Robust Statistics Guide

Step 1 — Should you use robust methods?

Check these triggers:

Trigger	Why it matters
Clear outliers in histogram or boxplot	Mean and SD are pulled toward outliers; unreliable
Distribution is heavily right- or left-skewed	Mean no longer represents the “centre”
Small $n$ with fat-tailed or unknown distribution	Normal-based methods break down
Max SD / Min SD > 2 across groups	Equal variance assumption fails for t-test / ANOVA

Breakdown point: the fraction of contaminated data an estimator can tolerate before it becomes arbitrarily bad. Mean has breakdown point = 0 (one extreme outlier destroys it). Median has breakdown point = 50%.

---

Step 2 — Choosing a Location Estimator

Estimator	Formula	Breakdown point	When to use
Mean	$\bar{X}=\frac{1}{n}\sum X_i$	0%	Data is Normal with no outliers
Median	Middle value	50%	Heavy skew or many outliers
$\gamma$-Trimmed mean	Drop smallest and largest $g=\lfloor \gamma n\rfloor$ values, average the rest	$\gamma$	Moderate outliers; recommended $\gamma \in (0,0.2]$
Winsorised mean	Replace smallest $g$ with $X_{(g+1)}$, largest $g$ with $X_{(n-g)}$, then average	$\gamma$	Same situations as trimmed mean; keeps sample size intact

Trimmed vs Winsorised: trimmed mean removes extreme values; Winsorised replaces them with the nearest kept value. Winsorised variance is used to compute the SE of the trimmed mean — they go together.

Use $\gamma =0.2$ (20% trim) as a sensible default when you suspect outlier contamination but don’t know the exact fraction.

---

Step 3 — Choosing a Scale Estimator

Estimator	Formula	For Normal, converts to $\hat{\sigma }$ by	Breakdown point
SD	$\sqrt{\frac{1}{n-1}\sum (X_i-\bar{X}{)}^2}$	already $\hat{\sigma }$	0%
IQR	$Q_3-Q_1$	$\hat{\sigma }\approx \text{IQR}/1.35$	25%
MAD	$\text{median}(	X_i - \text{median}(X)	)$	$\hat{\sigma }\approx 1.4826\times \text{MAD}$	50%

Rule of thumb: if you’re using a robust location estimator, use a robust scale estimator too. Pair trimmed/Winsorised mean with IQR or MAD.

---

Step 4 — Quick Flowchart

Is your data approximately Normal with no obvious outliers?
│
├─ Yes → Mean + SD
│
└─ No
    ├─ Moderate outliers or mild skew
    │   └─ γ-Trimmed mean (γ ≤ 0.2) + IQR or MAD
    │
    ├─ Heavy skew or many outliers
    │   └─ Median + MAD
    │
    └─ Unknown distribution / can't tell
        └─ Median + MAD (safest default)
            or Bootstrap CI (see Simulation & Bootstrap Guide)

---

Step 5 — Asymptotic Relative Efficiency (ARE)

ARE tells you how many extra observations the non-robust method needs to match the robust one.

• $\text{ARE}(\tilde{\theta };\hat{\theta })={\lim }_{n\to \infty }\frac{\text{Var}(\hat{\theta })}{\text{Var}(\tilde{\theta })}$
• Trimmed mean vs mean on pure Normal: ARE ≈ 88% (you “lose” 12% efficiency for robustness)
• Trimmed mean vs mean on 10%-contaminated Normal: ARE > 100% (robust estimator is better)

The ARE trade-off: robust estimators are slightly less efficient on clean Normal data, but dramatically better once contamination exists. For real data where the distribution is unknown, robustness is usually worth the cost.

---

See also: L5 Robust Statistics · EDA Guide · Simulation & Bootstrap Guide