Mann Whitney U Test Calculator – Free Online Tool

A Mann Whitney U test calculator compares two independent groups using ranks instead of means, so your data never has to be normally distributed.

Rank every value from both groups together, add the ranks inside group 1 to get R1, then subtract n1(n1 + 1) / 2. That is U1. The smaller of U1 and U2 is the reported U. A small U means one group sits almost entirely above the other. In the worked example on this page, U = 8.00 with n1 = n2 = 10, the exact p value is .0007, and the rank biserial effect size is .84, which is large.

Rule of thumb: if p is below .05 the two groups differ. Always report an effect size beside it, because with a large sample a trivial difference still produces a small p value.

The Mann Whitney U test answers one plain question: when you pick a value at random from group 1 and a value at random from group 2, is one of them reliably bigger than the other? It answers that without ever looking at the means. Instead it throws both groups into one pile, sorts them, and replaces every value by its position in that sorted pile. Those positions are the ranks. If one group holds most of the high ranks, the groups differ.

That single move, from values to ranks, is what makes the test so useful. A salary of 2 million and a salary of 200 thousand become rank 20 and rank 19. The extreme value stops dominating the answer. The same is true of ordinal data such as a 1 to 7 satisfaction scale, where the gap between 5 and 6 is not necessarily the same size as the gap between 1 and 2, so averaging is questionable but ranking is fine.

What the calculator reports:

• U1, U2 and U, the test statistic in all three forms so you can match any textbook or software output.
• Rank sums R1 and R2 and the mean rank of each group, which is what SPSS prints.
• Exact p value by full enumeration of the U distribution for small samples, and the normal approximation with tie correction for large ones.
• z score, with an optional continuity correction.
• Rank biserial correlation and r = z / √N, the two standard effect sizes.
• Common language effect size, the probability that a randomly drawn value from one group exceeds one from the other.
• Hodges Lehmann estimator, the median of every pairwise difference, which is the location shift the test is really estimating, with a confidence interval.
• Tie count and tie correction, plus a Noether sample size estimate for planning a replication.

A worked one-liner: with group 1 = 52, 48, 55, 61, 47, 50, 58, 53, 49, 56 and group 2 = 64, 59, 54, 66, 62, 57, 74, 60, 63, 69, the rank sums are R1 = 63 and R2 = 147. So U1 = 63 − 55 = 8, U2 = 100 − 8 = 92, and U = 8. The exact two tailed p value is .0007.

Who uses it: medical researchers compare pain scores and symptom scales between two treatment arms. Psychologists compare Likert responses between conditions. Ecologists compare counts between two sites where the data are heavily skewed. Education researchers compare exam performance between two teaching methods when class sizes are small. Economists compare income between two regions, where a few very high earners would wreck a t test.

Mann Whitney U test versus the nearest alternatives

4.1 What the U statistic actually counts

U is a counting statistic, not an average of anything. U1 counts how many times a value in group 1 beats a value in group 2 across every possible pairing, with a tie counted as half a win. If group 1 has 10 values and group 2 has 10 values there are 100 pairings, so U1 can range from 0 to 100. U1 = 0 means group 1 lost every single comparison. U1 = 100 means it won every one. U1 = 50 is the perfectly balanced middle.

That is why U carries information the p value does not. A U of 8 out of a possible 100 tells you the separation is nearly total. A U of 44 out of 100 tells you the groups are barely distinguishable, no matter what the p value says.

4.2 Reading the p value correctly

The p value is the probability of getting a U at least as extreme as yours if the two groups really came from the same distribution. It is not the probability that the groups are the same, and it is not the probability that you are wrong. A p of .0007 means: if there were truly no difference, a separation this clean would show up in about 7 samples out of every 10,000.

Compare p against your alpha, which is normally .05. If p is smaller, you reject the null hypothesis of no difference. If p is larger, you fail to reject it, which is not the same as proving the groups are identical. A non-significant result with a small sample most often means you did not have enough data to detect anything, not that nothing is there.

4.3 Exact versus approximate p values, and why they differ

For small samples this calculator enumerates the complete distribution of U by dynamic programming, so the exact p value is the true probability with no approximation involved. For larger samples it standardises U into a z score and reads the normal curve. The two will not agree to the last decimal, and the exact value is the one to quote whenever it is available.

In the worked example the exact p is .0007 while the normal approximation gives .0017. Both lead to the same decision, but they are not interchangeable in a write up. When ties are present the exact calculation is no longer valid, because the tie changes the set of possible rank arrangements, so the calculator automatically falls back to the corrected normal approximation and tells you it did.

4.4 The effect size is the part reviewers actually read

A p value tells you whether a difference is detectable. An effect size tells you whether it matters. The rank biserial correlation r_rb runs from minus 1 to plus 1 and has a beautifully direct reading: it is the probability that a group 1 value beats a group 2 value minus the probability of the reverse. An r_rb of .84 means group 1 wins 92 percent of the pairwise comparisons and loses 8 percent.

The second effect size, r = z divided by the square root of N, is the one many journals still expect. It is not identical to r_rb and will usually be a little smaller. Report whichever your field expects, and say which one you used.

4.5 The common language effect size, for non-statisticians

The common language effect size, sometimes written CLES or the probability of superiority, is U1 divided by n1 times n2. It answers a question anyone can follow: if you pick one person at random from each group, what is the chance the group 1 person scores higher? A CLES of .92 means 92 times out of 100. This is the single most useful number to put in an abstract or a slide for a non-technical audience.

4.6 The Hodges Lehmann estimator, the size of the shift

People often want to know how big the difference is in the original units, not in ranks. The Hodges Lehmann estimator gives you that. It takes every possible pair of one value from each group, calculates the difference, and reports the median of all those differences. It is the location shift the Mann Whitney U test is genuinely estimating, and it is far more robust than the difference between the two sample medians.

The calculator also gives a distribution free confidence interval around it, built from the ordered pairwise differences. If that interval excludes zero, the result is significant at the matching alpha, which is a useful cross check on the p value.

4.7 Medians, or something subtler?

This is the most misunderstood part of the test, and it is worth getting right. The Mann Whitney U test compares medians only if you are willing to assume the two distributions have the same shape and spread, and are simply shifted relative to each other. That assumption is often reasonable and rarely checked.

If the shapes clearly differ, for example one group is tightly clustered and the other is widely spread, the test still works but it is testing something broader: whether values from one group tend to be larger than values from the other. Two groups can share an identical median and still give a highly significant Mann Whitney result, because the test is sensitive to the whole distribution. When the shapes differ badly, say so in your write up, or switch to the Brunner Munzel test.

4.8 What the mean ranks tell you

SPSS reports mean ranks rather than U, which confuses people moving between packages. The mean rank of a group is its rank sum divided by its sample size. Under the null hypothesis both mean ranks should sit near (N + 1) / 2, where N is the combined sample size. In the worked example N = 20, so the expected mean rank is 10.5, and the observed values of 6.3 and 14.7 sit a long way either side of it. That distance is the effect, expressed in the units the test actually works in.

4.9 Ties, and what they cost you

Ties are values that appear more than once across the pooled sample. They are unavoidable with Likert scales, rounded measurements and small integer counts. Each tied group gets the average of the ranks it spans, which shrinks the variance of U. The tie correction restores the right variance, and without it your p value would be too large, meaning you would miss real effects.

Heavy ties are a signal, not just a nuisance. If more than about a quarter of your values are tied, your measurement is coarse relative to the differences you are hunting, and no test will rescue that. Consider a finer scale, or an ordinal regression model that treats the categories properly.

4.10 A decision guide for what to do next

4.11 Common mistakes this calculator prevents

• Reporting the wrong U. SciPy returns U1, R returns W, SPSS returns the smaller U. The calculator prints all three so you can never quote a mismatched pair of numbers.
• Skipping the tie correction. Doing the normal approximation by hand without it inflates the p value. It is applied automatically here.
• Using the normal approximation on tiny samples. With five per group it is unreliable. The calculator switches to the exact method for you.
• Calling it a median test with no shape check. The assumption panel checks the two spreads and warns you when a median interpretation is not safe.
• Reporting p alone. Every run prints two effect sizes and the shift in original units.
• Halving a two tailed p to fake a one tailed test. The alternative hypothesis is chosen before you run it, and the output states which one you used.

4.12 How confident should you be?

The Mann Whitney U test is powerful, but it is one test on one sample. A p value just under .05 in a study of ten people per group is weak evidence, whatever the asterisk implies. A clean separation with a large effect size, replicated in a second sample, is strong evidence. Read the confidence interval on the Hodges Lehmann shift: if it stretches from a trivial difference to an enormous one, your study simply has not pinned the effect down yet, and that is the honest thing to write.

Five templates, one for every place a Mann Whitney U test normally lands. Run the analysis and each one fills with your own numbers, ready to copy. Before you run it, they show the worked example so you can see the shape of a correct sentence.

Seven cards, each one step further from the raw ranks. Card 1 is the statistic itself, and every card after it builds on the one above.

Excel has no MANNWHITNEY function and the Analysis ToolPak does not include this test either, so you build it from five ordinary formulas. It takes about three minutes. Every screenshot below uses the same worked example as sections 9, 10 and 12, so all four agree on U = 8.00 and an exact p of .0007.

8.1 The Excel functions you need

8.2 Step by step

8.3 The awkward cases

When you have ties. The simple sigma is too large, so the p value comes out too conservative. Replace it with the tie corrected version. Put the count of each distinct value in a helper column with COUNTIF, then use:

where H holds the tie group sizes. It is fiddly, and it is the main reason people give up on Excel for this test.

When your groups are in two side by side columns. You cannot rank them properly that way. Restack them into long format first: copy group 2 underneath group 1 in column B and type the labels into column A.

When a group name has a trailing space. SUMIF will silently return 0 and your R1 will be wrong. Wrap the criteria in TRIM, or use Find and Replace to clean the labels first.

When you want the exact p value. Excel cannot do it without a macro. Use this calculator, or R, or Python. For n1 and n2 both at or below 20 the exact value is materially different from the approximation, as this example shows.

8.4 Why Excel, R and Python can disagree

8.5 Charting it in Excel

A Mann Whitney result is best shown as a box plot, because the test is about the whole distribution rather than one summary number. In Excel 2016 and later, select both value columns, then Insert, Statistic Chart, Box and Whisker. Excel draws the median as the line inside the box, which is exactly what you should be discussing. Add the individual points on top by inserting a scatter series with a small horizontal jitter, so a reader can see the sample size at a glance. Do not use a bar chart of means: it hides the very skew that made you choose this test.

8.6 Error messages and what they mean

R has the test built in as wilcox.test, because R uses the Wilcoxon rank sum name. The one liner is wilcox.test(x, y). The script below does the full job: the test, both p value methods, the effect sizes, the Hodges Lehmann shift, a plain English verdict and one 300 dpi figure. It uses base R only, so nothing needs installing.

9.1 The complete script

R · runs top to bottom, change only the two data lines

# Mann Whitney U test in R (also called the Wilcoxon rank sum test)
# Base R only. install.packages("effsize") is optional, the effect size is computed by hand below.

set.seed(42)

# ---- 1. Data. Change only these two lines. ----
placebo   <- c(52, 48, 55, 61, 47, 50, 58, 53, 49, 56)
treatment <- c(64, 59, 54, 66, 62, 57, 74, 60, 63, 69)
name1 <- "Placebo"; name2 <- "Treatment"

# Reading from a CSV instead (long format: a group column and a value column):
# df <- read.csv("data.csv")
# df <- df[!is.na(df$value), ]
# placebo   <- df$value[df$group == "Placebo"]
# treatment <- df$value[df$group == "Treatment"]

placebo   <- placebo[!is.na(placebo)]     # drop missing values
treatment <- treatment[!is.na(treatment)]

n1 <- length(placebo); n2 <- length(treatment); N <- n1 + n2

# ---- 2. The test. R reports W, which is U1 for the first group. ----
fit_exact <- wilcox.test(placebo, treatment, alternative = "two.sided", exact = TRUE, correct = FALSE)
fit_norm  <- wilcox.test(placebo, treatment, alternative = "two.sided", exact = FALSE, correct = TRUE)

U1 <- unname(fit_norm$statistic)
U2 <- n1 * n2 - U1
U  <- min(U1, U2)

pooled <- c(placebo, treatment)
ranks  <- rank(pooled)
R1 <- sum(ranks[1:n1]); R2 <- sum(ranks[(n1 + 1):N])

counts   <- table(pooled)
tie_term <- sum(counts^3 - counts)
n_ties   <- sum(counts[counts > 1] - 1)

mu    <- n1 * n2 / 2
sigma <- sqrt(n1 * n2 / 12 * ((N + 1) - tie_term / (N * (N - 1))))
z     <- (abs(U - mu) - 0.5) / sigma

use_exact <- (n_ties == 0 && n1 <= 20 && n2 <= 20)
p      <- if (use_exact) fit_exact$p.value else fit_norm$p.value
method <- if (use_exact) "exact enumeration" else "normal approximation"

# ---- 3. Effect sizes and the location shift ----
r_rb  <- 1 - 2 * U / (n1 * n2)
r_z   <- z / sqrt(N)
cles  <- U1 / (n1 * n2)
diffs <- sort(as.vector(outer(placebo, treatment, "-")))
hodges_lehmann <- median(diffs)

label <- if (abs(r_rb) >= .5) "large" else if (abs(r_rb) >= .3) "medium" else if (abs(r_rb) >= .1) "small" else "negligible"

# ---- 4. Plain English verdict ----
cat("Mann Whitney U test:", name1, "vs", name2, "\n")
cat(sprintf("  n1 = %d, n2 = %d, N = %d, ties = %d\n", n1, n2, N, n_ties))
cat(sprintf("  R1 = %.1f, R2 = %.1f   (check: %.0f = %.0f)\n", R1, R2, R1 + R2, N * (N + 1) / 2))
cat(sprintf("  U1 (R calls it W) = %.2f, U2 = %.2f, reported U = %.2f\n", U1, U2, U))
cat(sprintf("  z = %.4f, p = %.4f  (%s)\n", -z, p, method))
cat(sprintf("  rank biserial r = %.4f (%s), r = z/sqrt(N) = %.4f\n", r_rb, label, r_z))
cat(sprintf("  common language effect size = %.4f\n", cles))
cat(sprintf("  Hodges Lehmann shift = %.2f\n", hodges_lehmann))
cat(sprintf("  medians: %s = %.2f, %s = %.2f\n", name1, median(placebo), name2, median(treatment)))
if (p < 0.05) {
  cat(sprintf("  VERDICT: the two groups differ significantly (p = %.4f), effect is %s.\n", p, label))
} else {
  cat(sprintf("  VERDICT: no significant difference (p = %.4f).\n", p))
}

# ---- 5. One figure, 300 dpi ----
png("mann_whitney_boxplot.png", width = 7, height = 5, units = "in", res = 300)
boxplot(list(placebo, treatment), names = c(name1, name2),
        col = c("#86efac", "#93c5fd"), ylab = "Outcome value",
        main = sprintf("%s vs %s: U = %.2f, p = %.4f, r = %.2f", name1, name2, U, p, r_rb))
points(jitter(rep(1, n1), amount = .08), placebo,   pch = 19, col = "#334155")
points(jitter(rep(2, n2), amount = .08), treatment, pch = 19, col = "#334155")
grid(nx = NA, ny = NULL, col = "grey85")
dev.off()
cat("Figure saved to mann_whitney_boxplot.png\n")

9.2 What each line does

9.3 What the figure shows

A box plot for each group with every raw observation drawn on top with a small horizontal jitter. The heavy line inside each box is the median, which is what the Mann Whitney U test is about, and the box spans the interquartile range. Plotting the points as well as the box is deliberate: with ten per group a box plot alone hides how few observations there are, and a reader deserves to see that. The title carries U, the p value and the effect size, so the figure stands alone in a slide deck.

9.4 Problems and fixes

9.5 One liners worth remembering

Python does this through scipy.stats.mannwhitneyu. The important detail people miss is that SciPy returns U for the first argument, so it gives you U1, not the smaller U that textbooks report. The script below converts it for you and computes everything else.

Install with pip install numpy scipy matplotlib pandas. SciPy 1.7 or later is needed for the method="exact" argument.

10.1 The complete script

Python · runs top to bottom, change only the two data lines

# Mann Whitney U test in Python
# Requires: pip install numpy scipy matplotlib
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
from scipy.stats import mannwhitneyu, rankdata

np.random.seed(42)

# ---- 1. Data. Change only these two lines. ----
placebo   = np.array([52, 48, 55, 61, 47, 50, 58, 53, 49, 56], dtype=float)
treatment = np.array([64, 59, 54, 66, 62, 57, 74, 60, 63, 69], dtype=float)
name1, name2 = "Placebo", "Treatment"

# Reading from a CSV instead (long format: one column of groups, one of values):
# import pandas as pd
# df = pd.read_csv("data.csv").dropna(subset=["value"])
# placebo   = df.loc[df["group"] == "Placebo",   "value"].to_numpy(dtype=float)
# treatment = df.loc[df["group"] == "Treatment", "value"].to_numpy(dtype=float)

placebo   = placebo[~np.isnan(placebo)]       # drop missing values
treatment = treatment[~np.isnan(treatment)]

n1, n2 = len(placebo), len(treatment)
N = n1 + n2

# ---- 2. The test. SciPy returns U for the FIRST argument, so this is U1. ----
res_exact = mannwhitneyu(placebo, treatment, alternative="two-sided", method="exact")
res_norm  = mannwhitneyu(placebo, treatment, alternative="two-sided", method="asymptotic")

U1 = res_norm.statistic
U2 = n1 * n2 - U1
U  = min(U1, U2)

pooled = np.concatenate([placebo, treatment])
ranks  = rankdata(pooled)
R1, R2 = ranks[:n1].sum(), ranks[n1:].sum()

_, counts = np.unique(pooled, return_counts=True)
tie_term = sum(int(c) ** 3 - int(c) for c in counts)
n_ties   = int(sum(c - 1 for c in counts if c > 1))

mu    = n1 * n2 / 2
sigma = np.sqrt(n1 * n2 / 12 * ((N + 1) - tie_term / (N * (N - 1))))
z     = (abs(U - mu) - 0.5) / sigma

p = res_exact.pvalue if (n_ties == 0 and n1 <= 20 and n2 <= 20) else res_norm.pvalue
method = "exact enumeration" if (n_ties == 0 and n1 <= 20 and n2 <= 20) else "normal approximation"

# ---- 3. Effect sizes and the location shift ----
r_rb = 1 - 2 * U / (n1 * n2)
r_z  = z / np.sqrt(N)
cles = U1 / (n1 * n2)
diffs = np.sort(np.subtract.outer(placebo, treatment).ravel())
hodges_lehmann = float(np.median(diffs))

# ---- 4. Plain English verdict ----
label = "large" if abs(r_rb) >= .5 else "medium" if abs(r_rb) >= .3 else "small" if abs(r_rb) >= .1 else "negligible"
print("Mann Whitney U test:", name1, "vs", name2)
print(f"  n1 = {n1}, n2 = {n2}, N = {N}, ties = {n_ties}")
print(f"  R1 = {R1:.1f}, R2 = {R2:.1f}   (check: {R1 + R2:.0f} = {N * (N + 1) / 2:.0f})")
print(f"  U1 = {U1:.2f}, U2 = {U2:.2f}, reported U = {U:.2f}")
print(f"  z = {-z:.4f}, p = {p:.4f}  ({method})")
print(f"  rank biserial r = {r_rb:.4f} ({label}), r = z/sqrt(N) = {r_z:.4f}")
print(f"  common language effect size = {cles:.4f}")
print(f"  Hodges Lehmann shift = {hodges_lehmann:.2f}")
print(f"  medians: {name1} = {np.median(placebo):.2f}, {name2} = {np.median(treatment):.2f}")
if p < 0.05:
    print(f"  VERDICT: the two groups differ significantly (p = {p:.4f}), effect is {label}.")
else:
    print(f"  VERDICT: no significant difference (p = {p:.4f}).")

# ---- 5. One figure, 300 dpi ----
fig, ax = plt.subplots(figsize=(7, 5))
bp = ax.boxplot([placebo, treatment], widths=.55,
                patch_artist=True, medianprops=dict(color="black", linewidth=2))
ax.set_xticks([1, 2])
ax.set_xticklabels([name1, name2])
for patch, colour in zip(bp["boxes"], ["#86efac", "#93c5fd"]):
    patch.set_facecolor(colour)
for i, group in enumerate([placebo, treatment], start=1):
    jitter = np.random.uniform(-.08, .08, len(group))
    ax.plot(i + jitter, group, "o", color="#334155", alpha=.75, markersize=5)
ax.set_ylabel("Outcome value")
ax.set_title(f"{name1} vs {name2}: U = {U:.2f}, p = {p:.4f}, r = {r_rb:.2f}")
ax.grid(axis="y", alpha=.3)
fig.tight_layout()
fig.savefig("mann_whitney_boxplot.png", dpi=300)
print("Figure saved to mann_whitney_boxplot.png")

10.2 What each line does

10.3 What the figure shows

The same box plot as the R script, drawn with matplotlib: one box per group, the median as a heavy black line, and every raw observation jittered on top so the sample size is visible. The title carries U, p and the rank biserial effect size, which makes the image usable on its own in a poster or a slide.

10.4 Problems and fixes

10.5 One liners worth remembering

Six lookup tables. The critical values below were generated by full enumeration of the U distribution, so they match the published Mann Whitney tables exactly.

Table 1. Critical values of U, two tailed test at alpha = .05

What to conclude: reject the null hypothesis when your calculated U is less than or equal to the tabled value. This is the opposite direction to most tests, where you need a statistic larger than the critical value. A dash means no result can reach significance at this alpha with samples that small.

Table 2. Critical values of U, two tailed test at alpha = .01

What to conclude: the same rule, reject when U is at or below the tabled value. Notice how many cells are dashes: with fewer than about five per group you cannot reach the .01 level no matter how cleanly the groups separate.

Table 3. Critical values of U, one tailed test at alpha = .05

What to conclude: use this table only if you predicted the direction of the difference before you collected the data. The values are larger than the two tailed table, which is exactly why a one tailed test chosen after the fact is a form of cheating.

Table 4. Effect size interpretation bands

What to conclude: these bands come from Cohen's conventions and are a starting point, not a standard. In a field where a 2 percent improvement saves lives, a small effect can be the most important number in the paper.

Table 5. Sample size per group for 80 percent power, two tailed at alpha = .05

What to conclude: read down the first column using the effect you expect from a pilot or from the literature, never from the study you have just run. Add 10 to 20 percent for dropout. These come from the Noether formula in card 7 of section 6.

Table 6. Which test should you actually be running?

What to conclude: the commonest error on this list is running several Mann Whitney tests across three or more groups. Use Kruskal Wallis first, then run pairwise Mann Whitney tests with a Bonferroni or Holm adjustment only if it is significant.

Eight worked examples covering every situation you are likely to meet, from a textbook clean result to a null result, a borderline case, heavy ties and a heavily skewed sample. Every number below was computed with the same engine that powers the calculator above.

Design statement: a two group between subjects design. Each unit of observation, whether that is a person, an animal, a plot or a product, is measured once and belongs to exactly one group. The outcome must be ordinal or better, meaning it can at least be ranked from lowest to highest.

How the data sheet should look

Minimum sample: five per group is the mathematical floor for a two tailed test at alpha = .05, and it only works with perfect separation. Ten per group is a sensible practical minimum. Twenty per group and above lets you use the normal approximation comfortably.

Pre-registration: lodge the analysis plan before you collect the data. State the alternative hypothesis, the alpha, the p value method, how ties will be handled and which effect size you will report. Template wording is in section 5, card 5.

Common mistakes in data collection

• Measuring the same subject twice and treating the two values as independent. That is a paired design and needs the Wilcoxon signed rank test.
• Rounding to whole numbers when the instrument gives decimals, which manufactures ties and throws away power.
• Collecting until the result looks good. This inflates the false positive rate badly.
• Dropping inconvenient extreme values. The whole point of a rank test is that it tolerates them.
• Recording group membership only in the file name, so the labels are lost the moment the sheets are merged.

Four situations, and what to do in each

A quick decision tree

The four assumptions

Limitations you should state in your write up

• Less power than a t test when the data really are normal. The efficiency is about 95 percent for normal data, so you lose a little. If the data are normal, use the t test. If they are not, this test can be far more powerful.
• It is not a median test. Without the shape assumption it tests stochastic dominance. If the shapes clearly differ, say so, or use the Brunner Munzel test.
• It cannot adjust for covariates. There is no way to control for age or baseline severity inside this test. Use quantile regression or the ordinal logistic regression calculator.
• It gives no estimate of the difference on its own. The p value tells you nothing about size, which is why this calculator adds the Hodges Lehmann shift and a confidence interval.
• It is sensitive to differences in spread as well as location. Two groups with the same centre but very different variability can produce a significant result, which is easy to misread as a difference in typical value.
• The exact p value is unavailable with ties. The approximation is good, but it is an approximation, and it should be named as such.
• Very small samples cannot reach significance at all. Below five per group, no two tailed result at alpha = .05 is possible. Reporting a non-significant result from four per group as evidence of no effect is a serious error.
• Grouping a continuous predictor to use it loses information. If your grouping variable was originally continuous, a regression model will almost always tell you more.

If you also used the R or Python script from sections 9 and 10, cite the software as well: R Core Team for R, and Virtanen et al. (2020) for SciPy.

Twenty peer-reviewed journal articles, every one with a resolvable DOI link.

1. Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. The Annals of Mathematical Statistics, 18(1), 50-60. https://doi.org/10.1214/aoms/1177730491
2. Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80-83. https://doi.org/10.2307/3001968
3. Fay, M. P., & Proschan, M. A. (2010). Wilcoxon-Mann-Whitney or t-test? On assumptions for hypothesis tests and multiple interpretations of decision rules. Statistics Surveys, 4, 1-39. https://doi.org/10.1214/09-SS051
4. Divine, G. W., Norton, H. J., Barón, A. E., & Juarez-Colunga, E. (2018). The Wilcoxon-Mann-Whitney procedure fails as a test of medians. The American Statistician, 72(3), 278-286. https://doi.org/10.1080/00031305.2017.1305291
5. Hodges, J. L., & Lehmann, E. L. (1963). Estimates of location based on rank tests. The Annals of Mathematical Statistics, 34(2), 598-611. https://doi.org/10.1214/aoms/1177704172
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