Geography Reference
In-Depth Information
in the data set then
i
could take a value of 1, 2, 3, 4, or 5. h e number of observations
is indicated by
n
and, in the example given,
n
= 5. h e population (where population
indicates it is assumed we have all possible values and not just a sample) mean average,
m
(the Greek lower case letter mu), can be given by:
1
n
Â
m
=
z
(3.1)
i
n
i
=
1
h e only term not yet explained is S (the Greek upper case letter sigma). S indicates
summation. Below S is the term
i
= 1 and above it is
n
. h is means start at the i rst
observation (
i
= 1) and step through all values up to and including the last value (
i
=
n
).
Note that other sources may use other letters for the variables, but the use of letters
and symbols here is consistent throughout the text.
Â
n
i
z
indicates that all values of
z
i
=
1
i
should be added together.
In the example above
z
1
is taken i rst, then
z
2
is added to it and so on until all values
have been added together. h e end result is then multiplied by
1
/
n
(1 being the numerator
(top part of the fraction) and
n
the denominator (bottom part of the fraction)). h is
gives the mean average of the values and is the same as dividing the summed values
by
n
. As an example, if we have i ve values (
z
1
to
z
5
) and they are 11, 14, 13, 9, and 6 then
their sum is 53,
1
/
5
= 0.2
and the mean average is given by 0.2 ¥ 53 = 10.6.
Given these values, Equation 3.1 can be given as:
1
5
Â
m
=
z
=
0.2
¥
(
z
+
z
+
z
+
z
+
z
)
=
0.2
¥
(11
+
14
+
13
+
9
+
6)
=
0.2
¥
53
=
10.6
i
1
2
3
4
5
5
i
=
1
Other averages include the median (the middle value when all values are ordered
from smallest to largest) and the mode (the most frequently occurring value). When
there is an even number of values, the median is the mean average of the two values in
the middle of the distribution (e.g. values 40 and 41 out of a total of 80 values with
values ordered from smallest to largest). h e mean average is very sensitive to outliers
(i.e. unusually large or small values) and one benei t of using the median or mode is
that the impact of outliers is reduced or non-existent.
h e dispersion of a distribution is ot en of interest, i.e. how much variation is there
in the values? h e range—the absolute dif erence between the minimum and maxi-
mum value—is one simple measure. As noted above, the median is the middle of the
ranked values. h e value that falls 25% of the way along the list of ranked values (e.g.
the mean average of values 20 and 21 of a total 80 values) is called the lower quartile
and the value that falls 75% of the way along the ranked list (e.g. the mean average
values of 60 and 61 of a total 80 values) is called the upper quartile. Together, the mini-
mum, lower quartile, median, upper quartile, and maximum provide a summary of
the distribution.
h e dispersion around the mean—the degree to which values are close to the mean
average—is given by the standard deviation. h e standard deviation is small when the
Search WWH ::
Custom Search