Population Variance

Population Variance the average of the squares of the distance each value is from the mean.

That is for a population data set with \(n\) elements, i.e. \(P=\{x_1,...,x_n\}\) with population mean \(\mu\) we calculate the population variance as

\[\sigma^2=\frac{\Big(\sum_{i=1}^n\big(x-\mu\big)^2\Big)}{n}\]

Note that we denote the population variance with the Greek letter \(\sigma^2\) "sigma squared"

Finding the Population Standard Deviation

Step 1

Find the mean for the data.

\[\mu=\frac{\Big(\sum_{i=1}^nx_i\Big)}{n}\]

Step 2

Find the "deviation" of each data value.

\[x_i-\mu\]

Step 3

Square each of the deviations (from step 2)

\[(x_i-\mu)^2\]

Step 4

Find the sum of the squares (from step 3)

\[\sum_{i=1}^n(x_i-\mu)^2\]

Step 5

Divide by the number of data, \(n\)

\[\sigma^2=\frac{\Big(\sum_{i=1}^nx_i-\mu\Big)}{n}\]