U1 - U2 Confidence Interval Calculator

Main interval:

(x̄1 - x̄2) ± Critical Value × SE

Welch standard error:

SE = sqrt(s1² / n1 + s2² / n2)

Welch degrees of freedom:

df = (s1²/n1 + s2²/n2)² / [(s1²/n1)²/(n1-1) + (s2²/n2)²/(n2-1)]

Pooled standard deviation:

sp = sqrt(((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2))

Pooled standard error:

SE = sp × sqrt(1/n1 + 1/n2)

Known sigma standard error:

SE = sqrt(σ1² / n1 + σ2² / n2)

1. Select summary statistics or raw data mode.
2. Choose Welch, pooled, or known sigma method.
3. Enter sample sizes, means, and standard deviations.
4. Use raw data mode when original observations are available.
5. Enter the confidence level, such as 90, 95, or 99.
6. Press the calculate button.
7. Review the interval, margin, standard error, and chart.
8. Download the CSV or PDF for reporting.

What the Interval Means

A U1 minus U2 confidence interval estimates the difference between two population means. It is useful when two independent groups are compared. The groups may be treatments, classrooms, machines, stores, or time periods. The interval gives a likely range for the true mean difference.

Choosing a Method

This calculator supports three common approaches. Welch method is the safest default. It allows unequal sample variances. Pooled method assumes both populations share the same variance. Z method is useful when population standard deviations are known. Each method returns the estimated difference, standard error, critical value, margin of error, limits, and degrees of freedom when needed.

Reading the Sign

A positive interval means group one is likely higher. A negative interval means group two is likely higher. An interval that crosses zero means the observed difference may not be statistically clear at the chosen confidence level. This does not prove equality. It only says the data do not separate the means strongly enough.

Precision and Sample Size

The confidence level controls the width. A ninety nine percent interval is wider than a ninety five percent interval. Larger samples usually narrow the interval. Smaller standard deviations also narrow it. Outliers, wide variation, and small samples make results less precise.

Summary Data and Raw Data

Use summary statistics when you already know the means and standard deviations. Use raw values when you have original observations. Raw mode calculates the sample size, mean, and sample standard deviation automatically. Clean the data first. Remove text labels and use commas, spaces, or new lines between values.

Reporting the Result

For reporting, include the method, confidence level, mean difference, interval limits, and sample sizes. Add units when the variable has units. Always describe which group is U1 and which group is U2. That detail controls the sign of the answer. A clear label prevents reversed interpretations.

Good Practice

Confidence intervals are practical decision tools. They show both size and uncertainty. They are often more informative than a simple yes or no test.

Assumption Checks

Before using any interval, check basic assumptions. The groups should be independent. Measurements should represent the target populations. Very skewed samples may need larger sample sizes. When in doubt, compare raw data with the chart and review context carefully. Document choices so readers can reproduce the result.