Reading

Student (1908). The probable error of a mean. Biometrika, 6(1), 1–25.
Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). SAGE.
Bortz, J. & Schuster, C. (2010). Statistik für Human- und Sozialwissenschaftler (7th ed.). Springer.