Chebyshevs theorem (Chebyshev’s inequality) stated precisely
In descriptive statistics, chebyshevs commonly refers to Chebyshev’s theorem (also called Chebyshev’s inequality), which gives a guaranteed minimum proportion of observations within a certain number of standard deviations from the mean.
For any distribution with finite mean \(\mu\) and standard deviation \(\sigma>0\), and for any \(k>1\),
\[
P\!\left(\,|X-\mu|
The bound does not assume normality; it applies to any distribution with finite variance.
Minimum proportion within 2 and 3 standard deviations
Apply the formula \(1-\frac{1}{k^2}\) for the requested \(k\) values.
Case \(k=2\)
\[ P\!\left(|X-\mu|<2\sigma\right)\ge 1-\frac{1}{2^2} =1-\frac{1}{4} =\frac{3}{4} =0.75. \]
At least 75% of observations lie within \(2\sigma\) of the mean.
Case \(k=3\)
\[ P\!\left(|X-\mu|<3\sigma\right)\ge 1-\frac{1}{3^2} =1-\frac{1}{9} =\frac{8}{9} \approx 0.8889. \]
At least \(\frac{8}{9}\approx 88.89\%\) of observations lie within \(3\sigma\) of the mean.
Finding \(k\) to guarantee at least 90% within \(k\sigma\)
Require the Chebyshev bound to be at least 0.90: \[ 1-\frac{1}{k^2}\ge 0.90. \]
Solve step-by-step: \[ 1-0.90 \ge \frac{1}{k^2} \quad\Longrightarrow\quad 0.10 \ge \frac{1}{k^2} \quad\Longrightarrow\quad k^2 \ge 10 \quad\Longrightarrow\quad k \ge \sqrt{10}\approx 3.162. \]
A value of \(k=\sqrt{10}\) is the smallest threshold that guarantees at least 90% of observations within \(k\sigma\) by Chebyshev’s theorem.
Quick reference table
| \(k\) | Minimum proportion within \(k\sigma\) | Minimum percent |
|---|---|---|
| 2 | \(\displaystyle 1-\frac{1}{2^2}=\frac{3}{4}\) | 75% |
| 3 | \(\displaystyle 1-\frac{1}{3^2}=\frac{8}{9}\) | 88.89% |
| \(\sqrt{10}\approx 3.162\) | \(\displaystyle 1-\frac{1}{10}=\frac{9}{10}\) | 90% |
| 4 | \(\displaystyle 1-\frac{1}{16}=\frac{15}{16}\) | 93.75% |
Visualization: the guaranteed interval \(\mu \pm k\sigma\)
The interval from \(\mu-k\sigma\) to \(\mu+k\sigma\) always contains at least \(1-\frac{1}{k^2}\) of the distribution (for \(k>1\)). The shaded center segment represents the guaranteed proportion.
Interpretation and a common comparison
- Chebyshev’s theorem provides a lower bound; the true proportion within \(k\sigma\) is often larger.
- For approximately normal data, the empirical rule (about 95% within \(2\sigma\)) is typically much stronger than the Chebyshev guarantee (75% within \(2\sigma\)).
- When distribution shape is unknown or skewed, Chebyshev’s bound remains valid and conservative.