Example Data Table
| Example |
Mean 1 |
SD 1 |
N 1 |
Mean 2 |
SD 2 |
N 2 |
Suggested test |
| Training score comparison |
82.4 |
9.6 |
28 |
76.1 |
8.9 |
26 |
Welch test |
| Equal variance classroom trial |
71.2 |
6.8 |
35 |
68.5 |
7.1 |
33 |
Pooled test |
| Before and after treatment |
Mean difference: 5.4 |
22 pairs |
SD difference: 7.2 |
0.05 alpha |
Paired test |
Formula Used
Welch Two Sample Test
t = ((mean1 - mean2) - hypothesized difference) / sqrt((sd1² / n1) + (sd2² / n2))
df = ((sd1² / n1 + sd2² / n2)²) / (((sd1² / n1)² / (n1 - 1)) + ((sd2² / n2)² / (n2 - 1)))
Equal Variance Pooled Test
sp² = (((n1 - 1)sd1²) + ((n2 - 1)sd2²)) / (n1 + n2 - 2)
t = ((mean1 - mean2) - hypothesized difference) / sqrt(sp²(1 / n1 + 1 / n2))
df = n1 + n2 - 2
Paired Means Test
t = (mean of paired differences - hypothesized difference) / (SD of differences / sqrt(n))
df = n - 1
Confidence Interval
CI = observed difference ± t critical × standard error
How to Use This Calculator
Select summary statistics when you already know sample means, standard deviations, and sample sizes. Select raw data when you want the calculator to compute those values. Choose Welch for most independent samples. Choose pooled only when equal variance is reasonable. Choose paired when both values come from matched subjects.
Set alpha, confidence level, tail direction, and the hypothesized difference. A value of zero tests whether both population means are equal. Press Calculate. The result appears above the form. Download the result as CSV or PDF for notes, reports, or records.
About the T Test for Two Means
A t test for two means helps compare two average values. It is often used when population standard deviations are unknown. The method checks whether the observed difference is large enough to be considered statistically meaningful. This calculator supports independent samples and paired samples. It also supports summary data and raw data.
Choosing the Right Test
Welch testing is usually the safest independent sample option. It does not require equal variances. The pooled test is useful when both groups have similar spread and similar measurement conditions. A paired test is different. It is used when each observation in one group matches an observation in the other group.
Understanding the Output
The t statistic measures distance from the null hypothesis in standard error units. The p value gives evidence against the null hypothesis. A small p value means the observed difference would be unlikely under the null model. The confidence interval shows a practical range for the true mean difference.
Using Results Carefully
Statistical significance does not prove practical importance. Review the mean difference, interval width, sample size, and effect size together. Cohen d helps describe the size of the difference. Hedges g gives a small sample correction. Always check study design before interpreting any result.
Good Data Practice
Use independent tests only when observations are unrelated. Use paired tests for repeated measures, twins, matched cases, or before and after designs. Remove data errors before testing. Do not remove valid outliers without a clear reason. Report the test method, degrees of freedom, t value, p value, confidence interval, and conclusion.
FAQs
1. What does this calculator test?
It tests whether two population means differ. It uses sample data to calculate a t statistic, degrees of freedom, p value, confidence interval, and decision.
2. Should I use Welch or pooled testing?
Use Welch for most independent samples. It handles unequal variances better. Use pooled testing only when equal variance is a reasonable assumption.
3. When should I use a paired test?
Use a paired test when values are matched. Common examples include before and after scores, repeated measurements, or matched subjects.
4. What does the p value mean?
The p value shows how unusual the observed result is under the null hypothesis. Smaller values give stronger evidence against that null hypothesis.
5. What is the null hypothesis?
The null hypothesis usually says the true mean difference equals zero. You can change that value using the hypothesized difference input.
6. What does confidence interval mean?
The confidence interval gives a likely range for the true mean difference. Wider intervals usually mean less precise evidence.
7. Can I paste raw data?
Yes. Choose raw data mode. Enter values separated by commas, spaces, semicolons, or new lines. The calculator computes means and deviations.
8. What is Cohen d?
Cohen d is an effect size. It shows the difference between means using standard deviation units, helping judge practical importance.