How do range, variance, standard deviation, and MAD relate to each other?

Question

AcadiFi · Accepted Answer

They all measure dispersion but differ in how they treat outliers and units.

- Range = max - min. Simple but unstable — a single outlier dominates it.
- Mean Absolute Deviation (MAD) = (1/n) Σ|xᵢ - x̄|. Treats positive and negative deviations equally in a linear way.
- Variance (σ²) = (1/n) Σ(xᵢ - x̄)². Squares deviations, so large errors are penalized disproportionately.
- Standard Deviation (σ) = √variance. Same units as original data.

For returns {-3%, -1%, 2%, 4%, 8%} with mean 2%, the MAD is 2.8%, variance is 15.2%², and standard deviation is ~3.9%.

Why the difference matters: variance and standard deviation are used in CAPM, VaR, Sharpe ratio, and option pricing because they plug directly into normal-distribution mathematics and least-squares regression. MAD is more robust to fat tails but doesn't have the same analytical tractability.

```mermaid
flowchart TB
    A[Raw Returns] --> B[Deviation from mean]
    B --> C[Square Variance]
    B --> D[Absolute Value MAD]
    C --> E[Root Std Dev]
```

On the CFA exam, if asked about 'risk' in a normal-distribution context, use standard deviation. If asked about robustness to outliers or non-normal data, MAD may be more appropriate. Also remember that variance has squared units (%-squared, $-squared), which is why standard deviation is usually reported.

How do range, variance, standard deviation, and MAD relate to each other?

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