# standard deviation

Over 68% of a population exhibiting a normal distribution
for some property lie within one standard deviation (*σ*)
of the mean (*μ*). Over 95% lie within *μ*±2*σ* and over 99% within *μ*±3*σ*. The total
population in this example is unity (1).

Standard deviation is a measure of the spread of a set of data. For a normal distribution, the standard deviation gives a measure of the width of the tails of the distribution function.

The **sample standard deviation** is the positive square root
of the sample variance *s*^{2},
where variance is defined as the arithmetic
mean of the squares of the deviations of the members of a sample from
the arithmetic mean of the sample. The **population standard deviation** *σ* is given by the square root of the population variance:

where *n* is the number of observations, *μ* is the mean,
and *x _{i}* the

*i*th value.

In a normal or Gaussian distribution, about 68.3% of the population lies within one standard deviation of the mean.