Nonparametric Tests

Q: What is the difference between Mann-Whitney and Wilcoxon rank-sum?

They are the same test, described with different statistics. Mann-Whitney is often reported with U, and Wilcoxon rank-sum with W, which is the sum of the ranks of one group. Minitab calls it Mann-Whitney and reports W. The Wilcoxon signed-rank test is a different test, for paired data.

Q: Do nonparametric tests compare medians?

Only if the groups have a similar shape. In general they test whether values in one group tend to be larger than in the other. With similarly shaped distributions, that is the same as a difference in medians. Plot the groups first.

Q: Are nonparametric tests less powerful?

Slightly, when the data really are normal: the Mann-Whitney test needs about 5% more data than the t-test for the same power. When the data are skewed or have outliers, they can be more powerful than the t-test.

Q: What sample size do I need?

The same planning logic applies: choose the smallest difference that matters and size the sample for it. A rough approach is to calculate the sample for the t-test and add about 15%. Very small groups cannot give a significant result however large the difference.

Q: What if I have only a few values?

Use exact p-values, which Minitab gives for small samples with no ties. Remember that with groups of 3 and 3, the smallest possible p-value is 0.10, so a significant result is impossible.

Q: When should I use the sign test?

The sign test counts only which way each difference goes and ignores their size. It needs the fewest assumptions but has the least power. Use it when the data are only ordinal or the size of the differences is not meaningful.

Question it answers: Do groups differ, without assuming normal data?
Data needed: Continuous or ordinal data; independent observations
Method: Replace values with ranks, then test the ranks
Key output: Rank-sum statistic, p-value, difference in medians with an interval
Use when: Skewed or small samples, outliers, or ordinal data
Excel: Built from RANK.AVG, SUMIF, and the normal or chi-square functions
Minitab: Stat > Nonparametric > Mann-Whitney, 1-Sample Wilcoxon, Kruskal-Wallis
Prerequisite: Normality tests

The Idea in Plain Language

Most of the tests you meet first, such as the t-test and ANOVA, work with the actual values and assume the data are roughly normal. Nonparametric tests do not assume a particular distribution. Most of them replace the values with their ranks: the smallest value gets rank 1, the next rank 2, and so on. The test then asks whether the ranks in one group are systematically higher than in another.

Ranks have two useful properties. An extreme value counts only as the next step up, so outliers do little damage. And the pattern of ranks does not depend on the shape of the distribution, so the tests stay valid for skewed data, small samples, and ordinal ratings.

Ranking turns an extreme value into just another step. An outlier of 120 has the same influence on a rank test as a value of 9 would.

Not “assumption-free”. These tests still need independent observations, and for the simplest interpretation, groups whose distributions have a similar shape. They trade some power for robustness: when the data really are normal, they detect a difference slightly less often than the t-test or ANOVA (about 95% as efficiently), but when the data are skewed they can be much better.

When to Use Which Test

Question	Parametric test	Nonparametric alternative	Minitab menu
One sample against a target	1-sample t	1-sample Wilcoxon signed-rank (or sign test)	Stat > Nonparametric > 1-Sample Wilcoxon
Two paired measurements	Paired t	Wilcoxon signed-rank on the differences	Stat > Nonparametric > 1-Sample Wilcoxon
Two independent groups	2-sample t	Mann-Whitney (Wilcoxon rank-sum)	Stat > Nonparametric > Mann-Whitney
Three or more independent groups	One-way ANOVA	Kruskal-Wallis (or Mood’s median)	Stat > Nonparametric > Kruskal-Wallis
Blocked or repeated measures	Two-way ANOVA with blocks	Friedman test	Stat > Nonparametric > Friedman
Association between two variables	Pearson correlation	Spearman rank correlation	Stat > Basic Statistics > Correlation

Use a nonparametric test when	Because
The data are clearly skewed and the groups are small	Parametric p-values may be wrong; ranks do not depend on the shape
There are outliers you cannot remove	Ranks limit their influence
The data are ordinal (ratings 1 to 5, severity levels)	The distances between ratings are not meaningful
The measurement has a ceiling or floor (values pile up at a limit)	Ranks are less distorted
A transformation does not fix the problem	No other simple remedy

With large samples, or near-normal data, prefer the parametric test. It has more power and gives estimates, such as means and intervals, that are easy to explain. Nonparametric tests earn their place when normality fails and the sample is not large.

Mann-Whitney Test: Two Independent Groups

A maintenance team compares repair times (hours) after two diagnostic methods. Method A has 8 repairs and Method B has 9. Repair times are skewed, with a few long jobs.

Repair times by method. The gold bars are the medians. Method B is higher and has the long tail.

By hand

Hypotheses. H₀: the two methods give the same distribution of repair times. H₁: one method tends to give longer times. α = 0.05.
Rank all 17 values together, from smallest (rank 1) to largest, ignoring the groups. Average the ranks when values tie (there are none here).
Add the ranks in each group. W_A = 44 and W_B = 109. As a check they add to N(N + 1)/2 = 153.
Convert to U. U_A = W_A − n_A(n_A + 1)/2 = 44 − 36 = 8, and U_B = n_An_B − U_A = 64. The smaller, 8, is the test statistic.
Find the p-value. Under H₀, W_A has mean n_A(N + 1)/2 = 72 and standard deviation √(n_An_B(N + 1)/12) = 10.39. The normal approximation with continuity correction gives z = (28 − 0.5) / 10.39 = 2.65, p = 0.0081. The exact p-value for these sample sizes is 0.0055.
Estimate the size of the difference. The Hodges-Lehmann estimate is the median of all 72 pairwise differences (A − B): -13.0 hours, with an approximate 95% interval of -24 to -5 hours.

All 17 repair times in order, with their ranks and groups
Rank	Time (h)	Method
1	9	A
2	11	A
3	12	A
4	14	A
5	15	A
6	17	A
7	18	B
8	20	A
9	22	B
10	24	B
11	25	B
12	27	B
13	30	B
14	33	B
15	38	A
16	45	B
17	60	B

Conclusion. Method A repairs were shorter than Method B repairs (median 14.5 h against 27.0 h; Mann-Whitney W = 44, p = 0.008; estimated median difference -13.0 h, 95% CI -24 to -5 h). For comparison, a Welch t-test on the same data gives p = 0.018: the long right tail inflates the t-test’s standard error, so its evidence is weaker.

Run Mann-Whitney in Excel and Minitab

ExcelStep by step

Put the two groups in A2:A9 (Method A) and B2:B10 (Method B). In column D, stack all 17 values; in column E, mark the group (A or B).
Rank all values together: =RANK.AVG(D2,$D$2:$D$18,1) in column F. It averages the ranks of ties.
Sum the ranks for Method A: =SUMIF(E2:E18,"A",F2:F18) gives W_A.
Calculate z: =(ABS(W-n1*(N+1)/2)-0.5)/SQRT(n1*n2*(N+1)/12) and the p-value =2*(1-NORM.S.DIST(z,TRUE)).
Median of each group: =MEDIAN(A2:A9).

Excel has no built-in rank-sum test, so the steps above build it. The normal approximation is good for samples of about 8 or more per group; for smaller samples use exact tables or Minitab.

MinitabStep by step

Put the data in two columns, one for each method (Method A and Method B).
Choose Stat > Nonparametric > Mann-Whitney. Set First sample and Second sample.
Set the Confidence level (95%) and the Alternative (not equal, unless you decided on a direction in advance), and click OK.
Read the difference in medians with its interval, the W-value, and the p-value.
For several groups, use Stat > Nonparametric > Kruskal-Wallis.

Minitab reports the p-value from the normal approximation, adjusted for ties when there are any. Its interval for the difference may differ slightly from the one shown here in the last digit.

Minitab session window (typed excerpt, simplified)

Method

η₁: median of Method A
η₂: median of Method B
Difference: η₁ - η₂

Descriptive Statistics

Sample      N  Median
Method A    8    14.5
Method B    9    27.0

Estimation for Difference

          CI for  Achieved
Difference  Difference  Confidence
  -13.0  (-24.0, -5.0)      96.14%

Test

Null hypothesis         H₀: η₁ - η₂ = 0
Alternative hypothesis  H₁: η₁ - η₂ ≠ 0

Method                W-Value  P-Value
Not adjusted for ties    44.0   0.0081

Wilcoxon Signed-Rank Test: Paired Data

A new fixture is meant to cut handling time. Nine operators were timed with the old and the new fixture. The differences are small in number and not clearly normal, so the team uses a rank test on the paired differences.

Handling time (s) for nine operators before and after a new fixture
Operator	Before	After	Difference	\|Difference\|	Rank of \|difference\|	Signed rank
1	34	30	-4	4	4	−4
2	28	27	-1	1	1	−1
3	45	38	-7	7	7	−7
4	39	41	+2	2	2	+2
5	52	41	-11	11	9	−9
6	31	28	-3	3	3	−3
7	47	39	-8	8	8	−8
8	36	31	-5	5	5	−5
9	41	35	-6	6	6	−6

Difference each pair (after − before). Drop any zero differences.
Rank the absolute differences from smallest to largest, and give each rank the sign of its difference.
Add the ranks of the positive and the negative differences. W⁺ = 2 and W⁻ = 43, which add to n(n + 1)/2 = 45.
The test statistic is the smaller sum, 2. For n = 9, the exact two-sided p-value is 0.0117, so the change is significant at 0.05. (A paired t-test gives p = 0.0060.)

Conclusion. Handling time was lower with the new fixture for 8 of 9 operators (median change -5 s; Wilcoxon signed-rank W = 2, n = 9, p = 0.012).

In Excel: compute the differences, then =RANK.AVG(ABS(D2),ABS_range,1) for the ranks and =SUMIF(diff_range,">0",rank_range) for W⁺. In Minitab: make a column of differences and choose Stat > Nonparametric > 1-Sample Wilcoxon with a test median of 0.

Kruskal-Wallis Test: Three or More Groups

Kruskal-Wallis is the rank version of one-way ANOVA. Setup times on three machines were recorded for six runs each.

Setup times (min) on three machines, with ranks across all 18 values
Run	X	Rank	Y	Rank	Z	Rank
1	12	2	20	10	16	6
2	15	5	25	15	19	9
3	11	1	22	12	17	7
4	18	8	30	17	35	18
5	14	4	27	16	21	11
6	13	3	24	14	23	13
Rank sum		23		84		64

H = 12 / (N(N + 1)) × Σ R_i² / n_i − 3(N + 1)

With N = 18, rank sums 23, 84, and 64, and six values per machine: H = 12 / (18 × 19) × (23² + 84² + 64²) / 6 − 3 × 19 = 11.31. Compare with the chi-square distribution on k − 1 = 2 degrees of freedom: p = 0.0035. Machine setup times are not all alike. (One-way ANOVA on the same data gives F = 8.48, p = 0.0034, the same conclusion.)

As with ANOVA, a significant result does not say which machines differ. Follow up with Mann-Whitney tests on each pair and adjust the p-values for the number of comparisons (Bonferroni: multiply each by 3), or use Dunn’s test where available.

Mood’s median test is a simpler alternative. It counts how many values in each group fall above the overall median (19.5) and tests the counts with chi-square. Here it gives p = 0.002. It is more robust to extreme outliers but has less power than Kruskal-Wallis.

In Excel: rank with =RANK.AVG, sum the ranks per machine with =SUMIF, calculate H as above, and get the p-value with =CHISQ.DIST.RT(H,2). In Minitab: Stat > Nonparametric > Kruskal-Wallis (Response and Factor columns), or Stat > Nonparametric > Mood's Median Test.

Assumptions and Limits

Assumption or issue	Why it matters	What to do
Independent observations	Every test needs it	Think about how the data were collected
Similar shape in the groups	Needed to read the result as a difference in medians; if shapes differ, the test shows only that one group tends to be higher	Compare the dot plots or box plots
Many tied values	Ties reduce the information in the ranks	Use the version adjusted for ties; improve measurement resolution
Very small samples	Little power; the smallest possible p-value may be above 0.05	Use exact p-values; collect more data (two groups of 3 can never give p below 0.1)
The question is about the mean	Rank tests compare the whole distributions or the medians, not the means	State the question as medians, or use a transformation and a t-test
Estimates and intervals	Ranks do not give means and standard errors	Report medians and the Hodges-Lehmann interval

Reading and Reporting

Name the test and why: “Because repair times were skewed, a Mann-Whitney test was used.”
Report the medians, not the means, with the estimated difference and its interval.
Report the statistic and the p-value (W, U, H, or the signed-rank sum).
Check the shape assumption before you call it a difference in medians.
Say what it means in practice, in the units of the data.

A sentence you can use. Repair time was shorter with Method A than with Method B (median 14.5 h versus 27.0 h; estimated difference -13.0 h, 95% CI -24 to -5 h; Mann-Whitney W = 44, p = 0.008).

Common Mistakes

Mistake	Why it misleads	Better
Using a rank test on large, near-normal samples	Loses a little power for no gain	Use the t-test or ANOVA, after checking the plots
Reporting means with a rank test	The test does not compare means	Report medians
Choosing the test after seeing which gives p < 0.05	Inflates false alarms	Decide from the data type and shape before testing
Ignoring ties	The p-value can be off	Use the tie-adjusted version
Comparing groups with different shapes as if only the medians differed	A significant result may be about spread or shape	Plot the groups; describe the difference honestly
Many pairwise rank tests with no adjustment	False alarms accumulate	Adjust, or use Kruskal-Wallis first
Assuming nonparametric means “no assumptions”	Independence still matters	Check independence and shape

Try It Yourself

Customer waiting times (minutes) were recorded for two service desks: Desk 1: 4, 6, 3, 8, 5. Desk 2: 9, 12, 7, 15, 10, 11.

Rank all 11 values together and find the rank sum for Desk 1.
Is Desk 1 faster at α = 0.05? Use the exact p-value for groups this small.

Show the answer

Ranks of Desk 1: 2, 4, 1, 6, 3. The rank sum is 16 (the minimum possible is 15). The exact two-sided p-value is 0.0087, so Desk 1 is significantly faster at the 5% level: the median waiting time was 5 minutes at Desk 1 and 10.5 at Desk 2.

With groups this small, the exact p-value matters. Normal approximations can be inaccurate when both groups have fewer than about 8 values.