Formula Used
Mean difference: d = x̄1 - x̄2
Welch standard error: SE = sqrt(s12/n1 + s22/n2)
Pooled standard error: SE = sp sqrt(1/n1 + 1/n2)
Confidence interval: d ± tcritical × SE
Test statistic: t = (d - d0) / SE
Welch degrees of freedom use the Satterthwaite approximation. Pooled degrees of freedom equal n1 + n2 - 2.
How To Use This Calculator
- Enter group labels so the report is easy to read.
- Use summary statistics, or check raw data and paste values.
- Select Welch when variances or sample sizes differ.
- Choose pooled only when equal variance is justified.
- Set the confidence level and hypothesized difference.
- Press Calculate, then review the interval and p value.
- Use CSV or PDF buttons to download the same results.
Example Data Table
| Study Item |
Group 1 |
Group 2 |
Suggested Setting |
| Sample size |
16 |
14 |
Welch |
| Mean score |
82.4 |
76.1 |
95% confidence |
| Standard deviation |
9.6 |
8.2 |
Two-sided test |
About This Independent T Test
This calculator compares two separate groups and estimates the difference between their means. It supports summary statistics and raw observations. You can choose Welch testing when variances may differ. You can choose pooled testing when equal variance is a reasonable study assumption. The page also reports a confidence interval, t statistic, degrees of freedom, p value, standard error, margin of error, and effect size.
Why Confidence Intervals Matter
A confidence interval gives a practical range for the true mean difference. A narrow interval suggests precise estimation. A wide interval warns that the sample may be limited, noisy, or unbalanced. The interval is often more useful than a single p value because it shows direction and size. If the interval crosses zero, the observed difference may not be statistically clear at that confidence level.
Welch Versus Pooled Choice
Welch testing is usually the safer default. It adjusts the degrees of freedom when group spreads or sample sizes are unequal. The pooled method combines both standard deviations into one estimate. It works best when the two populations have similar variance and the study design supports that assumption. When unsure, select Welch and explain the reason in your report.
Interpreting The Results
The mean difference is Group 1 minus Group 2. A positive value means Group 1 has the higher sample mean. A negative value means Group 2 has the higher sample mean. The t statistic measures how many standard errors separate the observed difference from the hypothesized difference. The p value depends on your selected alternative hypothesis.
Good Data Practice
Use independent observations only. Do not use this tool for paired before and after data. Check for obvious data entry mistakes. Review outliers, study design, and measurement quality before presenting results. Statistical output should support judgment, not replace it. Use results carefully, and support decisions with study context.
Report Ready Outputs
The download buttons keep a clear record of each analysis. The CSV file helps spreadsheet review. The PDF file helps share a compact summary. Save the settings with the results, so another reader can repeat the same test. Include method labels beside exported values for quick review. Clear notes make statistical reporting easier for future readers.
FAQs
1. What does this calculator estimate?
It estimates the confidence interval for the difference between two independent group means. It also reports the t statistic, degrees of freedom, p value, standard error, and effect size.
2. When should I use Welch testing?
Use Welch testing when sample sizes differ, standard deviations differ, or equal variance is not strongly supported. It is often the safer independent samples option.
3. When is the pooled method suitable?
The pooled method is suitable when both groups appear to come from populations with similar variance. This assumption should come from study knowledge and data review.
4. What does the confidence interval show?
It shows a likely range for the true mean difference. If it excludes zero, the difference is statistically clear at the selected confidence level.
5. Can I use raw data?
Yes. Check the raw data option and paste numbers for both groups. The calculator will compute sample size, mean, and sample standard deviation automatically.
6. What is the hypothesized difference?
It is the comparison value used in the t statistic. Most tests use zero, meaning the null hypothesis assumes no mean difference.
7. What does Cohen d mean?
Cohen d standardizes the mean difference by the pooled standard deviation. It helps describe practical size, not just statistical significance.
8. Can this handle paired data?
No. This tool is for independent groups only. Use a paired t test calculator for before and after measurements on the same subjects.