# Chebyshev Inequality For Standard Deviation

The Chebyshev inequality goes like this. Suppose that \(x\) is an \(n\)-vector, and that \(k\) of its entries satisfy \(|x_i| \geq a\), where \(a > 0\). Then \(k\) of its entries satisfy \(x_i^2 \geq a^2\). It follows that \[||x||^2 = x_1^2 + \cdots + x_n^2 \geq ka^2\] since \(k\) of the numbers in the sum are at least \(a^2\), and the other \(n - k\) numbers are nonnegative. We conclude that \(k \leq ||x||^2 / a^2\), which is the Chebyshev inequality. …