Standard Deviation

Study guides: Exam P-1, Exam MAS-I, Exam MAS-II, Exam GI-301

Standard Deviation $\sigma$ is the positive square root of Variance, measuring the typical spread of a random variable $X$ around its mean in the same units as $X$ .

The standard deviation is directly interpretable because it shares units with $X$ , unlike variance
It satisfies $\sigma(aX + b) = |a|\,\sigma(X)$ , so location shifts do not affect spread

$\sigma = \sqrt{\text{Var}(X)}$

$= \sqrt{E[X^2] - (E[X])^2}$

Standard Deviation of a Claim Amount Claim amounts $X$ follow a distribution with $E[X] = 200$ and $E[X^2] = 50{,}000$ . Find the standard deviation of $X$ .

Answer First compute variance: $\text{Var}(X) = E[X^2] - (E[X])^2 = 50{,}000 - 200^2 = 50{,}000 - 40{,}000 = 10{,}000$ Then: $\sigma = \sqrt{10{,}000} = 100$ The standard deviation of the claim amount is $100.