Find Chebyshev’s distribution-free coverage bound or the interval width required for a target, using a population mean and standard deviation.
How to use this tool
- Choose the event or model and review the assumptions below.
- Enter the labeled inputs. Probability inputs are percentages from 0 to 100, not decimal fractions.
- Select Calculate and read the labeled outputs.
- Change an input and calculate again; Reset restores the example inputs.
Calculation method
P(|X−μ| < kσ) ≥ 1−1/k² for k > 1; required k = 1/√(1−coverage).
Assumptions and limits
Finite population mean and variance, σ > 0. This is a conservative probability bound, not a normal-distribution estimate, confidence interval or guarantee about an unseen finite sample.
Worked example
For mean 100, standard deviation 15 and k = 2, at least 75% of the probability is in the open interval (70,130). A target lower bound of 95% instead requires k = √20, approximately 4.4721, giving (32.9180,167.0820). These are conservative population bounds. The tool does not use a normal-distribution approximation or estimate a confidence interval for the mean.
Worked reading and practical questions
- Chebyshev vs the Empirical Rule: Which Coverage Can You Claim?
- Build a Distribution-Free Interval from a Target Coverage
Browse Statistics calculators or explore other tools on the GreenCalc homepage.
Frequently asked questions
What does this calculator assume?
Finite population mean and variance, σ > 0. This is a conservative probability bound, not a normal-distribution estimate, confidence interval or guarantee about an unseen finite sample.
How do I enter and update the inputs?
Use the labeled fields, then select Calculate. Percent fields use 0 to 100. After editing, calculate again; Reset restores the example inputs.