Seven titration readings, one of them clearly apart from the rest. Q is 0.80769 against a 95% critical value of 0.5693, and the simulated p-value is 0.00045. The gap between 12.34 and its nearest neighbour is 0.21; the whole range is 0.26. Now switch to the next preset, which adds a single further reading.
7 values · r10 · high end
Q = 0.80769, p = 0.00045
The value on trial is 12.340000, at the high end. Rejecting it needs Q above 0.5693 at 95% or 0.6871 at 99%. The other end gives Q = 0.03846. On this data the suspect value is separated from the rest by more than chance comfortably explains.
Q statistic
0.80769
high end, r10
p-value
0.00045
simulated, not tabled
Value on trial
12.340000
of 7 readings
Critical at 95%
0.5693
99%: 0.6871
The statistic at the opposite end is 0.03846. Only one end is in question here, which is the situation this test is built for.
The sorted readings
The sorted sample with the gap to each neighbour
Rank
Value
Gap to previous
Share of range
1
12.080000
—
—
2
12.090000
0.010000
3.8%
3
12.100000
0.010000
3.8%
4
12.110000
0.010000
3.8%
5
12.120000
0.010000
3.8%
6
12.130000
0.010000
3.8%
7
12.340000 — on trial
0.210000
80.8%
Q for this sample size is the gap to the next value over the full range. The whole test is one gap divided by one span, which is why a second outlier next to the first destroys it.
Which ratio applies at which n
The four Dixon ratios and the sample sizes they apply to
n
Ratio
What it measures
This sample
3 to 7
r10
gap to the next value over the full range
←
8 to 10
r11
gap to the next value, ignoring the far extreme
11 to 13
r21
gap to the third value, ignoring the far extreme
14 and up
r22
gap to the third value, ignoring the two far extremes
An actual p-value All four ratios Masking made visible One value, once
What this tool shows
Seven titrations with one suspect reading give Q = 0.80769 and p = 0.00045. Add a second suspect reading beside the first and Q falls to 0.07143 with p = 0.96700. Same experiment, one extra measurement, and the test now calls the largest value entirely unremarkable. Both presets are shipped here, because masking is the failure mode that decides whether this test is safe to use at all.
Dixon’s Q with all four ratios — r10, r11, r21 and r22 — selected by sample size
An actual p-value from a simulated null, not a pass/fail against three tabled thresholds
Critical values at 90%, 95% and 99%, computed rather than transcribed
The statistic at the opposite end of the sample, so a two-sided problem is visible
The sorted readings with every gap and its share of the range
Two presets differing by one value that demonstrate masking directly
Real p-values All four ratios Masking shown Both ends reported
A statistical outlier is not a wrong measurement. Deleting data needs a reason.
Updated 13 September 2026 · Works in any browser, no installation
Dixon’s Q asks whether the most extreme reading is further from its neighbour than the spread of the data can explain. It is one gap divided by one span, which is what makes it usable on three observations where nothing else is — and what makes it fragile. It assumes the rest of the data is normal, it tests exactly one value, and it must be applied once. Running it again after deleting a point is not a second test; it is a different test with no stated error rate.
At a glance
Formula shown
For 3 ≤ n ≤ 7, Q = |x₂ − x₁| / (xₙ − x₁) at the low end and its mirror at the high end. From n = 8 the denominator drops the far extreme (r11), from n = 11 the numerator reaches to the third value (r21), and from n = 14 the denominator drops two extremes (r22) — each change is there to stop a second outlier from contaminating the statistic. The null distribution is simulated under normality with a fixed seed, so the same input always returns the same p.
Scenario support
Analytical chemistry replicates, instrument calibration runs, small laboratory datasets where a normality test is impossible, quality control on three to ten measurements, and any situation where one reading looks wrong and the decision is whether to repeat the measurement.
Educational estimate
Planning support from the values you enter — not professional advice.
Two outliers hide each other completely
This is the reason Dixon’s Q gets misused, and the two shipped presets differ by exactly one number so it cannot be argued with.
Seven readings with one value apart from the rest: Q = 0.80769, p = 0.00045. Decisive, at only seven observations.
Add one more high reading beside it: Q = 0.07143, p = 0.96700. The test now reports that the largest value is about as ordinary as a value can be.
The cause is in the formula. Q is the gap between the extreme and its neighbour, divided by the range. A second outlier next to the first makes that gap tiny while leaving the range wide.
Which is why the ratio changes with n. From eleven observations r21 reaches past the second value to the third, precisely so one companion outlier cannot do this — but at n = 7 there is no such protection, and a great many laboratory datasets have exactly seven points.
A p-value instead of three thresholds
Dixon’s Q is normally used against a printed table with three columns, which answers a narrower question than most people want.
A table tells you whether Q cleared 90%, 95% or 99%. It cannot tell you whether you are at p = 0.051 or p = 0.49.
The null distribution here is simulated under normality with a fixed seed, so the p-value is an actual probability and repeats exactly for the same input.
The simulation reproduces the published table. At n = 3 to 7 the simulated 95% points come out within 0.006 of the standard values of 0.970, 0.829, 0.710, 0.625 and 0.568, which the verification suite asserts on every build.
It also settles a convention question. The published values match the two-sided form — testing whichever end is more extreme — and not the one-tailed form, which at n = 3 would give 0.942 rather than 0.970. That is why this calculator tests both ends and reports the larger.
Once. Not once per suspect value
The most common misuse is sequential: reject the extreme, recompute, reject the next one. Every step after the first has no stated error rate.
The 5% in “significant at 5%” is the chance of rejecting a good value once. Run the test three times on the same data and that guarantee is gone.
And the second test is run on data chosen to make it pass. Deleting the extreme shrinks the range, which inflates Q for whatever is now extreme.
If two values look suspect, Dixon’s Q is the wrong test. The generalised extreme studentised deviate test is built for an unknown number of outliers; a Grubbs test has the same single-outlier limitation but a documented masking probe.
The opposite-end statistic is printed here for this reason. When both ends look extreme, the tool says so rather than quietly testing one of them.
The assumption you cannot check
Q’s null distribution is derived under normality, and the sample sizes it is used at make that assumption untestable.
At seven observations no normality test has meaningful power.Shapiro–Wilk on seven points will accept almost anything.
And genuinely skewed data produces extreme values by construction. On a log-normal measurement, the largest of seven readings being far from the rest is the expected outcome, not a fault.
So the assumption has to come from the measurement process, not from the seven numbers in front of you. Replicate analytical measurements usually earn it; concentrations, times and counts often do not.
If the process is known to be skewed, test the logarithms instead, which is a different and usually better-behaved question.
Significant does not mean delete
A rejected value is a statistical statement about a gap. It is not a finding about the measurement.
An outlier is sometimes the most informative point in the set. Contamination, an instrument fault, a transcription error and a real effect all look identical to this test.
The defensible sequence is to investigate first and delete only on a found cause. “Q rejected it” is not a cause.
If no cause is found, report both analyses. With and without the point, and let the reader see how much the conclusion depends on one number.
Or use a method that does not need the decision. A trimmed mean or a rank-based summary downweights extremes without anyone having to choose which ones to remove.
Reporting a Q test
Four items, and the second is what stops the result being unreproducible.
Give n, Q and the p-value or the threshold used. Q alone is meaningless: 0.5 is decisive at n = 20 and unremarkable at n = 4.
Say which ratio and which convention. r10 and r22 are different statistics, and one-tailed and two-sided tables differ by more than 0.02 at small n.
Say the test was run once. If it was run more than once, the stated significance level is not the one that applies.
And say what happened to the value. Kept, deleted for a documented cause, or reported both ways — all three are defensible; silently dropping it is not.
Sources and methodology
References for the Q statistic and its critical values.
Method. The four Dixon ratios are selected by sample size exactly as Dixon specified, and both ends of the sorted sample are computed so the larger can be tested and the smaller reported alongside it. The null distribution is not a table: it is simulated under normality with a fixed seed and 20,000 draws, memoised per sample size, so the same input always returns the same p-value and the smallest p it can report is about 5×10⁻⁵. That simulation reproduces the published two-sided critical values at n = 3 to 7 to within 0.006, which the suite asserts against 0.970, 0.829, 0.710, 0.625 and 0.568 — and which also fixes the convention, since the one-tailed values differ by up to 0.03. The suite further asserts that Q is unchanged by shifting and rescaling the data, that negating every value swaps which end is on trial while leaving Q identical, that Q always lies between 0 and 1, and that a constant sample returns no result rather than a division by zero. That engine is verified on every change against 157 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Grubbs' TestIterated, with a masking probe: one value at 14.5 is caught at p = 3.09e-8 while a PAIR at 16 and 16.5 — both further out — is not caught at all.
OutlierRuns four outlier rules together and says when each is failing — the 3-sigma rule cannot flag anything below n=11, two outliers hide each other, and the quartile convention alone can decide the answer.
Normality TestShapiro-Wilk, Anderson-Darling and Jarque-Bera with a Q-Q plot, plus a resampled sweep answering the question the tests cannot: was your sample size big enough to detect anything?
Trimmed MeanTrimmed and Winsorized means at any fraction, plotted across the full sweep — because a 0% trim is the arithmetic mean and a 50% trim is exactly the median, so the trim fraction is a dial between them.
Z-ScoreA z-score from your data or from a known mean and SD — with the normal-table percentile checked against the share of your data that actually falls below it, and a warning when they disagree.
Benford's LawLeading-digit frequencies against Benford's law with MAD conformity bands, a chi-square test and per-digit z scores.
An educational tool. Dixon’s Q tests exactly one value, assumes the rest of the data is normal — an assumption that cannot be checked at the sample sizes it is used at — and fails silently when two outliers sit near each other. A significant result is a reason to investigate a measurement, not a licence to delete it.
Launched Dixon Q with all four ratios selected by sample size and both ends of the sample computed.
Replaced the usual three-row table with a simulated null, giving an actual p-value; the simulated 95% points match the published ones to 0.006 at n = 3 to 7.
Used that agreement to fix the convention, since the published values match the two-sided form and not the one-tailed one.
Shipped two presets differing by a single reading where p moves from 0.00045 to 0.96700 — masking, demonstrated rather than described.
Printed the opposite-end statistic so a two-outlier sample is visible before the result is trusted.
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