Group A beats group B in 82 of the 100 pairwise comparisons and loses 15, so δ = 0.6700. Cohen’s d on the same data is −0.3223 — it says group B is higher, because one value of 300 drags B’s mean to 39.50 and its standard deviation to 91.59. Switch to the next preset, which changes that single value to 17.
10 × 10 = 100 pairwise comparisons
δ = 0.6700 (large)
A wins 82, loses 15 and ties 3, so a value drawn from A beats one drawn from B 83.50% of the time with ties counted as half. The 95% interval is 0.0607 to 0.9155. Cohen's d on the same data is -0.3223 — the opposite sign, because a mean and a standard deviation can be dominated by a single value and a count of wins cannot.
Cliff's δ
0.6700
large
P(A beats B)
83.50%
ties count as half
95% interval
0.061 to 0.916
SE 0.2109
Cohen's d
-0.3223
for comparison
Every pairwise comparison, counted
How the pairwise comparisons between the two groups split
Outcome
Pairs
Share
Counts as
A greater
82
82.00%
+1
B greater
15
15.00%
−1
Tied
3
3.00%
0
δ
(82 − 15) / 100
0.670000
Changing any single value in group B can move δ by at most 2/nB = 0.2000, because that value can win or lose at most 10 comparisons either way. A mean has no such bound, which is the whole argument for using this statistic on messy data.
Magnitude bands
Romano’s magnitude thresholds for Cliff’s delta
|δ|
Label
Roughly
This sample
under 0.147
negligible
d ≈ 0.11
0.147 to 0.33
small
d ≈ 0.33
0.33 to 0.474
medium
d ≈ 0.52
0.474 and up
large
d ≈ 0.80
←
These thresholds were chosen to line up with Cohen’s conventions for d under normality, and they inherit every objection to those conventions. Read the interval first: this sample’s runs from 0.0607 to 0.9155, which covers more than one band.
No mean, no SD Works on ordinal data Bounded influence per value Ignores how far apart
What this tool shows
Change one value in the shipped sample from 17 to 300 and Cohen’s d goes from +1.7774 to −0.3223 — a sign flip. Cliff’s delta moves from 0.7900 to 0.6700. It cannot do worse than that: one value can shift delta by at most 2/n, because it can win or lose a bounded number of comparisons. Both presets are on the page, and d is printed beside delta throughout.
Cliff’s delta from the full pairwise comparison count, with wins, losses and ties shown
A confidence interval using DeLong’s structural components, checked against a bootstrap
P(A beats B) — the same statistic read as a probability, and the area under an ROC curve
Romano’s magnitude bands, with the interval printed so the band can be judged
Cohen’s d on the same data, so the divergence is visible rather than asserted
Ordinal data handled honestly, because only the ordering is ever used
Rank based Bounded influence DeLong interval Pair counts shown
Updated 13 September 2026 · Works in any browser, no installation
Cliff’s delta is how often a value from one group beats a value from the other, minus how often it loses. Every value in group A is compared against every value in group B, and the answer is the net win rate on a −1 to +1 scale. No mean, no standard deviation, no assumption about the shape of either distribution — which is why it survives outliers and works on ordinal ratings where a standardised mean difference is meaningless.
At a glance
Formula shown
δ = (#(a > b) − #(a < b)) / (n_A·n_B) over all pairs, with ties contributing nothing to either count. Equivalently δ = 2θ − 1, where θ = P(a > b) + ½·P(a = b) is exactly the Mann–Whitney U divided by n_A·n_B and exactly the area under the ROC curve of the two samples. The interval comes from DeLong’s structural components — the variance of each observation’s own win rate against the other group — placed on a logit so it cannot escape [−1, 1].
Scenario support
Comparing two groups on Likert or ordinal ratings, benchmark and latency comparisons where a few slow runs dominate the mean, reaction times and other skewed distributions, small samples where normality cannot be checked, and any comparison that has to survive a reviewer asking what happens if the largest value is a data-entry error.
Educational estimate
Planning support from the values you enter — not professional advice.
One value flips d and barely moves delta
The first two presets are the same data apart from a single number, and they are the argument for this statistic in its entirety.
With group B ending in 17: Cohen’s d is +1.7774 and delta is 0.7900. Both say group A is clearly higher.
With that 17 replaced by 300: d is −0.3223 and delta is 0.6700. d has changed sign. It now reports that group B is higher, on data where A wins 82 of the 100 comparisons.
The mechanism is that 300 moves two things at once. B’s mean goes from 11.20 to 39.50 and its standard deviation from about 4 to 91.59, so the numerator flips while the denominator explodes.
Delta is bounded by construction: one element of B can win or lose at most nA comparisons, so it can move delta by at most 2/nB— 0.20 here, whatever value it takes. It moved 0.12.
The same number as three other statistics
Cliff’s delta looks like a fringe statistic and is in fact a relabelling of three familiar ones, which is worth knowing before choosing between them.
It is exactly the rank-biserial correlation for two independent groups, so a Mann–Whitney test already produces it.
It is exactly 2θ − 1 where θ is the probability of superiority, the Mann–Whitney U over nAnB.
And θ is exactly the area under the ROC curve of the two samples, treating one as cases and the other as controls.
All three are computed here by unrelated code in three different engine files, and the verification suite asserts they agree to machine precision on 250 generated pairs. That is why the interval can borrow DeLong’s variance, which was derived for the ROC area.
Ordinal data, where a mean is fiction
Likert ratings get averaged constantly, and the average assumes something the scale does not provide.
“Agree” to “strongly agree” is not the same distance as “neutral” to “agree”, and nothing in the data says what either distance is.
A mean of 4.33 has no units. Neither does a standard deviation on that scale, which makes a standardised mean difference a ratio of two undefined quantities.
Delta only ever asks which of two ratings is higher, which the scale does define. The shipped Likert preset gives 0.7639: pick one rating from each group and A is higher 88.19% of the time.
Ties are handled explicitly rather than ignored. That preset has 22 tied pairs out of 144, each counted as half a win, which is what keeps delta unbiased on coarse scales.
The interval is wide, and that is the honest part
Delta is usually reported bare, with a magnitude label attached. The label is far more confident than the estimate.
The outlier preset gives δ = 0.6700, labelled “large”. Its 95% interval runs from 0.0607 to 0.9155.
The overlapping preset gives δ = −0.0600, labelled “negligible”, with an interval from −0.5333 to 0.4419 — covering medium effects in both directions.
Ten per group simply does not pin a rank statistic down. That is a fact about the design, and no choice of statistic repairs it.
The interval is built on a logit transform of θ, which keeps it inside [−1, 1] even when delta is close to a boundary — a symmetric interval would run past 1 on the separated preset.
Delta saturates and d does not
The robustness has a cost, and the fourth preset shows it plainly.
When every value in A beats every value in B, delta is exactly 1 and the standard error is exactly 0. It has nothing left to report.
Cohen’s d on that same data is 8.1650 and keeps rising as the groups separate further. Two groups a mile apart and two groups barely separated both score δ = 1.
So delta answers “how reliably?” and d answers “by how much?”. Those are different questions and the right one depends on whether the measurement scale means anything.
Complete separation is common in small samples, and a delta of 1 from eight per group is far weaker evidence than it looks — which is why the interval collapsing to a point should be read as a warning rather than as precision.
Where the 0.147 / 0.33 / 0.474 thresholds come from
The magnitude bands are quoted everywhere and their provenance is rarely mentioned.
They were chosen to line up with Cohen’s d conventions under normality: d = 0.11, 0.33 and 0.52 map to delta = 0.147, 0.33 and 0.474.
Which means they inherit every objection to Cohen’s conventions, including that Cohen described them as a last resort when no better basis exists.
And they assume normality to make the translation — on data non-normal enough to justify reaching for delta, the correspondence no longer holds.
The probability reading is more useful and needs no convention: “a value from A beats one from B 88% of the time” is interpretable without any band at all.
Reporting Cliff’s delta
Four items, and the third is what makes the result reproducible.
Give delta with its confidence interval. At ten per group the interval is wide enough to change the conclusion, and it is the part a band cannot convey.
Give the probability reading alongside it. “A beats B 83.5% of the time” travels further than “δ = 0.67”.
Give both group sizes and the number of ties. Ties are handled differently across implementations, and on coarse scales they can be a quarter of all pairs.
And say why you chose a rank statistic. “Skewed with a long right tail” is a reason; switching to delta after d came out non-significant is not.
Sources and methodology
References for delta, its variance and the magnitude bands.
Method. Delta is computed from the full nA×nBcomparison matrix rather than from ranks, so the win, loss and tie counts can be printed and the arithmetic checked by hand on a small sample. The confidence interval uses DeLong’s structural components — the variance of each observation’s own win rate against the opposing group — placed on a logit so the interval cannot leave [−1, 1]. That variance is not taken on trust: the suite compares it against a 6,000-draw bootstrap at two sample sizes and requires the agreement to be within 10% at ten per group, within 4% at sixty against eighty, and strictly better at the larger size, which is what consistency means. The suite also asserts on 250 generated pairs that delta equals the rank-biserial correlation from the site’s Mann–Whitney engine, that the probability reading equals the ROC area from the epidemiology engine, that swapping the groups flips the sign exactly, and that any strictly increasing transform of both samples leaves delta unchanged. That engine is verified on every change against 134 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Mann-Whitney UMann-Whitney U with the tie correction and both medians shown, because it is not a test of medians — two samples with identical medians of 4 are separated at p = 0.01.
Hedges' gBias-corrected standardised mean difference using the exact gamma correction, with Cohen's d beside it and a confidence interval.
Effect SizeCohen d, Hedges g and the overlap between groups, with a sample-size control that moves the p-value while leaving the effect size fixed — the same d gives t = 1.29 at n=30 and 23.57 at n=10,000.
Wilcoxon Signed-RankReports how many zero differences it dropped and gives the Hodges-Lehmann shift, because the classical and Pratt variants disagree on the same data and the median of the differences is not what this test estimates.
Kruskal-WallisApplies the tie correction and shows it against the uncorrected value, because on ordinal data it moves p from 0.054 to 0.027 — across the conventional threshold, on identical data.
Fleiss' KappaAgreement among any number of raters, with a per-category breakdown, a bootstrap interval over subjects and both agreement figures shown.
An educational tool. Cliff’s delta counts how often one group beats the other and never asks by how much, so it saturates at ±1 under complete separation and cannot distinguish a small separation from an enormous one. The magnitude bands are translated from Cohen’s d conventions under an assumption of normality, which is the assumption a rank statistic is usually chosen to avoid.