For a normal distribution, we know that approximately 0.68 of the data points will lie within 1 standard deviation of the mean, and 0.95 will fall within 2 standard deviations. But our distributions aren’t always normal. What if it’s skewed or bimodal?

Say we have a histogram that looks like this:

We’re given that the mean is 4.99 and the standard deviation is 3.13, and we want to find the proportion of observations that are within 2 standard deviations of the mean. Since our distribution is not normal, we can’t use our 0.95 figure. However, we can ues Chebyshev’s Theorem, which states:

For any sample, the proportion of observations within k standard deviations of the mean is at minimum

1 - (1 / k^2)

So here, we can calculate our proportion within 2 standard deviations to be at minimum **1 - (1/2^2) = 0.75**. For the example skewed distribution, the actual proportion is 0.89. 0.75 may seem pretty far from 0.89, but remember that it’s a lower bound. The beauty of Chebyshev’s Theorem is that it works for *all* distributions, no matter the shape. Even if it looks something like this: