ML Standard Deviation

ML Dispersion: Standard Deviation & Variance

Standard deviation ($\sigma$) and variance ($\sigma^2$) quantify the spread or dispersion of data points around their arithmetic mean.

Mathematical Formulae

• Variance ($\sigma^2$): $\sigma^2 = rac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2$
• Standard Deviation ($\sigma$): $\sigma = \sqrt{\sigma^2}$

Practical Code Example

import numpy as np
from typing import Tuple

def calculate_dispersion(data: np.ndarray, ddof: int = 0) -> Tuple[float, float]:
    """Calculate population (ddof=0) or sample (ddof=1) variance and standard deviation."""
    variance = float(np.var(data, ddof=ddof))
    std_dev = float(np.std(data, ddof=ddof))
    return variance, std_dev

if __name__ == "__main__":
    # Dataset A: Low Variance / Tight Cluster
    group_a = np.array([49, 50, 51, 50, 50])
    # Dataset B: High Variance / Widespread
    group_b = np.array([10, 30, 50, 70, 90])

    var_a, std_a = calculate_dispersion(group_a)
    var_b, std_b = calculate_dispersion(group_b)

    print(f"Group A -> Mean: {np.mean(group_a)}, Std Dev: {std_a:.2f}")
    print(f"Group B -> Mean: {np.mean(group_b)}, Std Dev: {std_b:.2f}")

Best Practices & Gotchas

• Bessel's Correction (ddof=1): Use ddof=1 when calculating sample standard deviation from a subset of a population. NumPy defaults to ddof=0 (population).
• Units Match Data: Standard deviation is in the same physical units as the original data, whereas variance is in squared units.
• 68-95-99.7 Rule: For normally distributed data, ~68% of samples lie within $1\sigma$, ~95% within $2\sigma$, and ~99.7% within $3\sigma$.

Self-Check Challenge

Compute the sample standard deviation (ddof=1) of [2, 4, 4, 4, 5, 5, 7, 9] using np.std().