Calculator Inputs
Example Data Table
This example uses independent samples and the Welch method.
| Group | Sample Mean | Sample SD | Sample Size | Role |
|---|---|---|---|---|
| Group 1 | 82.4 | 9.8 | 36 | First population mean |
| Group 2 | 78.1 | 10.5 | 34 | Second population mean |
| Difference | 4.3 | Calculated SE | Welch df | Interval estimate |
Formula Used
The calculator estimates the interval for the difference between two population means, written as μ1 − μ2.
Welch is usually safest when sample variances differ. Use pooled only when equal variance is a justified study assumption.
How to Use This Calculator
- Select independent samples or paired differences.
- Choose Welch, pooled, or known sigma for independent samples.
- Enter means, standard deviations, and sample sizes.
- Keep confidence at 95 percent, or enter another level.
- Set the null difference, usually zero.
- Click calculate and read the interval above the form.
- Use CSV or PDF buttons to save the result.
Understanding the 95 Percent Interval
What the interval means
A confidence interval gives a plausible range for μ1 − μ2. It starts with the observed sample difference. Then it adds and subtracts a margin of error. The margin uses sampling variation, sample size, and a critical value. A 95 percent interval uses the middle 95 percent of the sampling distribution.
Choosing the right method
Welch is the default choice for two independent samples. It does not require equal population variances. That makes it useful for many real studies. Pooled intervals assume the two populations share one common variance. Use that method only when the assumption is defensible. Known sigma is rare. It applies when population standard deviations are truly known.
Paired studies
Paired data need a different setup. Examples include before and after scores, twins, matched patients, or repeated measurements. For paired work, first compute each pair difference. Then use the mean and standard deviation of those differences. This removes much of the between-person variation.
Reading the result
If the full interval is above zero, group one is estimated higher. If the full interval is below zero, group two is estimated higher. If the interval crosses zero, the observed data are not strong enough to show a clear difference at that level. The null check follows the same idea. You can enter a nonzero benchmark when the practical question needs one.
Good practice
Always check the study design before entering numbers. Independent samples must not contain matched observations. Paired data must keep every difference in the same direction. Larger samples usually shrink the interval. Larger variation usually widens it. Report the method, confidence level, point estimate, margin, and interval bounds. Also explain the units. A statistical difference may still be too small to matter in practice.
FAQs
1. What is μ1 − μ2?
It is the difference between two population means. The calculator estimates that unknown difference using sample data from two groups or paired observations.
2. When should I use Welch?
Use Welch for most independent sample problems. It handles unequal sample variances and unequal sample sizes better than the pooled method.
3. When is the pooled method valid?
Use pooled only when equal population variances are reasonable. This should come from design knowledge, prior research, or a strong assumption.
4. What does a 95 percent interval mean?
It means the method captures the true mean difference in about 95 percent of repeated random samples under model assumptions.
5. What if the interval includes zero?
Including zero means the data do not show a clear difference between population means at the chosen confidence level.
6. Can I use raw data?
This page uses summary statistics. Calculate each group mean, standard deviation, and sample size first, then enter those values.
7. What is standard error?
Standard error measures expected sampling variation in the estimated difference. Smaller standard errors usually create narrower confidence intervals.
8. Why enter a null difference?
The null difference is a comparison value. Zero checks equality. Other values can test a practical benchmark or required improvement.