Key concepts 2 Statistics - Spatial Data Analysis: An Introduction for GIS Users - page 29

Given this, s is obtained:

1

s = ¥ 41.20 = 0.25 ¥ 41.20 = 3.21

51 -

h e mean and standard deviation are useful measures of a distribution if it is nor-
mal. A normal distribution is characterized by an equal proportion of small and large
values, with a peak of values in the middle ranges—the distribution is symmetric.
h is type of distribution is called bell-shaped. A distribution with a large number of
small values and a small number of large values is termed 'positively skewed, while a
distribution with a small number of small values and a large number of large values is
termed 'negatively skewed' (an example is the histogram in Figure 3.1). h e mean
average is 'pulled' in the direction of the skew, i.e. it is af ected by extreme values.
In a skewed distribution, the mode will be under the peak of values, the mean will be
closer to the 'tail' of extreme values, and the median will be in between the mode and
the mean. h e degree of skewness can be measured by a statistic called the coei cient
of skewness (note that dif erent measures of skewness exist; the measure below is as
implemented in Microsot ® Excel®). h is can be given by:

3

n
nn

n

Ê zz - ˆ

-- Â

i

skewness = (3.4)

Á ˜

( ( ) Ë s ¯

i = 1

In words, the right-hand side of Equation 3.4 is the sum of the cubed product of
dif erences between individual values and their mean average divided by the standard
deviation. Positive values of the skewness coei cient indicate positive skew and nega-
tive values indicate negative skew, while a value of zero indicates no skew. Some
examples are given in Figure 3.2.
Some distributions have two or more modes, i.e. peaks of values. It is important to
examine the distribution of a variable prior to further analysis. Examples of distributions
which are normal, positively skewed, and negatively skewed are given in Figure 3.2;
for the purposes of the discussion the data are treated as measurements of precipita-
tion amount in millimetres. If a distribution is normal (and we can perceive it as a
bell-shaped smooth curve, rather than a histogram with discrete bars, as shown in
Figure 3.1) then 68.26% of the values in the data set should fall within one standard
deviation of the mean. In other words, 68.26% of the area under the normal curve lies
within one standard deviation of the mean (i.e. above or below the mean), 95.46% of
the area lies within two standard deviations, and 99.73% lies within three standard
deviations. If the distribution is normal, the mean is 10.6, and the standard deviation
is 3.21 then 68.26% of the values should be 10.6 ± 3.21 (i.e. in the range 7.39 to 13.81).
Following the discussion above, the mean and standard deviation are not represen-
tative if the distribution is not close to normal—this potentially af ects many of the
procedures detailed in Chapters 8, 9, and 10 in particular. Various possible solutions
exist: the variables can be transformed (e.g. by taking the square root or the log of the
values) and the transformed variables may have a less skewed distribution. Analysis can