Free · Probability and statistical models

Chebyshev Tail Probability Bound

Bound the chance of a deviation from the mean using finite variance.

  • Formula & worked example
  • Private in your browser
  • No signup
Go to tool

Calculator inputs

Loading calculator…

How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Variance 4 and a deviation of 4 give a tail-probability upper bound of 0.25.

P(|X − μ| ≥ d) ≤ min(1, σ²/d²)

Worked example

Enter these known values and leave the other values blank.

Bound (k)
4
Variance (σ²)
4
Divergence
4
Probability
0.25

Assumptions and limitations

  • The distribution must have finite mean and variance.
  • This is an upper bound, not the observed probability. k must be positive and variance nonnegative.
  • The alternate mode supplies deviation in standard-deviation multiples.
  • The standard-deviation-multiple mode requires positive variance. Zero variance is supported only with a positive deviation in measurement units.

Common questions

What should I check before using this result?

The distribution must have finite mean and variance.

References

Enforce the mathematical input domain and include valid zero boundaries while preserving the captured arithmetic and control graph.

Explore probability and statistical models or browse all statistics calculators.