Breakdown Points: Which Summaries Survive Contamination

Move a contaminated fraction anywhere on the line and watch the mean chase it while the median refuses.

Ch 2 2.4

Explorer

Foundations & Data Quality

14 mindifficulty 2/5

Controls

Contaminated shareε4%

5 of 120 observations replaced.

Where the bad values sit40

Clean data is centred at 10. Drag this far away and watch which summaries follow.

Trim fraction10%

Each tail dropped before averaging.

Clean spreadσ1

Sample sizen120

Show breakdown curves

Estimate as a function of contamination share.

Random seed

Every result on this page is a deterministic function of the seed and the controls.

The sample

Clean data centred at 10, with a contaminated fraction moved to one location.

clean drawscontaminatedmeanmedian10% trimmed

Location estimators

Each curve as contamination grows from 0 to 50%.

meanmediantrimmed mean

Scale estimators

SD versus MAD and IQR under the same contamination.

SDMAD (×1.4826)IQR

Mean shift

1.24

clean 10.05 → 11.29

Median shift

0.12

clean 9.96 → 10.08

Trimmed-mean shift

0.07

breaks down past ε = 10%

6.09

clean 1.01

MAD

1.13

clean 1.12

SD / MAD

5.40

≈ 1 for clean normal data

Predict before you look

A sample of 120 clean measurements gets one value replaced by a number ten standard deviations away. Which is true?

Scenarios

Teacher-ready starting points.

Documentation

What this shows

A clean normal sample with a controllable fraction of its values replaced by a single arbitrary number. Every classical and robust summary is recomputed live, and the breakdown curves trace each estimator as contamination grows from 0 to 50%.

Breakdown point and influence

The finite-sample breakdown point of an estimator is the smallest fraction of observations that, moved arbitrarily far, can drive the estimate arbitrarily far. The mean's breakdown point is 1/n — one observation is enough. The median's is 1/2, the highest any translation-equivariant location estimator can achieve.

A trimmed mean interpolates: trimming a fraction α from each tail buys a breakdown point of exactly α, which is why the trimmed curve is flat until ε exceeds the trim setting and then bends sharply.

For scale, the sample SD has breakdown 1/n as well, because a single squared deviation is unbounded. The MAD, scaled by 1.4826, is consistent for σ under normality and has breakdown 1/2, so the ratio SD/MAD works as a contamination diagnostic.

ε*(T) = min{ m/n : sup_{x'} |T(x') − T(x)| = ∞ }

ε*(mean) = 1/n,  ε*(median) = 1/2,  ε*(trim_α) = α

MAD = 1.4826 · medᵢ |xᵢ − med(x)|

Try this

Set contamination to 1% and drag the outlier location from 10 to 140: record the mean shift and the median shift.
Set the trim fraction to 10% and contamination to 8%, then to 12%. Explain the sudden change in the trimmed mean.
Watch SD/MAD as contamination rises. At what ε does the ratio exceed 1.5?
Put the contaminated values at 9.5 instead of 40 — is contamination always detectable?
Raise n from 20 to 400 with ε fixed. Does more data protect the mean?

Common misconception

“If I have plenty of data, a few strange values cannot really matter.”

Sample size does not protect a non-robust estimator. The mean is a linear functional of the data, so one observation at distance d shifts it by d/n — and an adversarial or instrument-error value can have unbounded d. Robustness is a property of the estimator, not of the sample size. Conversely, trimming is not free: a 10% trimmed mean breaks down at exactly 10% contamination.

DSPA3 Chapter 2 — Data quality and robust summaries