How this calculation works
The bound treats both tails together and applies without assuming a normal distribution.
THE FORMULA
P(|X−μ| ≥ d) ≤ min(1,σ²/d²)A worked example
With σ = 4 and distance 10, the bound is 16/100 = 0.16.
Good to know
Use a distribution with finite variance.
Reference: OpenStax Introductory Statistics (opens in a new tab) ↗
Read the result
Bound the fraction that can lie at least a given distance from the mean using the standard deviation.
The primary result is labeled “Outside probability upper bound”. Keep the input definitions in view when comparing this result with another method.
Try a related calculation
Use a distribution with finite variance.
For a related exercise, try the statistics calculator.
A few questions, answered
How can I check the default inputs by hand?
With σ = 4 and distance 10, the bound is 16/100 = 0.16.
Which rule does this calculator apply?
P(|X−μ| ≥ d) ≤ min(1,σ²/d²)