Example Data Table
| Scenario |
Group 1 Mean |
Group 1 SD |
Group 1 n |
Group 2 Mean |
Group 2 SD |
Group 2 n |
Suggested Method |
| Training score comparison |
82.4 |
9.6 |
35 |
76.8 |
10.2 |
32 |
Welch |
| Machine output check |
51.2 |
4.5 |
25 |
48.9 |
4.1 |
25 |
Pooled |
| Website test conversion score |
14.8 |
3.2 |
60 |
13.1 |
4.8 |
55 |
Welch |
Formula Used
Observed difference: d = x̄1 - x̄2
Test statistic: t = (d - d0) / SE
Welch standard error: SE = √(s1² / n1 + s2² / n2)
Pooled standard deviation: sp = √(((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2))
Pooled standard error: SE = sp × √(1 / n1 + 1 / n2)
Confidence interval: (x̄1 - x̄2) ± tcritical × SE
Cohen d: d = (x̄1 - x̄2) / sp
Hedges g: g = Cohen d × correction factor
How to Use This Calculator
Select summary statistics or raw data. Choose Welch when variances may differ. Choose pooled only when equal variance is reasonable. Enter both group labels, sample sizes, means, and standard deviations. For raw data, paste values into each box. Select the tail, confidence level, alpha, and decimal places. Press Submit. Review the t statistic, degrees of freedom, p value, confidence interval, effect size, and decision.
What This Calculator Does
An independent samples t test compares two separate groups. It checks whether their means differ more than expected from random sampling error. This calculator supports summary statistics and raw observations. It also offers Welch and pooled variance methods. That makes it useful for research, classes, quality checks, marketing tests, and health studies.
Why Independent Samples Matter
Independent groups do not share the same members. One group might receive a treatment. Another group might be a control. Each person or item belongs to only one group. This separation is important. It keeps the test model valid. Paired data needs another test.
Choosing Welch or Pooled
Welch's method is safer when group variances differ. It adjusts the standard error and degrees of freedom. The pooled method assumes equal variances. It combines both sample variances into one estimate. Use pooled results only when that assumption is reasonable. For most real data, Welch is often a careful default.
Reading the Result
The t statistic measures the standardized difference. A larger absolute value means stronger evidence against the null difference. The p value gives the probability of seeing such evidence, assuming the null is true. A small p value suggests the group means are unlikely to be equal under that assumption.
Confidence Intervals and Effect Size
The confidence interval shows a likely range for the mean difference. If it excludes zero, the two sided test is usually significant at the related level. Cohen's d describes the difference in standard deviation units. Hedges g applies a small sample correction. These values help explain practical importance, not just statistical significance.
Good Data Practice
Check sample sizes before trusting results. Inspect unusual values. Confirm that measurements are independent. Very skewed data can affect small samples. Larger samples make the t test more stable. Report the method, tail choice, confidence level, t value, degrees of freedom, p value, interval, and effect size. Clear reporting helps readers understand both the calculation and the decision. This page also helps compare classroom examples. Try the example table first. Then replace values with your own data. Save exports for assignments or records. Recheck units and labels before sharing. Keep raw input clean. Use commas, spaces, or lines between observations.
FAQs
What is an independent samples t test?
It compares the means of two separate groups. Each observation belongs to only one group. It tests whether the observed mean difference is larger than expected from sampling variation.
When should I use Welch's method?
Use Welch's method when group variances or sample sizes differ. It is often a safer default because it does not require the equal variance assumption.
When is the pooled method suitable?
The pooled method is suitable when both groups have reasonably similar variances. It combines variances into one estimate and uses n1 plus n2 minus 2 degrees of freedom.
Can I paste raw data?
Yes. Select raw data mode. Paste each group into its own box. Separate values with commas, spaces, semicolons, or new lines.
What does the p value mean?
The p value measures evidence against the null hypothesis. A smaller p value means the observed result is less likely if the null difference is true.
What is the null mean difference?
It is the difference assumed by the null hypothesis. Most tests use zero. You can enter another value for nonzero difference testing.
What does Cohen d show?
Cohen d shows the mean difference in pooled standard deviation units. It helps describe practical size, not only statistical significance.
Why download CSV or PDF?
CSV is useful for spreadsheets and records. PDF is useful for reports, homework, client notes, or saving a readable summary of the test.