Grubbs' Test 2D: Dot-Plot Outlier Rejection

Sample

Sample size N 30

Injected outliers 2

Significance level α

Grubbs' Test

G statistic

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G critical

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Remaining N

Removed

Mean x̄

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Std dev s

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Press Step Test to evaluate the most extreme remaining point.

How it works

This is a 2D dot-plot rendition of Grubbs' test: instead of a number line in 3D space, each sample value is binned and stacked as a column of dots (a Wilkinson dot plot), and a second panel traces the test statistic across iterations as the algorithm runs. For each remaining point x_i, compute:

G = max|x_i − x̄| / s

Two-sided critical value:
G_crit = (N-1)/√N · √( t²_(α/2N, N-2) / (N-2 + t²_(α/2N, N-2)) )

where x̄ and s are the sample mean and standard deviation, N is the current remaining sample size, and t_{(α/2N, N-2)} is the upper-α/2N critical value of Student's t-distribution with N−2 degrees of freedom (found here via a Cornish–Fisher expansion of the normal quantile — the same closed-form approximation textbooks use in place of a t-table).

If G > G_crit, the most extreme point is a statistically significant outlier at level α — it drops out of the dot-plot into the rejection tray below the axis, and the G-trace chart logs the iteration as a rejection.
If G ≤ G_crit, no further points are significant and the procedure stops — the G-trace chart's final point marks the stopping iteration.
The critical value shrinks as α grows and as N grows, so watch the G_crit line in the trace chart step downward each time a point is removed and N shrinks.

Real-world relevance: this exact test is used to vet single extreme readings in lab instrumentation, quality-control measurements, and small sensor batches before a mean or control limit is computed from them — anywhere a single erroneous reading could otherwise distort a small dataset.

Normal point Candidate (max |z|) Rejected outlier

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