Grubbs test
G = 4.657 · Critical value 2.802 · p = 7.6 × 10⁻²⁰ · two-sided · α = 0.05
Flagged values: 28.95
The normality assumption may be questionable; an outlier itself can lower W.
Outlier Calculator
FigranPaste one group or a whole table to run Grubbs, Dixon Q, Tukey IQR and z-score side by side.
- Six tests on one dataset
- Several groups at once
- Figure, exact p and methods text
Data
One value per line, or a table with one group per column.
1 group · 24 values
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Grubbs
Options
28.95 is an outlier
Grubbs G = 4.657 · Critical value 2.802 · p < 0.001 · n = 24 · two-sided
Flagged by test: Grubbs 1 · ESD 2 · Dixon n/a · IQR 2 · Modified z 2 · z 1
Results
Flagged values
| Value | z | Modified z | Flagged by |
|---|---|---|---|
| 5.28 | 0.1887 | 3.601 | Generalized ESD, Tukey IQR, Modified z-score (MAD) |
| 28.95 | 4.657 | 48.57 | Grubbs test, Generalized ESD, Tukey IQR, Modified z-score (MAD), z-score |
Copper (ppm) · n = 24
Shapiro–Wilk normality reference: W = 0.3115 · p < 0.001. Outliers themselves can reduce W.
Generalized ESD
Flagged values: 28.95, 5.28
No exact p value; compare Rᵢ with λᵢ.
Generalized ESD critical values are approximate, especially for samples smaller than 25.
Show steps
| i | Removed | Ri | λi | Exceeds critical value |
|---|---|---|---|---|
| 1 | 28.95 | 4.657 | 2.802 | Yes |
| 2 | 5.28 | 3.016 | 2.780 | Yes |
| 3 | 2.2 | 1.724 | 2.758 | No |
| 4 | 2.2 | 1.910 | 2.734 | No |
| 5 | 2.4 | 1.741 | 2.708 | No |
| 6 | 2.4 | 1.956 | 2.681 | No |
| 7 | 2.5 | 2.023 | 2.652 | No |
| 8 | 2.7 | 1.835 | 2.620 | No |
| 9 | 2.8 | 1.828 | 2.586 | No |
| 10 | 2.9 | 1.810 | 2.548 | No |
Dixon Q test
Dixon’s Q test is tabulated for n = 3 to 10; use Grubbs’ test or the generalized ESD instead
Tukey IQR
Q1 = 2.750 · Q3 = 3.700 · Lower fence = 1.325 · Upper fence = 5.125 · k = 1.5
Flagged values: 5.28, 28.95
Modified z-score (MAD)
MAD = 0.3550 · Modified z threshold = 3.5
Flagged values: 5.28, 28.95
z-score
Largest possible |z| = 4.695 · z threshold = 3
Flagged values: 28.95
Assumes approximately normal observations apart from outliers. Rule of thumb, not a hypothesis test.
Report
Methods Selected screening methods: Grubbs test (two-sided; α = 0.05); Generalized ESD (α = 0.05; r = Auto); Dixon Q test (r10; α = 0.05); Tukey IQR (Exclusive; k = 1.5); Modified z-score (MAD) (Modified z threshold = 3.5); z-score (z threshold = 3). No values were removed. Compare conclusions with all observations and with flagged observations excluded; investigate exclusions and report both analyses. Dixon Q test was not applied to Copper (ppm): n = 24 is outside its tabulated range (3–10).
Results Copper (ppm), n = 24: Grubbs test (G = 4.657, p < 0.001) and z-score flagged 28.95; Generalized ESD, Tukey IQR (Lower fence = 1.325, Upper fence = 5.125), and Modified z-score (MAD) flagged 5.28 and 28.95; Dixon Q test could not be applied. Compare results with all observations retained and with flagged values excluded; no data were removed automatically.
Outlier calculator examples
Explore different kinds of measurements and compare methods.
Copper in flour: one determination far from the rest
Compare six methods on 24 copper determinations.
Mass spectrometer readings: the textbook Grubbs example
A small sample shows why a z-score threshold of 3 can miss an outlier.
Three outliers at one end: the generalized ESD
A later ESD step reveals three values even when the first step does not exceed its critical value.
Replicate titration: the classic Q test says keep it
Compare the largest gap with the tabulated Q10 critical value.
Two high values hide each other (masking)
Compare a single-value test with generalized ESD.
Five runs of one experiment, screened separately
Analyze five experiments independently from one table.
Six replicate injections: no outlier, and why
A small, consistent set of chromatographic peak areas.
Plant growth by treatment: three groups
Screen each treatment group separately.
How to test for outliers
Screen your measurements in four steps.
1Paste your data
Enter one group or a table of groups.
2Choose tests
Choose the tests and significance level.
3Read the results
Compare flagged values and applicability.
4Export your work
Download the figure or copy the methods.
Grubbs test and Dixon Q test
| Test | How many | Sample size | Assumption | p value | Typical use |
|---|---|---|---|---|---|
| Grubbs test | One | ≥ 3 | Approximate normality | Yes | One suspected value |
| Generalized ESD | Several | ≥ 5 | Approximate normality | No | Multiple suspected values |
| Dixon Q test | One | 3–10 | Approximate normality | Table only | One suspected value |
| Tukey IQR | Any number | ≥ 4 | Descriptive rule | No | Distribution screening |
| Modified z-score (MAD) | Any number | ≥ 3 | Descriptive rule | No | Distribution screening |
| z-score | Any number | ≥ 3 | Descriptive rule | No | Distribution screening |
In analytical chemistry, Dixon’s Q test compares an extreme gap with the range. Grubbs uses the mean and sample standard deviation; generalized ESD can screen for several outliers.
Why the z-score fails on small samples
For n observations the largest possible absolute sample z-score is (n−1)/√n. At n = 8 it is 2.475, below the common threshold of 3.
Masking: two outliers hide each other
Two nearby extreme values can mask one another. Generalized ESD accounts for the possibility of multiple outliers.
One-sided or two-sided
The one-sided Grubbs test uses α/n in its critical value; the two-sided test uses α/(2n). For n = 8 and α = 0.05 these give 2.032 and 2.127.
References: NIST/SEMATECH 1.3.5.17 · Grubbs (1969) · Dixon (1951) · Rosner (1983) · Rorabacher (1991)
Outlier formulas and critical values
Grubbs test
G = max |xᵢ − x̄| / s
Gcrit = (n − 1)/√n · √[t² / (n − 2 + t²)]
p = min(1, 2n · P(T ≥ √[k(n − 2)/(1 − k)])), k = G²n/(n − 1)²
Dixon Q test
Qhigh = (xₙ − xₙ₋₁)/(xₙ − x₁)
Qlow = (x₂ − x₁)/(xₙ − x₁)
Tukey IQR
Q1 − k · IQR, Q3 + k · IQR
Modified z-score (MAD) / z-score
zᵢ = (xᵢ − x̄)/s
Mᵢ = 0.6745 · (xᵢ − median)/MAD
For a two-sided test, t uses the upper α/(2n) tail with n − 2 degrees of freedom. A one-sided test uses α/n and replaces 2n by n in the p formula.
For the copper example, G = 4.657 exceeds the two-sided critical value 2.802 at α = 0.05.
Critical value tables
| n | Two-sided α = 0.05 | Highest value α = 0.05 |
|---|---|---|
| 3 | 1.1543 | 1.1531 |
| 5 | 1.7150 | 1.6714 |
| 10 | 2.2900 | 2.1761 |
| 20 | 2.7082 | 2.5566 |
| 30 | 2.9085 | 2.7451 |
| 50 | 3.1282 | 2.9570 |
| 100 | 3.3841 | 3.2095 |
| n / α | 0.10 | 0.05 | 0.04 | 0.02 | 0.01 |
|---|---|---|---|---|---|
| 3 | 0.941 | 0.970 | 0.976 | 0.988 | 0.994 |
| 4 | 0.765 | 0.829 | 0.846 | 0.889 | 0.926 |
| 5 | 0.642 | 0.710 | 0.729 | 0.780 | 0.821 |
| 6 | 0.560 | 0.625 | 0.644 | 0.698 | 0.740 |
| 7 | 0.507 | 0.568 | 0.586 | 0.637 | 0.680 |
| 8 | 0.468 | 0.526 | 0.543 | 0.590 | 0.634 |
| 9 | 0.437 | 0.493 | 0.510 | 0.555 | 0.598 |
| 10 | 0.412 | 0.466 | 0.483 | 0.527 | 0.568 |
No p value is calculated from the Dixon table.
References: Harvey, Dixon Q10 · Tukey (1977) · Iglewicz & Hoaglin (1993)
What to do with a flagged outlier
Investigate an assignable cause
Review records, instrumentation and the sample; repeat the measurement if appropriate.
Do not silently delete values
Report results with and without the observation when relevant.
Document the decision
Record the test, side, α, value and reason for any exclusion.
Do not remove a point merely to make a test significant. Recalculate precision after any exclusion. t-test calculator · RSD calculator
References: NIST/SEMATECH · ASTM E178-21
Outlier calculator FAQ
Which outlier test should I use?
Use Grubbs for one suspected outlier and generalized ESD for several; Dixon Q is tabulated here for 3–10 observations. Tukey IQR and modified z-scores are descriptive rules.
Is the z-score useful for a small sample?
A threshold of 3 cannot flag any observation when n ≤ 10: the largest possible absolute sample z-score is (n−1)/√n.
What is the difference between Grubbs and Dixon Q?
Grubbs uses the sample mean and standard deviation and provides a p value. Dixon Q10 uses an extreme gap and a tabulated critical value for a single suspected outlier.
Why do critical values differ?
One-sided and two-sided Grubbs tests use different tail probabilities. For n = 8 at α = 0.05 they are approximately 2.032 and 2.127, respectively.
Can I test for more than one outlier?
Yes. Generalized ESD evaluates up to a chosen maximum and uses the last significant step to determine how many observations to flag.
Should I delete an outlier?
Do not delete a value solely because a test flags it. Investigate an assignable cause and report the effect of retaining or excluding it. To compare relative precision, use the RSD calculator.
How does this relate to a box plot?
The Tukey rule flags observations outside the selected IQR fences. The box plot maker uses the same quartile methods. box plot maker.
How do I run these tests in Excel or R?
In R, outliers::grubbs.test and EnvStats::rosnerTest provide related tests. In Excel, calculate the statistic and compare it with an appropriate critical value.
Does my data have to be normally distributed?
Grubbs, generalized ESD and Dixon Q assume the non-outlying observations are approximately normal. An outlier itself can affect a normality test.
Check your replicates before you report them
Compare the evidence and document your decisions.
Test for outliers