Statistics for ML #3 — Measures of Dispersion
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Two datasets can have identical means but completely different spreads. Dispersion measures capture how spread out values are around the center.
Range
\(\text{Range} = x_{\max} - x_{\min}\) Simple but highly sensitive to outliers. Only uses two data points.
Variance
\(\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2 \quad \text{(population)}\) \(s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2 \quad \text{(sample, Bessel's correction)}\)
Why n−1? The sample mean already uses the data, so we lose one degree of freedom. This makes s² an unbiased estimator of σ².
Standard Deviation
\(\sigma = \sqrt{\sigma^2}\) Same units as the original data — interpretable. The most widely used dispersion measure.
Empirical Rule (Normal distribution):
- ~68% of data within μ ± 1σ
- ~95% within μ ± 2σ
- ~99.7% within μ ± 3σ
Interquartile Range (IQR)
\(\text{IQR} = Q_3 - Q_1\) Robust to outliers. Used for outlier detection: values beyond Q1 − 1.5×IQR or Q3 + 1.5×IQR are flagged as outliers.
Coefficient of Variation (CV)
\(CV = \frac{\sigma}{\mu} \times 100\%\) Dimensionless — allows comparison across datasets with different units or scales.
Example: Comparing variability of ANC visits (mean=4, SD=1.5) vs. birth weight (mean=3000g, SD=500g):
- CV(ANC) = 37.5%
- CV(birth weight) = 16.7% → ANC visits are relatively more variable.
ML Relevance
| Concept | ML Application |
|---|---|
| Variance | Bias-variance tradeoff; PCA (maximises variance) |
| Standard deviation | Feature standardisation (Z-score normalisation) |
| IQR | Robust scaling: RobustScaler in scikit-learn |
| CV | Comparing feature variability before selection |
from sklearn.preprocessing import StandardScaler, RobustScaler
# Z-score (mean=0, std=1) — sensitive to outliers
scaler = StandardScaler()
# IQR-based — robust to outliers, better for health data
robust = RobustScaler() # subtracts median, divides by IQR
X_scaled = robust.fit_transform(X)
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