Coefficient of Variation — Relative Dispersion Across Different Scales

Coefficient of Variation (CV)

$CV = \frac{s}{\bar{x}} \times 100\%$

A dimensionless measure of relative dispersion. It answers: "How large is the standard deviation relative to the mean?"

Investment Risk Comparison

import numpy as np
import pandas as pd

np.random.seed(42)

portfolios = {
    'Tech ETF':    np.random.normal(150, 35, 252),   # high CV
    'Bond Fund':   np.random.normal(105,  6, 252),   # low CV
    'Gold':        np.random.normal(180, 28, 252),   # moderate
}

print(f"{'Asset':<15} {'Mean':>8} {'Std Dev':>10} {'CV%':>8} {'Risk Level':>12}")
print("-"*55)
for name, prices in portfolios.items():
    daily_ret = np.diff(prices)/prices[:-1]*100
    cv = np.std(daily_ret,ddof=1)/np.mean(prices)*100
    mu = np.mean(prices); sd = np.std(daily_ret, ddof=1)
    risk = "High" if cv>15 else ("Moderate" if cv>8 else "Low")
    print(f"{name:<15} {mu:>8.2f} {sd:>10.4f} {cv:>7.2f}% {risk:>12}")

Quality Control: Machine Precision

# Two machines producing 50mm bolts
np.random.seed(1)
machine_a = np.random.normal(50.0, 0.3, 500)   # precise
machine_b = np.random.normal(50.0, 0.9, 500)   # less precise

for name, data in [('Machine A', machine_a), ('Machine B', machine_b)]:
    cv = np.std(data,ddof=1)/np.mean(data)*100
    out_of_spec = np.sum(np.abs(data-50)>1.0)/len(data)*100
    print(f"{name}: CV={cv:.4f}%, Out-of-spec: {out_of_spec:.2f}%")

Limitations

# CV breaks near zero mean
near_zero = np.array([-2, -1, 0, 1, 2])
print(f"Mean={near_zero.mean()}, CV={np.std(near_zero,ddof=1)/near_zero.mean()*100:.2f}% ← nonsense")

Key Takeaways

1. CV = SD/mean × 100% is dimensionless — compare across any units
2. Low CV (<10%)=tight cluster; High CV (>30%)=highly variable
3. Requires ratio scale data with a meaningful non-zero positive mean
4. In finance, CV is equivalent to volatility-to-return ratio
5. In manufacturing, CV benchmarks machine precision regardless of target value
6. Never use CV for data that can be zero or negative