Grubbs Outlier Test

Grubbs Outlier Test (cross-cutting/numerics/grubbs-outlier-test)

Use when the task is deciding whether the single most extreme reading of a normally distributed sample is a spurious outlier: a telemetry channel with one wild sample, a calibration record with one bad measurement, or a test series that needs one suspect value screened before further statistics. This leaf implements the Grubbs test in pure Python, stdlib only (scripts/grubbs_outlier_test_logic.py): it computes the sample mean and the sample standard deviation (n-1 denominator), forms the G statistic as the largest absolute deviation from the mean divided by the standard deviation, and compares G with the two-sided 0.05 critical value for the sample size from an embedded reference table with linear interpolation between listed sizes. It pairs with cross-cutting/numerics/descriptive-statistics for summary measures and non-probabilistic screening of the same data, with cross-cutting/numerics/hypothesis-testing for parametric significance tests on group comparisons, and with rank-based-hypothesis-testing for rank procedures; this leaf tests ONE normal sample for ONE outlier, it is not a screening rule without a probability model, not a group-comparison test, and not an ordering-randomness test of a sequence (see runs-test for that procedure).

Domain quick reference

• Test question: under the null the sample is a single normal draw; a verdict of reject says the most extreme reading is a single outlier at the two-sided 0.05 significance level. Either tail counts, the largest deviation from the mean carries the test.
• Sample statistics: mean x_bar = sum(x) / n; sample standard deviation s = sqrt(sum((x - x_bar)^2) / (n - 1)) with the n-1 denominator (grubbs_statistic).
• G statistic: G = max(|x - x_bar|) / s over all readings. The flagged candidate is the reading achieving the maximum deviation, returned with its index (first occurrence on a tie).
• Embedded critical table (module constant GRUBBS_CRIT_05, two-sided alpha 0.05, listed sample sizes with critical G): n = 3: 1.155, 4: 1.481, 5: 1.715, 6: 1.887, 7: 2.020, 8: 2.032, 9: 2.215, 10: 2.290, 12: 2.412, 15: 2.549, 20: 2.709, 30: 2.908, 40: 3.036, 50: 3.128. Critical G grows with n: larger samples give the maximum deviation more chances under the null.
• Interpolation: an unlisted sample size n in [3, 50] takes the linear interpolation between the nearest listed sizes below and above it, e.g. n = 11 sits halfway between n = 10 and n = 12 and returns 2.351. Only alpha 0.05 is supported; n below 3 or above 50, a fractional n, and any other alpha raise ValueError.
• Verdict rule: reject when G > critical (strictly greater). The result dict carries g, critical, verdict ("reject" or "no-outlier"), rejected_value, rejected_index, mean and std; the rejected fields are None on a no-outlier verdict.
• Iteration: grubbs_remove_outliers removes the flagged value and repeats on the remainder until no outlier remains or fewer than 3 values are left. Each pass uses the critical value for the CURRENT sample size. An all-identical remainder (zero standard deviation) cannot carry an outlier and stops the loop.
• Minimum data: at least 3 values with nonzero spread. The verdict is a normal-model statement; the sample should look normal apart from the single suspect reading.
• NACA TR-824 is named as the numerics-pack reference; the relations above are standard statistical methodology, summary-only.

Workflow

1. Collect the measured sample as a list of numbers with at least 3 values and nonzero spread; confirm the normality assumption apart from the suspect reading.
2. Form the statistic: grubbs_statistic(sample) returns (g, mean, std, candidate, candidate_idx), where candidate is the reading farthest from the mean.
3. Look up the decision threshold: grubbs_critical(len(sample)) from the embedded 0.05 table with linear interpolation.
4. Run the full test: grubbs_test(sample) returns the result dict {g, critical, verdict, rejected_value, rejected_index, mean, std}.
5. Read the verdict next to the critical value: on "reject" quote the flagged reading and index with G and the threshold; on "no-outlier" state that no single outlier is supported at 0.05.
6. For several suspect readings, iterate with grubbs_remove_outliers(sample), which returns the clean list and the removed values, removing one outlier per pass.
7. Confirm the deterministic checks with the contract test scripts/test_grubbs_outlier_test.py.

Worked example

Eight resistance readings in ohms, one of them suspiciously high: [10.2, 10.1, 10.3, 10.0, 9.9, 10.2, 10.1, 12.5], n = 8.

• Sample statistics: mean = 10.4125, sample standard deviation = 0.8526 (spec bound 0.853). The 12.5 reading sits 2.0875 ohms from the mean, far beyond any other deviation (next largest 0.5125).
• G statistic: G = 2.0875 / 0.8526 = 2.4483 (module value 2.4483, spec anchor 2.448 within 0.01), candidate 12.5 at index 7.
• Critical value: grubbs_critical(8) = 2.032 (exact table hit).
• Verdict: 2.4483 > 2.032, so grubbs_test returns verdict "reject" with rejected_value 12.5 and rejected_index 7.
• Removal: grubbs_remove_outliers returns the clean list [10.2, 10.1, 10.3, 10.0, 9.9, 10.2, 10.1] with removed [12.5]; exactly the spurious reading is taken out and the iteration stops.
• Rerun on the 7-value remainder: G = 1.5930 stays below the n = 7 critical value 2.020, verdict "no-outlier": removing the flagged value and rerunning never flags it again.
• Clean control sample [10.0, 10.1, 10.2, 10.1, 10.0, 10.3, 10.2, 10.1]: G = 1.6907 below 2.032, verdict "no-outlier".

Verification

• Confirm grubbs_statistic on the worked sample returns mean 10.4125 within 1e-4, std 0.853 within 1e-3, and G = 2.448 within 0.01 with candidate 12.5 at index 7.
• Confirm grubbs_test verdict is "reject" with rejected_value 12.5, and that the 7-value remainder and the clean control sample both return "no-outlier" with None rejected fields.
• Confirm critical table spot checks: n = 8 -> 2.032, n = 5 -> 1.715, n = 20 -> 2.709, all within 1e-3, plus the boundaries n = 3 -> 1.155 and n = 50 -> 3.128.
• Confirm interpolation: grubbs_critical(11) = 2.351 (midpoint of the n = 10 and n = 12 entries) and grubbs_critical(16) = 2.581, with every interpolated value bounded by its table neighbors and the critical values monotone in n.
• Confirm the identity: a symmetric sample gives the same G whichever side is larger, e.g. [10.0, 11.0, 12.0] and [12.0, 11.0, 10.0] both give G = 1.0.
• Confirm iterative removal: the worked sample drops exactly the 12.5 and stops; a two-pass case [10.0, 10.0, 10.0, 10.0, 10.0, 10.0, 20.0, 50.0] removes 50.0 then 20.0; all-identical samples and an identical remainder left after a removal stop cleanly.
• Confirm ValueError rejection of fewer than 3 values, zero standard deviation, non-numeric values, n outside [3, 50], fractional n, non-integer n, and any alpha other than 0.05.
• Run the contract test offline: python3 scripts/test_grubbs_outlier_test.py (35 tests, deterministic).

Related leaves

• cross-cutting/numerics/descriptive-statistics: summary measures and fence-based suspect flagging without a probability model, the screening layer that hands a suspect reading to this test.
• cross-cutting/numerics/hypothesis-testing: the parametric significance-test family for group comparisons of means and variances, for questions beyond one sample and one outlier.
• cross-cutting/numerics/rank-based-hypothesis-testing: nonparametric two-sample and paired procedures built on ranks, used when the normal assumption does not hold.
• cross-cutting/numerics/runs-test: the ordering-randomness test for a two-sign sequence (wave-36 neighbor), for questions about order rather than extreme values.

Pitfalls

• Quoting G without its critical value: the threshold depends on the sample size (2.032 at n = 8, 2.709 at n = 20), so a fixed rule of thumb has no meaning; always report G against grubbs_critical(n).
• Dividing by a population standard deviation: the n-1 denominator is part of the contract; an n-denominator std shrinks G and can flip a reject into a no-outlier.
• Testing several outliers in one pass: the G test flags ONE extreme reading per run. Removing one outlier changes both the mean and the standard deviation, so the iteration must re-derive the critical value at the current sample size on every pass.
• Extrapolating the table: n below 3 or above 50 has no embedded support and raises ValueError instead of inventing a threshold. Small samples also carry low power, a reject at n = 3 needs G above 1.155 only.
• Forgetting the normal-model assumption: the verdict is a statement about a normal sample with one contaminant. Heavily skewed or multi-modal data should go to descriptive-statistics screening or a rank-based procedure instead.
• Reading the boundary as inclusive: the contract rejects strictly when G > critical; equality keeps the no-outlier verdict.
• Interpolating with the wrong neighbors or a non-0.05 alpha: unlisted sizes interpolate only between the nearest listed sizes inside [3, 50], and the table exists only for two-sided alpha 0.05.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_grubbs_outlier_test.py

The test covers the worked-example anchors (mean 10.4125, std 0.853, G = 2.448 at the 12.5 reading, critical 2.032, verdict "reject"), the clean control and the post-removal remainder giving "no-outlier" with None rejected fields, exact critical-table hits with the n = 8, 5, 20 spot checks, midpoint and general linear interpolation with neighbor-bounding and monotonicity, the symmetric-tie G identity, the iterative removal contract (exactly the 12.5 removed and stopped; the two-pass case; the all-identical guard), determinism across calls, the exact result dict key set, and ValueError rejection of every non-physical input (short, zero-spread, or non-numeric samples; n outside [3, 50]; fractional and non-integer n; alpha other than 0.05).

Compliance

• Standards referenced, not reproduced: NACA TR-824 is named as the numerics-pack reference; the Grubbs test relations and the embedded critical table (module constant GRUBBS_CRIT_05) are standard statistical methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.