Two Population Means for Independent Samples, σ₁ and σ₂ Unknown but Equal

Statistics • Estimation and Hypothesis Testing, Two Populations

Written by STEM Calculators Team Published December 25, 2025 Updated February 24, 2026

Task

Two independent samples, σ₁ and σ₂ unknown, but assumed equal (σ₁ = σ₂). Uses the t distribution with df = n₁ + n₂ − 2 and the pooled standard deviation.

Sample mean x̄₁

Sample mean x̄₂

Sample SD s₁

Sample SD s₂

Sample size n₁

Sample size n₂

Hypotheses

Null difference δ₀ (from H₀)

Alternative H₁

Decision settings

Significance level α

Show decision by

The Conclusion follows your selected method.

Ready

t model visualization

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The shaded area updates after calculation.

Key formulas (equal but unknown σ)

Use the pooled standard deviation s_p and a t model with df = n₁ + n₂ − 2.

Pooled SD and standard error

\[ \begin{aligned} s_p &= \sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}} \\ SE &= s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \\ df &= n_1+n_2-2 \end{aligned} \]

When is this appropriate?

Samples must be independent. If both samples are large, the method is very robust. If one or both samples are small, the populations should be approximately normal.

Enter values and click Calculate.

Batch mode: paste CSV data (compute many rows at once)

Paste rows as CSV (comma-separated). Header is optional. Supported columns: x1, x2, s1, s2, n1, n2, and (optional) conf, alpha, delta0, alt (two/gt/lt), task (ci/ht).

CSV input

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What this calculator does

Computes a confidence interval or performs a hypothesis test for the difference of two population means μ₁ − μ₂ using two independent samples when the population standard deviations are unknown but assumed equal (σ₁ = σ₂).

When you can use it

The two samples are independent.
The population standard deviations are unknown but can be treated as equal.
If samples are small, the populations should be approximately normal; if both are large, the method is very robust.

Formulas used (pooled t procedure)

The calculator first computes the pooled standard deviation and the standard error.

\[ \begin{aligned} s_p &= \sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}} \\ SE &= s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}} \\ df &= n_1+n_2-2 \end{aligned} \]

Confidence interval for μ₁ − μ₂

Choose Confidence interval, select a confidence level, and enter x̄₁, x̄₂, s₁, s₂, n₁, n₂.

\[ (\bar{x}_1-\bar{x}_2)\pm t^\* \cdot SE \]

The critical value t* is taken from the t distribution with df = n₁ + n₂ − 2.

Hypothesis test for μ₁ − μ₂ (p-value approach)

Choose Hypothesis test, set the null difference δ₀ (usually 0), pick the alternative (two-tailed / left / right), and choose α. The test statistic is:

\[ t=\frac{(\bar{x}_1-\bar{x}_2)-\delta_0}{SE} \]

The calculator reports the p-value and (optionally) the critical-value decision. A typical decision rule is: reject H₀ when p-value ≤ α.

Batch mode (CSV)

Use the Batch mode section to compute many rows at once. You can paste a CSV table with columns x1,x2,s1,s2,n1,n2 and optionally task (ci/ht) plus conf or alpha,delta0,alt. The results table can be copied with one click.

Frequently Asked Questions

What does it mean to assume equal variances in a two-sample t procedure?

It means the two populations are assumed to share the same standard deviation (sigma1 = sigma2), even though that value is unknown. The method combines information from both samples using a pooled standard deviation.

How does the calculator compute the pooled standard deviation and standard error?

It computes sp from both sample variances using sp = sqrt(((n1-1)s1^2 + (n2-1)s2^2) / (n1+n2-2)). Then SE = sp x sqrt(1/n1 + 1/n2) for inference on mu1 - mu2.

What degrees of freedom are used for the pooled two-sample t model?

The pooled procedure uses df = n1 + n2 - 2. This df is used to obtain the t critical value and the p-value from the t distribution.

When should I use the equal-variance two-sample t method instead of the unequal-variance method?

Use the equal-variance method when the equal sigma assumption is reasonable and you want the pooled estimate of variability. If the group variances differ substantially or the assumption is doubtful, the unequal-variance two-sample t method is typically safer.

How do p-value and critical-value decisions differ in this hypothesis test?

The p-value approach compares the p-value to alpha and rejects H0 when p-value is less than or equal to alpha. The critical-value approach compares the test statistic to t critical cutoffs determined by alpha and the selected alternative.

Hypotheses

Decision settings

Key formulas (equal but unknown σ)

Calculation steps

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Frequently Asked Questions

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