Calculator
Example Data Table
| Case | Group 1 Score | Group 2 Score |
|---|---|---|
| 1 | 14 | 11 |
| 2 | 15 | 13 |
| 3 | 13 | 12 |
| 4 | 17 | 15 |
| 5 | 18 | 14 |
| 6 | 16 | 13 |
| 7 | 15 | 12 |
| 8 | 19 | 16 |
Formula Used
Welch standard error: SE = sqrt(s1² / n1 + s2² / n2)
Welch degrees of freedom: df = (s1²/n1 + s2²/n2)² / [((s1²/n1)²/(n1 - 1)) + ((s2²/n2)²/(n2 - 1))]
Pooled variance: sp² = [((n1 - 1)s1²) + ((n2 - 1)s2²)] / (n1 + n2 - 2)
Pooled standard error: SE = sqrt(sp²(1/n1 + 1/n2))
t statistic: t = [(mean1 - mean2) - hypothesized difference] / SE
Confidence interval: mean difference ± t critical × SE
Cohen d: d = (mean1 - mean2) / pooled standard deviation
How to Use This Calculator
Select raw data when you have every observation. Paste values for both independent groups. Select summary statistics when you already know n, mean, and sample standard deviation. Choose Welch for unequal variances or unequal sample sizes. Choose pooled variance only when equal variance is a reasonable assumption. Pick the alternative hypothesis before calculating. Enter the confidence level and hypothesized difference. Press Calculate to view the result above the form. Use the CSV or PDF buttons to save the same calculation.
Independent T Test Guide
What the Test Compares
An independent t test compares two separate groups. It checks whether their means differ more than random sampling noise suggests. This calculator supports raw data and summary statistics. That makes it useful for classes, labs, reports, audits, and practical research reviews.
Welch and Pooled Methods
Use Welch's method when group spreads or sample sizes differ. It adjusts the degrees of freedom. Use the pooled variance method when equal variance is a reasonable assumption. Both approaches estimate a standard error. Then they convert the observed mean difference into a t statistic.
Input Quality
Good input matters. Raw values should come from independent observations. One person or item should not appear in both groups. The data should be numeric. Extreme outliers can move the mean and standard deviation. Review the sample before trusting the final decision. Missing values should be removed with a clear rule. Keep units the same for both groups. Do not mix percentages, scores, and counts without planning. Always describe the source of each group, because group definitions affect the meaning of every comparison.
Reading the P Value
The p value answers a specific question. It estimates how unusual the observed t statistic would be if the null difference were true. A small p value supports rejection of the null hypothesis. It does not prove a large practical effect. For that reason, the calculator also reports Cohen's d and Hedges' g.
Confidence Interval
The confidence interval is often the most useful output. It shows a range of plausible values for the true mean difference. If a two sided interval excludes the hypothesized difference, the test usually rejects at the matching alpha level. Wider intervals show more uncertainty. Larger samples usually narrow the interval.
Alternative Hypothesis
Choose the alternative hypothesis before reviewing results. A two tailed test looks for any difference. A right tailed test checks whether group one is greater. A left tailed test checks whether group one is smaller. Changing the tail after seeing data weakens the conclusion.
Reporting Results
Interpret results with context. A statistically significant result may be too small to matter. A non significant result may still be important when samples are small. Report sample sizes, means, standard deviations, test type, t value, degrees of freedom, p value, and confidence interval. These details make your result reproducible and easier to review.
FAQs
1. What is an independent t test?
It compares the means of two separate groups. The groups should contain different people, items, or observations. It tests whether the observed mean difference is larger than expected by random sampling error.
2. When should I use Welch's method?
Use Welch's method when sample sizes differ, variances differ, or you are unsure about equal variance. It is often the safer default because it adjusts the degrees of freedom.
3. When is the pooled method acceptable?
The pooled method is acceptable when both groups have similar variances and equal variance is reasonable. It combines variance estimates into one pooled standard deviation.
4. Can I use raw data and summary data?
Yes. Use raw mode for full observations. Use summary mode when you know n, mean, and sample standard deviation for each independent group.
5. What does the p value mean?
The p value shows how unusual the observed result would be if the hypothesized mean difference were true. Smaller p values give stronger evidence against the null hypothesis.
6. What does Cohen's d show?
Cohen's d describes the mean difference in pooled standard deviation units. It helps judge practical size, not just statistical significance.
7. Why does the confidence interval matter?
The confidence interval gives plausible values for the true mean difference. It shows direction, size, and uncertainty in one range.
8. Can the test prove the groups are equal?
No. A non significant result means the calculator did not find strong evidence of a difference. It does not prove the means are identical.