How does Chebyshev's inequality work, and when should I use it instead of the empirical rule?

Chebyshev's inequality is a distribution-free bound that guarantees a minimum proportion of observations within k standard deviations of the mean for any distribution with a finite variance. The formula is:

P(|X - μ| < kσ) ≥ 1 - 1/k² for k > 1

The empirical rule (68-95-99.7) is tighter but only valid for normal distributions. Chebyshev is your safety net when you cannot assume normality.

Worked Example — Oakridge Small-Cap Fund

Suppose the Oakridge Small-Cap Fund has a mean annual return of 11.4% and a standard deviation of 7.8%. The return distribution is right-skewed (not normal) because small-cap stocks occasionally deliver outsized gains.

Question: What is the minimum proportion of annual returns falling within two standard deviations of the mean?

Using Chebyshev's inequality with k = 2:

P(|R - 11.4%| < 2 × 7.8%) ≥ 1 - 1/2² = 1 - 0.25 = 75%

So at least 75% of returns lie between -4.2% and 27.0%.

If the distribution were normal, the empirical rule would say roughly 95.4% of observations fall within two standard deviations — a much tighter estimate. But since the fund's returns are skewed, we cannot rely on that figure, and Chebyshev gives us the conservative floor.

Key Comparison Table:

Exam Tip: If a vignette describes a non-normal distribution (or says nothing about the shape), default to Chebyshev. If the question explicitly states normality, the empirical rule gives a better answer. The CFA Level I exam loves testing this distinction.

For more practice on quantitative methods, explore our CFA Level I question bank.