Definition 1.5.4. Variance and Standard Deviation.
The variance is the average squared deviation from the mean. If this data comes from the entire universe of possibilities then we call it a population variance and denote this value by \(\sigma^2\text{.}\) Therefore
\begin{equation*}
\sigma^2 = \frac{\sum_{k=1}^n ( x_k-\mu )^2}{n}
\end{equation*}
The standard deviation is the square root of the variance. If this data comes from the entire universe of possibilities then we call it a population standard deviation and denote this value by \(\sigma\text{.}\) Therefore
\begin{equation*}
\sigma = \sqrt{\frac{\sum_{k=1}^n ( x_k-\mu )^2}{n}}.
\end{equation*}
If data comes from a sample of the population then we call it a sample variance and denote this value by
\begin{equation*}
v = \frac{\sum_{k=1}^n ( x_k-\overline{x} )^2}{n}.
\end{equation*}
Sample data tends to reflect certain "biases". For example, a small data set is unlikely to contain a member of the data set that is far away from the major portion of the data. However, data values that are far from the mean provide a much greater contribution to the calculation of v than do values that are close to the mean. Technically, bias is defined mathematically using something called "expected value" and would be discussed in a course that might follow this one.
To account for this, we increase the value computed for v slightly by \(\frac{n}{n-1}\) to give the sample variance via
\begin{equation*}
s^2 = \frac{n}{n-1} v = \frac{n}{n-1}\frac{\sum_{k=1}^n ( x_k-\overline{x} )^2}{n} = \frac{\sum_{k=1}^n ( x_k-\overline{x} )^2}{n-1}.
\end{equation*}
and the sample standard deviation \(s\) similarly as the square root of the sample variance.