Sample Standard Deviation

The sample standard deviation is a measure of how widely the observations in a sample are scattered around their average. It is written $s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$, where $x_i$ are the observations, $\bar{x}$ is the sample mean and $n$ is the number of observations. Dividing by $n - 1$ instead of $n$, known as Bessel's correction, offsets the fact that deviations are measured from the sample's own mean, which would otherwise make the spread look too small. The population standard deviation divides by $n$.

For monthly returns of 2, -1, 3, 0 and 1 percent, the mean is 1. The deviations are 1, -2, 2, -1 and 0, their squares sum to 10, and $10 / 4 = 2.5$, so $s$ is about 1.58 percent. In finance this statistic, applied to returns and annualized, is the usual estimate of volatility, and it is the denominator of the Sharpe ratio. Its square is the sample variance.