U1 U2 Hypothesis Test Calculator

Enter Two Group Statistics

Sample mean 1

x̄1 for the first group.

Spread 1

Use sample or population spread based on method.

Sample size 1

Must be at least 2.

Sample mean 2

x̄2 for the second group.

Spread 2

Use the same spread type as group one.

Sample size 2

Must be at least 2.

Claimed difference Δ0

Usually 0 for equal means.

Alpha level

Common choices: 0.10, 0.05, 0.01.

Test method

Welch is safest for unequal variances.

Alternative hypothesis

Choose two tailed unless direction is planned first.

Formula used

Observed difference: d = x̄1 - x̄2

Test statistic: statistic = (d - Δ0) / SE

Welch SE: SE = √(s1² / n1 + s2² / n2)

Pooled SE: SE = sp √(1 / n1 + 1 / n2)

Confidence interval: d ± critical value × SE

Welch degrees of freedom use the Satterthwaite equation. The z method uses the normal distribution when population spreads are known.

How to use this calculator

Enter the mean, spread, and sample size for both groups.
Enter the claimed value for μ1 - μ2. Use zero for equality.
Select Welch, pooled, or z method based on your assumptions.
Choose the alternative hypothesis and alpha level.
Press calculate. Review the p value, decision, interval, and graph.
Use the CSV or PDF buttons to save the output.

Example data table

Scenario	x̄1	s1	n1	x̄2	s2	n2	Suggested method
Training score comparison	84.2	12.4	45	79.1	10.8	42	Welch t test
Machine output test	102.8	8.5	60	100.1	7.9	55	Welch t test
Known population spread	51.3	6.0	80	49.8	5.5	78	Two mean z test

Understanding the U1 U2 Hypothesis Test

Purpose

A U1 U2 hypothesis test compares two population means. It asks whether the difference between mean one and mean two is likely to equal a claimed value. The claimed value is often zero. That means the tool checks whether two groups appear equal on average.

Choosing a method

The calculator supports two common paths. Use the z option when population standard deviations are known. Use the t option when you only know sample standard deviations. The Welch t method is usually safer because it does not assume equal variances. The pooled t method is useful when equal variances are reasonable.

Inputs and tails

You enter each sample size, sample mean, and spread. Then choose the alternative hypothesis. A two tailed test checks for any difference. A right tailed test checks whether U1 minus U2 is greater than the claim. A left tailed test checks whether it is smaller.

Statistic and p value

The test statistic measures how far the observed difference is from the claimed difference. It uses standard error as the measuring unit. A larger absolute statistic gives stronger evidence against the null hypothesis. The p value converts that evidence into a probability scale.

Decision rule

The alpha level is your decision line. Common values are 0.10, 0.05, and 0.01. If the p value is less than or equal to alpha, reject the null hypothesis. Otherwise, fail to reject it. This does not prove equality. It only means the sample evidence was not strong enough.

Interval and reporting

The confidence interval shows a practical range for U1 minus U2. If a two tailed test uses alpha 0.05, the related interval is usually 95 percent. A range that excludes the claimed difference supports rejection.

Practical checks

The graph gives a quick visual check. It marks the test statistic and critical areas. The table helps compare example scenarios. Download options make reports easier. Always confirm that samples are independent, measurements are valid, and extreme outliers are reviewed before trusting results. Use results as decision support, not as automatic truth. Study design matters. Random sampling improves trust. Balanced group sizes improve precision. When assumptions look weak, try a nonparametric method or collect more data before making a final choice during early research planning and review.

FAQs

1. What does U1 U2 mean?

U1 and U2 represent two population means. The test studies the difference between them. Most users test whether μ1 - μ2 equals zero, which means both populations have the same average.

2. When should I use Welch t test?

Use Welch t test when group variances may differ. It is a strong default for independent samples because it does not require equal variance. It also adjusts degrees of freedom automatically.

3. When should I use the pooled t test?

Use the pooled t test only when equal variance is reasonable. It combines both sample spreads into one pooled estimate. Avoid it when spreads or sample sizes differ strongly.

4. When is the z test correct?

The z test is correct when population standard deviations are known. This is less common in practical work. If you only have sample standard deviations, use a t based method.

5. What is the null difference?

The null difference is the claimed value of μ1 - μ2. It is usually zero. You can enter another value when testing a planned margin or expected gap.

6. What does p value mean?

The p value measures how unusual the sample result is if the null hypothesis is true. A smaller p value gives stronger evidence against the null claim.

7. Does failing to reject prove equality?

No. Failing to reject means evidence was not strong enough. It does not prove both means are equal. More data or an equivalence test may be needed.

8. Why include effect size?

Effect size shows practical difference size. A result can be statistically significant but small. Cohen d and Hedges g help users judge real-world importance.