In English: "The probability that the outcome of an experiment with the random variable will fall more than standard deviations beyond the mean of , , is less than ."
Let be the sample space for a random variable, , and let stand for the pdf of . Let partition , such that for every sample point
since the term that evaluates to the variance in has been subtracted on the right-hand side.
For any sample point
Notice that the direction of the inequality changes since squaring causes the right-hand expression to become positive.
And for any sample point
So, for any sample point or , it can be said that , and so
Dividing each side of the inequality by results in
Or, in other terms
And thus the original claim is proven.