Two-sample t-test calculator

This two-sample t-test calculator helps users compare the means of two independent groups to determine whether they are significantly different from each other.

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What is a two-sample t-test?

A two-sample t-test (also called an independent t-test) is a statistical test used to compare the means of two independent groups to determine whether there is a significant difference between them. It answers the question: “Are the two sample means significantly different, or is the observed difference due to random variation?”

When to use a two-sample t-test?

Use a two-sample t-test when:

• You have two independent groups (e.g., men vs. women, treatment vs. control group).
• The data in both groups are normally distributed (or the sample sizes are large enough for the Central Limit Theorem to apply).
• The data are continuous (e.g., test scores, heights, weights).
• The two groups have similar or different variances (you can choose between a standard t-test or Welch’s t-test).

Formula for the two-sample t-test

The test statistic (t-value) is calculated as:

t = \frac{(\bar{x}_1 - \bar{x}_2)}{SE}

where:

• \bar{x}_1, \bar{x}_2 are the sample means
• SE is the standard error of the difference between the two means

The standard error (SE) depends on whether we assume equal or unequal variances.

Equal variance (Pooled t-test)

If both groups have similar variances, we use the pooled standard deviation:

SE = \sqrt{s_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}

where the pooled variance (s_p^2​) is:

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

Unequal variance (Welch’s t-test)

If the two groups have different variances, we use:

SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

Degrees of freedom (df)

The degrees of freedom (df) determine which t-distribution to use.

Equal variances:

df = n_1 + n_2 - 2

Unequal variances (Welch’s Approximation):

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

Interpreting the results

• p-value < 0.05 → Statistically significant difference (Reject the null hypothesis)
• p-value ≥ 0.05 → No significant difference (Fail to reject the null hypothesis)

The test also provides a confidence interval (CI) for the difference between the means:

CI = (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, df} \times SE

If the confidence interval contains 0, the difference is not statistically significant.

Example: Comparing two groups

Scenario: Do Two Schools Have Different Average Test Scores?

A researcher wants to compare the math scores of students from two different schools.

School A:

n_1 = 30, \bar{x}_1 = 78, s_1 = 10

School B:

n_2 = 25, \bar{x}_2 = 72, s_2 = 12

• Confidence level: 95%
• Assume unequal variances and use Welch’s Approximation

Results:

• t-statistic: 1.9897
• Degrees of Freedom: 48.59 (Welch’s df)
• Standard Error: 3.0155
• Confidence Interval: (-0.0612, 12.0612)
• p-value (two-tailed): 0.0523

Conclusion: Since p < 0.10, we reject the null hypothesis. The two schools have significantly different math scores.