Calculator
Example Data Table
| Group | Sample Size | Mean | Standard Deviation | Purpose |
|---|---|---|---|---|
| Training A | 12 | 82.4 | 7.8 | Compare first independent group |
| Training B | 10 | 76.1 | 6.9 | Compare second independent group |
| Settings | Both groups | Null difference 0 | Alpha 0.05 | Build test and graph |
Formula Used
Mean difference: d = x̄1 - x̄2
Welch standard error: SE = sqrt(s12 / n1 + s22 / n2)
Pooled variance: sp2 = ((n1 - 1)s12 + (n2 - 1)s22) / (n1 + n2 - 2)
Pooled standard error: SE = sqrt(sp2(1 / n1 + 1 / n2))
Test statistic: t = (d - d0) / SE
Confidence interval: d ± tcritical × SE
Effect size: Cohen d = d / sp. Hedges g applies a small sample correction.
How to Use This Calculator
Choose summary statistics or raw sample values. Enter both independent groups. Select Welch when variances may differ. Select pooled only when equal variance is reasonable. Choose the alternative hypothesis, confidence level, alpha, and decimal places. Press Calculate. Review the result above the form. Use CSV or PDF to save the report.
Two Sample T Test Graph Calculator Guide
A two sample t test compares the average of two independent groups. It asks whether the observed difference is larger than random sampling noise. This calculator supports summary statistics and raw values. It also draws a t curve, so the test statistic can be viewed beside the rejection region.
What the Test Measures
The main result is the t statistic. It scales the mean difference by its standard error. A large absolute t value means the groups are far apart relative to their variation. The p value shows how unusual that result would be when the null difference is true.
Welch and Pooled Choices
Welch’s method is usually safer. It does not assume equal population variances. The pooled method can be useful when the two groups have similar spread and the equal variance assumption is defensible. The calculator lets you switch methods before exporting the report.
Graph and Confidence Interval
The graph displays the reference t distribution, the observed statistic, and critical cut points. The confidence interval estimates a range for the true difference in means. When a two sided interval excludes the null difference, it agrees with a significant two sided test at the matching level.
Practical Interpretation
Statistical significance is not the full story. Effect size helps describe practical importance. Cohen’s d and Hedges g summarize the difference in standard deviation units. Always combine these values with subject knowledge, sample design, and measurement quality.
Good Data Practice
Use independent observations. Check for extreme outliers. Review whether the data are roughly continuous and measured consistently. Larger samples make the test more stable. Small samples need extra care, because one unusual value can change both spread and significance.
Reporting Results
A clear report should include group sizes, means, standard deviations, test type, t statistic, degrees of freedom, p value, confidence interval, and effect size. The CSV and PDF buttons help store the same results for later review, coursework, audits, or technical notes.
For best results, record units, sampling dates, and data sources. Avoid rounding inputs too early. Rounded means or deviations can slightly shift the final statistic, especially when samples are small or close to the decision boundary during final review checks too.
FAQs
What is a two sample t test?
It is a statistical test for comparing the means of two independent groups. It estimates whether the observed difference is large enough to be unlikely under a chosen null difference.
When should I use Welch’s test?
Use Welch’s test when group variances may differ or sample sizes are unequal. It is often the safer default because it does not require the equal variance assumption.
When is the pooled test suitable?
The pooled test is suitable when both groups are independent and population variances can reasonably be treated as equal. If that assumption is doubtful, use Welch’s method instead.
What does the p value mean?
The p value measures how unusual the test statistic is under the null hypothesis. A smaller p value gives stronger evidence against the null difference.
What does the graph show?
The graph shows the t distribution, observed t statistic, and critical cut points. It helps visualize whether the statistic falls in a rejection region.
Why are degrees of freedom different in Welch’s test?
Welch’s test estimates degrees of freedom from both sample variances and sample sizes. The result is often decimal because unequal spread changes the reference distribution.
What is Cohen’s d?
Cohen’s d expresses the mean difference in pooled standard deviation units. It helps describe practical effect size, not just statistical significance.
Can I use raw data instead of summaries?
Yes. Select raw sample mode and enter values separated by commas, spaces, semicolons, or new lines. The calculator will compute size, mean, and standard deviation.