Calculator Inputs
Example Data Table
| Case | Mean 1 | Mean 2 | SD 1 | SD 2 | N 1 | N 2 | Method |
|---|---|---|---|---|---|---|---|
| Training scores | 82 | 77 | 12 | 10 | 35 | 32 | Welch |
| Machine output | 104.6 | 101.2 | 8.4 | 8.1 | 50 | 50 | Pooled |
| Known process spread | 15.8 | 14.9 | 2.1 | 2.4 | 80 | 75 | Known deviation |
Formula Used
Observed difference: d = x̄1 - x̄2
Adjusted difference: d0 = (x̄1 - x̄2) - Δ0
Welch standard error: SE = sqrt(s12 / n1 + s22 / n2)
Pooled standard error: SE = sp sqrt(1 / n1 + 1 / n2)
Test statistic: t or z = d0 / SE
Welch degrees of freedom: df = (a + b)2 / [a2 / (n1 - 1) + b2 / (n2 - 1)], where a = s12 / n1 and b = s22 / n2.
How to Use This Calculator
- Enter both sample means, standard deviations, and sample sizes.
- Enter the null difference. Use zero when testing equal means.
- Choose Welch unless equal variance is justified.
- Select a two sided, greater, or less alternative.
- Set alpha and confidence level, then press Calculate.
- Use CSV or PDF export after a result appears.
Statistical Meaning
A test statistic for two means shows how far two sample averages are from a claimed difference. It divides the adjusted mean gap by its standard error. A large absolute value means the samples are not close to the null claim. The calculator supports equal variance, unequal variance, and known deviation cases. It also supports one sided and two sided alternatives.
Why This Calculator Helps
Manual work can cause small rounding errors. This tool keeps each step visible. It reports the observed difference, standard error, test statistic, degrees of freedom, p-value, decision, confidence interval, and effect size. These results help students, analysts, and researchers compare groups with more control.
Choosing A Method
Use the Welch option when sample spreads or sample sizes differ. It is a safe default for many independent samples. Use the pooled option only when equal variance is reasonable. That method combines both sample deviations into one shared estimate. Use the known deviation option when population standard deviations are truly known. That situation is less common in practice.
Reading The Output
The p-value measures how unusual the observed evidence is under the null claim. A small p-value means the sample result would be rare if the claim were correct. The decision compares the p-value with alpha. The confidence interval gives a practical range for the true mean difference. If a two sided interval excludes the null difference, the related two sided test usually rejects.
Good Data Practice
Enter sample sizes greater than one for sample based tests. Use positive standard deviations. Keep units the same for both groups. The calculator assumes independent samples. It does not replace study design checks, outlier review, or subject knowledge. Always confirm whether the samples were selected fairly. Also check whether the response variable is measured consistently.
Using Results Responsibly
A significant result is not automatically important. Review the mean difference and Cohen's d. A small effect may matter in large systems. A large effect may be uncertain with small samples. Report the method, alternative, alpha, p-value, interval, and sample summary. Clear reporting makes the comparison easier to audit and repeat. Use judgment before making decisions. Document assumptions clearly when sharing results with other people.
FAQs
What does this calculator test?
It tests whether two independent sample means differ from a selected null difference. Most users enter zero as the null difference, which checks whether both population means are equal.
Should I choose Welch or pooled?
Choose Welch when sample sizes or standard deviations differ. Choose pooled only when equal variance is reasonable. Welch is often the safer default for independent samples.
What is the null difference?
The null difference is the mean gap assumed by the null hypothesis. Use zero for equal means. Use another value when testing a specific expected difference.
What does the p-value mean?
The p-value shows how unusual your result is if the null hypothesis is true. Smaller values give stronger evidence against the null claim.
What is alpha?
Alpha is the rejection cutoff. A common value is 0.05. If the p-value is less than or equal to alpha, the calculator rejects the null hypothesis.
Can I use known standard deviations?
Yes. Select the known deviation method when the entered deviations are population standard deviations. If they are sample deviations, use Welch or pooled instead.
What does Cohen d show?
Cohen d shows effect size in pooled standard deviation units. It helps judge practical importance, not only statistical significance.
Does this work for paired data?
No. Paired data needs a paired mean difference test. This calculator assumes two independent groups with separate standard deviations and sample sizes.