Our Chebyshev’s Theorem Calculator will quickly help you determine the proportion of observations that fall within a specified number of standard deviations from the mean in a dataset.

## Chebyshev’s Theorem Calculator

### How to Use the Calculator

Enter a numerical value greater than 1 for *k* into the input box and press “Calculate”. The calculator will then display the percentage of values that lie within k standard deviations of the mean, according to Chebyshev’s Theorem.

### How It Calculates the Result

The calculator uses Chebyshev’s Theorem, which states that no more than (1/k^2) of the distribution’s values can be more than k standard deviations away from the mean. Therefore, at least ((1 – 1/k^2)) of the data values must lie within k standard deviations of the mean. It calculates this minimum percentage by subtracting (1/k^2) from 1, then multiplies by 100 to convert to a percentage.

### Limitations

While Chebyshev’s Theorem applies to all data distributions, it does not indicate how many values are exactly within k standard deviations of the mean, simply the minimum that must be. Additionally, it is more effective with larger k values. The theorem is a conservative estimate and distributions with a specific shape, like the normal distribution, may have tighter bounds.