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Grubbs' Test Calculator for Outliers

Run Grubbs' outlier test on a data list: the G statistic, the critical G for your α, the suspected outlier and whether to reject it — two-sided or one-sided (max or min).

Data values

Significance level α

Test

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How it works

G = max|xᵢ − x̄| / s (two-sided), (x_max − x̄)/s or (x̄ − x_min)/s (one-sided) · G_crit = ((n − 1)/√n)·√(t²/(n − 2 + t²)), t = t-quantile at α/(2n) (two-sided) or α/n (one-sided) with n − 2 df

Grubbs' test (the maximum normed residual test) checks whether the most extreme value in a normally distributed sample is an outlier. It measures how many sample standard deviations that value lies from the mean and compares G with a critical value derived from the t distribution. If G exceeds the critical value you reject the hypothesis of no outliers. Test only one outlier at a time: remove it and rerun if needed, and confirm the data are roughly normal first.

Worked example

For 8.2, 8.4, 8.3, 8.1, 8.5, 9.9: the mean is 8.566667 and s = 0.668331. The most extreme value is 9.9, so G = (9.9 − 8.566667)/0.668331 = 1.995. With n = 6 and α = 0.05 two-sided the critical G is 1.8871, so 9.9 is a significant outlier.

Sources: NIST/SEMATECH e-Handbook — Grubbs' Test for Outliers · Grubbs's test — Wikipedia

Frequently asked questions

How many data points do I need?

The test can run with 3 values, but it has little power below about 7. The maximum possible G is (n − 1)/√n, so with very small samples even an extreme point may not be flagged.

Can Grubbs’ test find more than one outlier?

It tests one value at a time. You can remove a detected outlier and repeat, but with several outliers they can mask each other; the generalized ESD test is designed for that case.

Two-sided or one-sided?

Use two-sided when you do not know in advance whether the suspect value is high or low. Use one-sided only when you specifically suspect the maximum (or the minimum).

Does the data need to be normal?

Yes. Grubbs’ test assumes the data, apart from the possible outlier, come from a normal distribution. For skewed data use the IQR rule or transform the data first.

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IQR Outlier Detection Calculator

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Standard Deviation Calculator

Paste a list of numbers and get the sample or population standard deviation, variance, mean, standard error, confidence interval, quartiles, z-scores, a frequency table and a histogram, with every step worked out.

Critical Value Calculator

Find critical values for the Z, t, chi-square and F distributions from the significance level α, the tail and the degrees of freedom — with the rejection region for your test.

This calculator is for educational purposes. Double-check important results before acting on them.