Chebyshev Interval Calculator
Apply Chebyshev’s theorem to any distribution — find the minimum share of data within k standard deviations of the mean and the interval bounds.
How it works
within k SD ≥ 1 − 1/k² · interval = μ ± kσ (for k > 1)
Chebyshev’s theorem gives a guaranteed lower bound on how much data lies near the mean, for any distribution whatsoever — no normal-shape assumption needed. At least 1 − 1/k² of the values fall within k standard deviations of the mean. The trade-off for that generality is looseness: the bound is conservative, and real distributions usually pack far more data into the interval than the theorem promises.
Worked example
With mean 100 and SD 15, take k = 2. The interval is 100 ± 30, i.e. 70 to 130, and Chebyshev guarantees at least 1 − 1/2² = 75% of the data falls inside it — regardless of the distribution’s shape.
Sources: Chebyshev’s inequality — Wikipedia
Frequently asked questions
How does this differ from the empirical rule?
The empirical rule (68-95-99.7) applies only to normal distributions and is fairly tight. Chebyshev’s theorem works for every distribution but is looser — at k = 2 it guarantees just 75%, versus the empirical rule’s 95% for normal data.
Why is there no bound for k ≤ 1?
The formula 1 − 1/k² is zero or negative when k ≤ 1, so the theorem simply promises “at least 0%.” It only becomes informative once k exceeds 1, and grows stronger as k rises.
Does Chebyshev give the exact percentage?
No — it gives a minimum. The true share within the interval is usually much higher; the theorem only tells you the worst case that no distribution can beat.
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For education — check the assumptions (sample size, distribution shape) before using statistics in real decisions.