Variance and standard deviation

The standard deviation measures the typical distance of values from their mean; the sample version divides the sum of squares by n − 1.

s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \xbar)^2, \qquad s = \sqrt{s^2}

Sample variance and standard deviation (the engine's default).

$s^2,\ s$: Sample variance and sample standard deviation.
$x_i - \xbar$: Deviation of value $i$ from the sample mean.
$n - 1$: Degrees of freedom: the deviations sum to zero, so only $n - 1$ are free.

\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2, \qquad \sigma = \sqrt{\sigma^2}

Population variance and standard deviation (divide by N).

$\sigma^2,\ \sigma$: Population variance and standard deviation.
$\mu,\ N$: Population mean and size.

When to use

Quantitative data, alongside the mean. For skewed data or outliers also report the IQR or MAD.
Use $s$ (divisor $n - 1$ ) when the data are a sample used to learn about a larger population — almost always. Use $\sigma$ (divisor $N$ ) only when the data are the entire population of interest.

How to interpret

The SD is in the units of the data ("delivery times vary by about 4 minutes around the mean"). The variance is in squared units and is mainly a building block. For roughly bell-shaped data about 68% of values lie within 1 SD of the mean and about 95% within 2 SD. Dividing by $n - 1$ makes $s^2$ an unbiased estimator of $\sigma^2$ .

Common mistakes

Using the population formula for sample data (spreadsheet STDEV.P instead of STDEV.S); it underestimates spread, especially for small $n$ .
Forgetting the square root (reporting the variance as the SD).
Confusing the SD (spread of individual values) with the standard error (spread of a statistic).
Expecting the 68–95 rule to hold for strongly skewed data.

Example

Data 2, 4, 4, 4, 5, 5, 7, 9: $\xbar = 5$ , sum of squared deviations $= 32$ . Sample: $s^2 = 32/7 = 4.571$ , $s = 2.138$ . Population formula: $\sigma^2 = 32/8 = 4$ , $\sigma = 2$ .