The variance σ2 of a dataset measures the spread of data around the mean.

The standard deviation σ is the square root of the variance. The advantage of the standard deviation is that is has the same units as the original data.

When you use a calculator to find standard deviation, you will see two values: Sx and σx. We use Sx when the data is a sample of a large population, and σx when the data represents the entire population.

The difference is due to the fact that a sample will usually have a smaller variance than the population, because there are fewer elements. The value Sx is larger than σx and attempts to correct for this difference based on the number of items in the sample.

Example 1

Find the variance of [1,5,6,7,11,−1].

Using technology we find Sx=4.31 and σx=3.93. Since this is the whole dataset and not a sample, we use the (smaller) value σx=3.93. The variance is then the square of this so σ2=15.4.

Example 2

A scientist samples the length of fish in an aquarium. He measures the length of 5 fish, and finds [21cm,19cm,7cm]. Estimate the variance in the length of fish in the aquarium.

Using technology we find Sx=7.57 and σx=6.18. Since we have a only a sample of the population, we use Sx=7.57⇒s2=57.3.