Two Sample t Test Calculator
Calculate independent Two-Sample $t$-test statistic (Welch's $t$), mean difference ($\bar{x}_1 - \bar{x}_2$), $df$, $p$-value, and 95% confidence intervals.
Calculate Independent Two-Sample t-Test
Enter sample mean, SD, and size for Group 1 and Group 2.
Calculation Results
Calculated using Welch\'s unequal variances t-test: t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
Quick Summary
The Two Sample t Test Calculator evaluates independent group mean differences ($\bar{x}_1 - \bar{x}_2$), Welch's $t$-statistic, degrees of freedom ($df$), and $p$-values.
Formula Explanation
t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
df_{\text{Welch}} = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}}How It Works
An independent 2-sample t-test compares the means of two distinct groups. The Two Sample t Test Calculator applies Welch's t-test, which handles unequal group variances and different sample sizes ($n_1 \neq n_2$).
Step-by-Step Worked Example
Practical Problem: Group 1 ($n_1 = 20, \bar{x}_1 = 78, s_1 = 10$) vs Group 2 ($n_2 = 20, \bar{x}_2 = 72, s_2 = 12$). Calculate Welch's $t$-statistic and test for significant difference ($\alpha = 0.05$).
- Step 1: Calculate mean difference ($\bar{x}_1 - \bar{x}_2$): $78 - 72 = \mathbf{+6.0000}$.
- Step 2: Calculate variance ratios ($s_1^2/n_1$ and $s_2^2/n_2$): $100/20 = 5.0$; $144/20 = 7.2$.
- Step 3: Calculate standard error of difference ($\text{SE}_{\text{diff}}$): $\sqrt{5.0 + 7.2} = \sqrt{12.2} = \mathbf{3.4928}$.
- Step 4: Divide mean difference by $\text{SE}_{\text{diff}}$: $t = 6.0 / 3.4928 = \mathbf{+1.7178}$.
- Step 5: Determine Welch df ($df \approx 36.8$) & p-value: $p$-value $= \mathbf{0.0943\text{ (Fail to reject } H_0 \text{ at } \alpha=0.05)}$.
Real-World Calculation Examples
Scenario 1: Test Scores (x̄1 = 78, x̄2 = 72, n = 20)
Parameters: Group 1 vs Group 2
Result: t = +1.7178, p = 0.0943 (Not significant at 5%).
Scenario 2: Clinical Drug vs Placebo Patient Trial
Parameters: Drug (x̄1 = 120, s1 = 8, n1 = 50) vs Placebo (x̄2 = 135, s2 = 10, n2 = 50)
Result: t = -8.29, p < 0.0001 (Highly significant).
Scenario 3: Website A/B Testing Conversion Rate
Parameters: Variant A vs Variant B revenue per user
Result: Independent 2-sample Welch test.
Scenario 4: Salary Comparison (Male vs Female)
Parameters: Salary dataset comparisons
Result: Welch 2-sample t-test evaluation.
Key Benefits of Using This Calculator
Robust Welch's t-Test
Uses Welch's t-test to eliminate equal variance assumption bias.
Welch-Satterthwaite Degrees of Freedom
Computes exact fractional degrees of freedom $df$.
95% CI for Difference
Outputs 95% confidence interval for mean difference $(\bar{x}_1 - \bar{x}_2)$.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a Two-Sample t-test?
A Two-Sample t-test compares the means of two independent groups to determine if they differ significantly.
What is difference between Student's t-test and Welch's t-test?
Student's t-test assumes equal population variances ($\sigma_1^2 = \sigma_2^2$); Welch's t-test does NOT assume equal variances and is recommended by modern statisticians.
How do I test equal variance assumption (Homogeneity of Variance)?
Use Levene's Test or F-test of equal variances ($F = s_1^2 / s_2^2$).
How do I calculate a 2-sample t-test in Excel?
Use formula =T.TEST(array1, array2, tails, type) (Type 3 for Welch unequal variance).
What is Cohen's d for two independent samples?
$\text{Cohen's d} = \frac{\bar{x}_1 - \bar{x}_2}{s_{\text{pooled}}}$.
Can sample sizes be different ($n_1 \neq n_2$)?
Yes — Welch's t-test handles unequal sample sizes seamlessly.
What is null hypothesis ($H_0$) in 2-sample t-test?
$H_0: \mu_1 - \mu_2 = 0$ (no mean difference between groups).
What non-parametric test replaces 2-sample t-test?
The Mann-Whitney U test (Wilcoxon rank-sum test) replaces 2-sample t-test for non-normal data.
What is pooled variance formula for Student's 2-sample t-test?
$s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$.
When should I use Paired t-test instead of 2-Sample t-test?
Use Paired t-test when measurements are taken on the same subjects (before/after) or matched pairs.