Statistics

Chebyshev's Theorem Calculator

Find the minimum fraction of data within k standard deviations.

Calculated Output

At least within k SD (%) 75

At most outside k SD (%) 25

Formula: fraction ≥ 1 − 1/k²

Step-by-Step Breakdown

1. Values used:

2. Number of standard deviations (k) = 2

4. At least within k SD = 75%

5. At most outside k SD = 25%

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The math behind it

At least 1 − 1/k² of values fall within k standard deviations of the mean, regardless of shape.

k = 2 → at least 75% within 2 SD.

How does it compare to the empirical rule?

Chebyshev is weaker but works for any distribution, not just normal ones.