90 Confidence Interval for μ1 − μ2 Calculator

Calculator Inputs

Use this tool to estimate a confidence interval for the population mean difference μ1 − μ2. You can enter summary values or paste raw sample data.

Example Data Table

Formula Used

The basic confidence interval for the difference between two population means is:

(x̄1 − x̄2) ± critical value × standard error

For Welch’s method, the standard error is: SE = √(s1² / n1 + s2² / n2)

Welch degrees of freedom are: df = (s1²/n1 + s2²/n2)² / [(s1²/n1)²/(n1−1) + (s2²/n2)²/(n2−1)]

For equal variances, the pooled standard deviation is: sp = √(((n1−1)s1² + (n2−1)s2²) / (n1+n2−2))

Then: SE = sp × √(1/n1 + 1/n2)

How to Use This Calculator

1. Select summary statistics or raw sample data.
2. Choose Welch, pooled, known sigma, or large-sample method.
3. Enter means, standard deviations, and sample sizes.
4. Keep the confidence level at 90 for a 90 percent interval.
5. Click calculate to see the interval above the form.
6. Download the result as a CSV or PDF file.

Understanding the 90 Percent Interval

What the Calculator Measures

This calculator estimates the difference between two population means. The target is μ1 − μ2, not a single group average. That difference is useful when two groups, treatments, machines, classes, or time periods must be compared. A positive result means the first population mean is estimated to be higher. A negative result means the second population mean is estimated to be higher.

Why the 90 Percent Level Matters

A 90 percent confidence interval is narrower than a 95 percent interval. It gives a more compact range, but it accepts more uncertainty. In repeated sampling, about 90 percent of correctly built intervals would contain the true population mean difference. The interval does not say there is a 90 percent chance that one fixed interval contains the value.

Choosing the Right Method

Welch’s method is often the safest default. It does not require equal population variances. The pooled method is best when the two groups have similar variability and the equal variance assumption is reasonable. The known sigma method is used when population standard deviations are known. This is uncommon in real studies. The large-sample option uses a normal critical value and is useful when both samples are large.

Reading the Output

The calculator reports the mean difference, standard error, critical value, margin of error, lower limit, and upper limit. If zero is inside the interval, the data do not show a clear difference at the selected confidence level. If zero is outside the interval, the estimated difference is stronger. Always combine the interval with study design, sample quality, and subject knowledge.

Good Data Practice

Independent samples are usually required. Each group should represent its population fairly. Extreme outliers can distort means and standard deviations. Raw data mode helps check sample summaries quickly. For small samples, inspect the data shape before trusting the interval. A calculator gives the arithmetic, but sound interpretation still needs statistical judgment.

FAQs