Rank Sum Calculator

Calculator Inputs

Enter independent samples using spaces, commas, semicolons, or line breaks between numbers.

Sample A Label

Sample B Label

Alternative Hypothesis

Significance Level (α)

Displayed Decimals

Apply continuity correction

Sample A Values

Use independent observations only.

Sample B Values

Ties are handled with average ranks.

Example Data Table

You can load these values directly with the example button above.

Observation	Method A	Method B
1	18	11
2	22	14
3	25	15
4	27	18
5	31	20
6	33	21

Formula Used

The calculator follows the Wilcoxon rank sum approach, commonly reported through the Mann–Whitney U statistic.

1) Rank every combined observation

Pool both samples, sort the values, and assign ranks from smallest to largest. Tied values receive the average of their occupied ranks.

2) Compute the rank sums

R₁ = sum of ranks for Sample A
R₂ = sum of ranks for Sample B

3) Convert rank sums to U statistics

U₁ = R₁ - n₁(n₁ + 1) / 2
U₂ = R₂ - n₂(n₂ + 1) / 2

4) Expected value and tie-corrected variance

E(U) = n₁n₂ / 2

Var(U) = (n₁n₂ / 12) × [ (N + 1) - Σ(t³ - t) / (N(N - 1)) ]

Here, t is the size of each tie group and N = n₁ + n₂.

5) Z statistic with optional continuity correction

z = (U - E(U) ± 0.5) / √Var(U)

The p value is obtained from the standard normal distribution. This is an approximation and is strongest for moderate or large sample sizes.

6) Effect size

Probability of superiority = U₁ / (n₁n₂)
Rank biserial correlation = 2 × [U₁ / (n₁n₂)] - 1

How to Use This Calculator

Enter a label for each sample so the report reads clearly.
Paste the two independent samples into their text boxes.
Choose a two-sided or one-sided alternative hypothesis.
Set the significance level and your preferred decimal display.
Turn continuity correction on when you want a normal approximation adjustment.
Click Calculate Rank Sum to place the result summary above the form.
Review rank sums, U statistics, p value, effect size, and the detailed rank table.
Use the CSV or PDF buttons to export the current result set.

FAQs

1) What does this calculator test?

It compares two independent samples using their ranks rather than raw values. This helps when data are skewed, ordinal, or less suitable for a standard two-sample t test.

2) Is rank sum the same as Mann–Whitney U?

They are two forms of the same underlying test. Rank sums are converted into U statistics, and both lead to the same inferential conclusion.

3) Can I include tied values?

Yes. Tied observations receive averaged ranks, and the variance is adjusted with a tie correction factor. That keeps the approximation more realistic.

4) When should I use a one-sided alternative?

Use a one-sided test only when your research question was directional before seeing the data. Otherwise, a two-sided test is safer and more defensible.

5) What does the rank biserial correlation show?

It gives an effect-size view of sample separation. Values near zero suggest little difference, while values farther from zero indicate stronger directional separation.

6) Does this calculator use exact p values?

No. It uses the normal approximation with tie correction. That works well for many practical datasets, especially when samples are not extremely small.

7) What if every value is tied?

If all observations are effectively identical, the variance can collapse to zero. In that case, the calculator reports the issue instead of forcing a misleading z statistic.

8) Can I use this for paired data?

No. Paired observations require a signed-rank method instead. This calculator is designed only for two independent samples.