Pooled T Interval Calculator

Calculator Form

Example Data Table

Formula Used

The calculator estimates a confidence interval for the difference between two population means. It uses a pooled variance model. This model assumes both populations have equal variance.

Pooled variance: Sp² = [((n1 - 1)s1²) + ((n2 - 1)s2²)] / (n1 + n2 - 2)

Standard error: SE = Sp × sqrt((1 / n1) + (1 / n2))

Interval: (x̄1 - x̄2) ± t critical × SE

Degrees of freedom: df = n1 + n2 - 2

How to Use This Calculator

Enter the mean for each sample. Add each sample standard deviation. Then enter both sample sizes. Choose the confidence level. Use 95 for a common interval. Use 90 or 99 when your study needs a wider or narrower risk choice. Press calculate. The result appears above the form and below the page header. Use CSV for spreadsheet work. Use PDF for a simple report.

Guide to Pooled T Intervals

What the Calculator Does

A pooled t interval compares two independent sample means. It estimates the likely range for the true difference between two population means. This tool is useful when both groups measure the same type of outcome. It is often used in statistics, testing, quality checks, education, and controlled experiments.

Why Pooled Variance Matters

The pooled method combines both sample variances into one shared estimate. This approach is suitable when the two population variances can be treated as equal. A shared variance can improve precision when that assumption is reasonable. It also gives one clear standard error for the mean difference.

Inputs You Need

You need two sample means, two sample standard deviations, and two sample sizes. The confidence level controls the interval width. A higher confidence level usually makes the interval wider. A lower confidence level usually makes it narrower. The decimal field controls only the final display.

How to Read the Interval

The interval shows lower and upper limits for group one minus group two. If the entire interval is above zero, group one is likely higher. If it is below zero, group two is likely higher. If it crosses zero, the data does not show a clear difference at that confidence level.

Best Use Cases

Use this calculator for balanced studies, classroom examples, conversion comparisons, product tests, and process checks. It works best when samples are independent. It also works best when distributions are roughly normal or sample sizes are large enough for stable estimates.

Important Notes

The calculator does not prove equality or cause. It gives an estimated range from sample data. Always check study design, data quality, variance similarity, and outliers before using the final result. For strongly unequal variances, a Welch interval may be better.

FAQs

1. What is a pooled t interval?

It is a confidence interval for the difference between two independent population means. It uses a shared variance estimate from both samples.

2. When should I use pooled variance?

Use it when both populations can reasonably be assumed to have equal variances. The samples should also be independent.

3. What does the interval mean?

It gives a likely range for the true mean difference. The difference is calculated as sample one mean minus sample two mean.

4. What if the interval includes zero?

If zero is inside the interval, the data does not show a clear mean difference at the selected confidence level.

5. Is a 99% interval always better?

No. A 99% interval gives more confidence but is wider. Wider intervals may be less precise for practical decisions.

6. Can sample sizes be different?

Yes. The pooled t interval can handle different sample sizes. Each sample size must be at least two.

7. What is the pooled standard deviation?

It is a combined estimate of standard deviation. It uses both sample variances and their degrees of freedom.

8. Should I use Welch instead?

Use Welch when the two variances look very different. Welch does not require the equal variance assumption.