t test for two means

T-Test for Difference Between Two Means (Most Common)

Use this calculator to test if two population means differ when population standard deviations are unknown (the typical scenario).

When to Use

Two independent samples from different populations
Population SDs unknown (estimated from samples)
Testing if means differ between groups
Choose variance assumption: Test with Levene’s test or use Welch’s (unequal) as default

How to Use

Step 1: Enter both sample means and sample sizes

Step 2: Enter both sample standard deviations

Step 3: Select variance assumption: Equal or Unequal (Welch’s)

Step 4: Enter level of significance (α, typically 0.05)

Step 5: Select tail type (two-tailed for comparing)

Step 6: Click “Calculate”

Interpret: If p-value < α, means differ significantly

t test Calculator for two means
	Sample 1	Sample 2
Mean
Standard Deviation
Sample Size
Variances	Equal	Unequal
Level of Significance ($\alpha$)
Tail	Left tailed Right tailed Two tailed

Results
Standard Error of Diff. of Means:
Test Statistics t:
Degrees of Freedom:
t-critical value(s):
p-value:

Test Formula (Equal Variances - Pooled)

$$t = \frac{(\overline{x}_1 - \overline{x}_2)}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

Where:

$s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}$ = pooled standard deviation
$df = n_1 + n_2 - 2$

Welch’s version: Use individual SEs if variances unequal

When to Use Each Assumption

Equal variances: Use Levene’s test first, or if sample SDs are similar Unequal variances (Welch’s): Use by default when unsure - more robust

Worked Example

Scenario: Testing if training improves test scores. Control (n=20, x̄=75, s=8) vs Trained (n=20, x̄=82, s=9).

Hypotheses:

H₀: μ₁ = μ₂ (no difference)
H₁: μ₁ ≠ μ₂ (training makes difference)

Calculation (pooled):

Pooled s ≈ 8.54
SE ≈ 2.70
t ≈ 2.59, df = 38
p-value ≈ 0.013

Decision: p-value (0.013) < α (0.05), reject H₀. Significant evidence that training improves scores.