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Chebyshev's Inequality

Lower-bound the probability within kσ of the mean and the interval.

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P(|X−μ|
k standard deviations from the mean

📖 Tutorial | Chebyshev's Inequality

1. Definition
Chebyshev's inequality holds for any distribution: P(|X−μ|≥kσ) ≤ 1/k², so the probability within kσ is at least 1−1/k², with no shape assumption.
2. Symbols
SymbolMeaning
μmean
σstandard deviation
knumber of sd from the mean
1−1/k²probability lower bound
3. How it works
  • Holds for any distribution;
  • Interval [μ−kσ, μ+kσ];
  • Probability inside is at least 1−1/k²;
  • More conservative than the normal empirical rule.
4. Steps
  1. Enter μ, σ, k;
  2. Click Calculate;
  3. Read the interval and lower bound;
  4. Remember it is a lower bound, usually smaller than reality.
5. Example
Example: μ=100, σ=15, k=2. Interval (70,130), probability at least 1−1/4=75%.
6. Pitfalls
k must be greater than 1;
This is a conservative bound, not the exact probability;
For a known normal, the 68/95/99.7 rule is tighter.

❓ FAQ | Chebyshev's Inequality

Strength of Chebyshev?
It needs no distributional assumption; it holds for any distribution.
Why is it a “lower bound”?
It guarantees a minimum probability; the true probability is usually larger.
What is k=2?
At least 75%; for normal it is about 95%.
Difference from the empirical rule?
The empirical rule is normal-only and tighter; Chebyshev is universal but conservative.
When to use it?
When the shape is unknown and you only need a safe lower estimate.