Standard Deviation (SD)

The standard deviation (\sigma) is the positive square root of the Variance, measuring the spread of a Random Variable in the same units as the variable itself.

\text{SD}(X) = \sigma

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

• Because it is in the original units (not squared), \sigma is more directly interpretable than the variance.
• For a linear transform, \text{SD}(aX + b) = |a|\,\sigma.
• Standardizing via Z = (X - \mu)/\sigma expresses outcomes in standard-deviation units.

Standard Deviation of a Loss Distribution The variance of a loss distribution is 400. Find the standard deviation and interpret it.

Answer \sigma = \sqrt{400} = 20 Individual losses deviate about 20 units from the mean on average. Because \sigma shares the loss's units — unlike the variance, which is in squared units — it is the more interpretable measure of spread.