basic

Chebyshev's Inequality Probability Bound Calculator

P(|X − μ| ≥ k·σ) ≤ 1/k².

Compute Chebyshev's inequality bound on the tail probability of any distribution with finite variance: P(|X − μ| ≥ k·σ) ≤ 1/k² (for k > 1). Distribution-free; conservative but universal. For k = 2 it bounds tail prob ≤ 25 %; for k = 3 ≤ 11.1 %; for k = 4 ≤ 6.25 %.

Daniel Oh By Daniel Oh
Published Last reviewed 1 min read

Inputs

Results

Enter values and click Calculate to see results.
Was this helpful?

How to use this calculator

  1. Fill in the inputs above using the units you already have.
  2. Values update automatically as you type — no submit button needed.
  3. Hover any result row for the underlying formula and intermediate values.

Formula

P(|X−μ| ≥ kσ) ≤ 1/k².

In depth

Compute Chebyshev's inequality bound on the tail probability of any distribution with finite variance: P(|X − μ| ≥ k·σ) ≤ 1/k² (for k > 1). Distribution-free; conservative but universal. For k = 2 it bounds tail prob ≤ 25 %; for k = 3 ≤ 11.1 %; for k = 4 ≤ 6.25 %.