A ruler spanning two boundary markers with a central green marker. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Start from the guaranteed fraction
To obtain a Chebyshev lower bound c, solve c = 1−1/k² for k. The result is k = 1/√(1−c), with c strictly between zero and one. The corresponding interval is μ ± kσ. This constructs a conservative population probability interval from a finite mean and variance; it is not a confidence interval for an unknown mean.
Build a 95% bound numerically
Let the population mean be 100 and standard deviation 15. For c = 95%, k = √20, approximately 4.47213595. The half-width is about 67.0820393, giving the open interval (32.9179607, 167.0820393). Under the model, at least 95% of the probability lies strictly inside that interval. Using 100 ± 2×15 would only guarantee 75% by Chebyshev, even though a normal model would yield a different coverage estimate.
Recognize the cost of approaching certainty
At 99% target coverage, k = 10. At 100%, no finite k follows from the generic finite-variance bound. The interval becomes wider as you ask it to cover more probability without additional shape information. A very wide result does not indicate a broken calculator; it shows how little the first two population moments alone can guarantee.
Write the units and assumptions beside the interval
Mean, standard deviation and interval endpoints share the measured variable's unit. Negative lower endpoints may be mathematically valid even for a quantity that is physically nonnegative; an additional support restriction can be reported separately, rather than silently changing the theorem. Use the Chebyshev calculator in target-coverage mode to reproduce the example. Read Chebyshev versus the empirical rule before swapping the distribution-free interval for a normal approximation. If the standard deviation is zero, the variable is degenerate under the population model; the ordinary kσ expression used by this tool is not the appropriate input case.
Check the model behind your result
Use the Chebyshev's Theorem Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26