#### 3.1.2 Measures of Dispersion

Measures of dispersion are the quantities that characterize the
‘spread’ of the data, such as range, inter–quartile range, or
standard deviation.

##### The range

The range *R *is the difference between the maximum and minimum value
of *X*_{i}. This measure is unstable since it depends upon the two
extreme values of the data and not the entire set. The extreme values can
result from exceptional observations, but the range is useful to show the
extent (or limits) of the data.

In order to reduce the influence of the extreme values, inter–quartile
range (IQR) or inter–decile range is often used as indicators of the
dispersion of the data.

Inter–quartile range = *q*_{3}
– *q*_{1}

Inter–decile range = *q*_{9 }–
*q*_{1}

##### Standard deviation

Standard deviation (denoted by *s* or *s.d.*) is the root mean
square of the deviations from the arithmetic mean. The standard deviation
indicates the average distance of the observations from the mean of the data
set.

To get a better estimate of the standard deviation of the population
(denoted by s), standard deviation is
often computed with *n*–1 instead of *n* in the denominator.
However, for large values of *n* (*n **³ *30) there is practically no difference between the two
definitions. Variance is the square of the standard deviation.

The standard deviation is an absolute measure of deviation that expresses
variation in the same units as the original data. *Coefficient of Variation*
(*C V*) is a relative measure that indicates the magnitude of variation
relative to the magnitude of the mean. Coefficient of variation is computed as
follows:

*CV= s/m *