Lab-Tacular

Outlier Test

Tests whether the most extreme value in a set of replicates is an outlier.Paste your values and pick a test.

One per line, or paste a column from a spreadsheet.

Tests whichever value sits furthest from the mean, in either direction.

How often a value this extreme would be flagged by chance alone.

The formulas this test uses

Test statistic
Test statistic — Grubbs — either endG=max |xi − x̄|s
Critical value
Critical value — Grubbs — either endGcrit=(n − 1)√n·t2α/2n, n−2n − 2 + t2α/2n, n−2

x̄ is the mean and s the sample standard deviation (n−1). t is the value whose upper-tail area under the t distribution with n−2 degrees of freedom is α/2n.

How to use

  1. Decide which test and which significance level you are using, and write it into the protocol — before you look at the numbers.
  2. Paste the replicate measurements, one per line.
  3. Pick the test. Grubbs works at any count; Dixon's Q is for three to ten values.
  4. Read the statistic against the critical value. The p value, where there is one, says how unusual that statistic is.

💡 Exponents can be entered with e — for example, 1.5×10⁻⁵ is entered as 1.5e-5.

Formulas and cautions

Both tests compare one extreme value against the spread of the rest, and both assume the remaining values are roughly normal.

Grubbs

The distance from the mean, in standard deviations. Its critical value comes from the t distribution, so it is exact at any count and any level rather than read off a printed row.

Two-sided
G = max|xi − x̄| / s
Smallest
G = (x̄ − xmin) / s
Largest
G = (xmax − x̄) / s

Dixon's Q

The gap between the suspect value and its neighbour, over the full range. Critical values are read from a published table, which is why the count is limited and why no p value is shown.

Q = gap / range

Cautions

  • A test result is not permission to delete data. Decide the rejection rule before the experiment, apply it once, and report what you did either way.
  • An outlier is often the finding. A contaminated well and a real effect produce the same shaped number, and only the bench knows which happened.
  • Both tests are for one suspect value. Running them repeatedly on the same set, or removing a value and testing again, inflates the error rate well past the level you chose.
  • Both assume the rest of the data is roughly normal. On strongly skewed data they flag the tail rather than anything unusual.

FAQs

References

Where the critical values come from. The test statistics themselves (G = |x − mean| / SD, Q = gap / range) are definitions and are not cited.

  1. NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.17.1 Grubbs' Test for Outliers. National Institute of Standards and Technology.https://www.itl.nist.gov/div898/handbook/eda/section3/eda35h1.htmThe expression that builds Grubbs' critical value from the t distribution, the ten critical values (n = 3–20, α = 0.05) used as reference values, and the worked example (eight mass measurements, G = 2.4687)
  2. Rorabacher DB. (1991) Statistical treatment for rejection of deviant values: critical values of Dixon's "Q" parameter and related subrange ratios at the 95% confidence level. Anal Chem 63(2):139–146.doi:10.1021/ac00002a010The 24 Dixon Q critical values (n = 3–10; α = 0.10, 0.05, 0.01). The paper is paywalled, so the numbers were read from two independent sources that cite it (the LibreTexts analytical chemistry appendix and Wikipedia); all 24 agreed to three decimals

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