Example Data Table
| Case |
Training Group |
Control Group |
| 1 |
82 |
75 |
| 2 |
86 |
79 |
| 3 |
91 |
83 |
| 4 |
77 |
72 |
| 5 |
88 |
80 |
| 6 |
84 |
78 |
Formula Used
The calculator compares two independent group means. It supports Welch and pooled variance methods.
Welch Test
Standard error = sqrt((s1² / n1) + (s2² / n2))
t = ((mean1 - mean2) - hypothesized difference) / standard error
df = ((s1² / n1 + s2² / n2)²) / (((s1² / n1)² / (n1 - 1)) + ((s2² / n2)² / (n2 - 1)))
Pooled Test
sp² = (((n1 - 1)s1²) + ((n2 - 1)s2²)) / (n1 + n2 - 2)
Standard error = sqrt(sp² × ((1 / n1) + (1 / n2)))
df = n1 + n2 - 2
Effect Size
Cohen's d = (mean1 - mean2) / pooled standard deviation
Hedges' g = Cohen's d × small sample correction
Glass delta = (mean1 - mean2) / group 2 standard deviation
How to Use This Calculator
- Select raw data or summary statistics.
- Choose Welch for unequal variance or pooled for equal variance.
- Select the alternative hypothesis.
- Enter the confidence level and hypothesized mean difference.
- Add group labels for cleaner reports.
- Enter both groups of values or summary fields.
- Press the calculate button.
- Review t, df, p value, interval, and effect sizes.
- Download the CSV or PDF report when needed.
Independent Samples T-Test Guide
An independent samples t-test compares means from two unrelated groups. It is useful when each participant, item, or observation belongs to only one group. The goal is to decide whether the observed mean difference is larger than expected sampling noise. This calculator accepts raw values or summary statistics. Raw mode is helpful for classroom data. Summary mode is useful for reports, studies, and copied software output.
When the test fits
Use this test when the outcome is numeric. The two groups should be independent. Measurements should be taken on a similar scale. Each group should have reasonable variation. Extreme outliers can distort the result. Check the data before trusting any p value. Welch’s option is usually safer when group variances differ. The pooled option is best when equal variance is a justified assumption.
What the result means
The t statistic measures the mean difference in standard error units. A large absolute t value suggests a stronger difference. The p value answers a probability question under the null hypothesis. It does not measure importance by itself. The confidence interval shows a likely range for the true mean difference. If a two-sided interval misses zero, the two-sided test is usually significant at that confidence level.
Why effect size matters
Statistical significance depends on sample size. A tiny difference can become significant with many observations. A useful difference may be missed with small samples. Cohen’s d and Hedges’ g express the gap in standard deviation units. Hedges’ g adds a small sample correction. Read these values with subject knowledge, not fixed labels only.
Good reporting practice
Report group means, standard deviations, sample sizes, t, degrees of freedom, p, confidence interval, method, and effect size. State whether Welch or pooled variance was used. Mention the alternative hypothesis. Keep raw data available when possible. Export the result for records. Use the example table to understand expected inputs before entering real study data. Always connect the number result with the study design.
Common limits
The test is not a proof of causation. Random assignment, sampling quality, and measurement rules still matter. Non-normal data can be acceptable with larger samples. With small samples, inspect plots and consider a nonparametric alternative when assumptions fail.
FAQs
What is an independent samples t-test?
It compares the means of two separate groups. Each observation belongs to only one group. The test checks whether the observed mean difference is unlikely under the null hypothesis.
When should I use Welch’s test?
Use Welch’s test when the groups may have unequal variances. It is also a safe default for many real datasets because it adjusts the degrees of freedom.
When should I use the pooled test?
Use the pooled test when equal variance is reasonable. This assumption should come from study design, past evidence, or variance checks. Do not choose it only because it gives a smaller p value.
What does the p value mean?
The p value shows how unusual the sample result is under the null hypothesis. A small value suggests stronger evidence against that null. It does not prove practical importance.
What does Cohen’s d show?
Cohen’s d expresses the mean difference in pooled standard deviation units. It helps judge practical size. Interpret it with the field, measurement scale, and study context.
Can I enter raw data?
Yes. Paste values separated by commas, spaces, semicolons, or line breaks. The calculator will compute sample size, mean, and sample standard deviation automatically.
Can I use summary statistics?
Yes. Select summary statistics mode. Then enter sample size, mean, and sample standard deviation for both groups. This is useful when raw data is unavailable.
What should I report from the test?
Report group sample sizes, means, standard deviations, t statistic, degrees of freedom, p value, confidence interval, test method, alternative hypothesis, and effect size.