# Standard Deviation and Variance (1 of 2)

The variance and the closely-related

**standard deviation** are measures
of how

spread out
a distribution is. In other words, they are measures of variability.

The
variance is computed as the average squared deviation of each number
from its mean. For example, for the numbers 1, 2, and 3, the mean is
2 and the variance is:

.

The formula (in

summation
notation) for the variance in a

population
is

where μ is the mean and N is the number of
scores.

When the variance is computed in a

sample,
the statistic

(where M is the mean of the sample) can be used. S² is a

biased estimate of σ², however. By far the most common
formula for computing variance in a sample is:

which gives an unbiased estimate of σ². Since samples
are usually used to estimate parameters, s² is the most commonly
used measure of variance. Calculating the variance is an important
part of many statistical applications and analyses. It is the first
step in calculating the standard deviation.

Standard Deviation

The standard deviation formula is very simple: it is the square root
of the

variance. It is the most commonly used measure
of spread.

An important attribute of the standard deviation as a measure of spread
is that if the mean and standard deviation of a

normal distribution
are known, it is possible to

compute the percentile
rank associated with any given score. In a normal distribution, about
68% of the scores are within one standard deviation of the mean and about
95% of the scores are within two standard deviations of the mean.

The standard deviation has proven to be an extremely useful measure
of spread in part because it is mathematically tractable. Many formulas
in

inferential statistics use the standard
deviation.

(See

next page for applications to risk analysis and stock portfolio volatility.)

How to compute the standard deviation
in SPSS.