Outlier Tests

Grubbs' Test

Identifies a single outlier in normally-distributed data.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > Outlier Test (Grubbs)

Description

Grubbs' test (a.k.a. maximum normed residual test) checks whether the value furthest from the mean is compatible with a normal distribution. The G statistic is the maximum standardised deviation; the critical value is derived from the t-distribution with n-2 degrees of freedom under a Bonferroni correction. Suitable for n ≥ 3, but should be applied only cautiously when iterated — for multiple suspected outliers prefer Generalized ESD.

Formulas

G = \frac{max _{i} ∣ x _{i} - x ˉ ∣}{s}

Test statistic (two-sided)

G_{crit}

G_{cr i t} = \frac{n - 1}{n} \frac{t _{α / (2 n), n - 2}^{2}}{n - 2 + t _{α / (2 n), n - 2}^{2}}

Critical value from the t-distribution (Bonferroni)

H₀

H_{0} : kein Ausrei ß er vs. H_{1} : ein Ausrei ß er

Null and alternative hypotheses

Assumptions

Data are independent and (apart from the suspected outlier) normally distributed
Sample size: n ≥ 3
Continuous data

Limitations

At most one outlier per call
Sensitive to deviations from normality — pre-test with e.g. Shapiro-Wilk
Masking is possible with multiple true outliers → use Generalized ESD

References

Grubbs, F. E. (1969), Procedures for detecting outlying observations in samples, Technometrics 11(1), 1–21
Stefansky, W. (1972), Rejecting outliers in factorial designs, Technometrics 14(2), 469–479
NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.17