Geography Reference
In-Depth Information
values are all quite similar to the mean and large when some values deviate markedly
from the mean. h e population standard deviation, indicated by
s
(lower case sigma),
is given by:
1
n
(
)
2
Â
s
=
z
-
m
i
n
(3.2)
i
=
1
Most of the notation is familiar from the equation for the mean average. In this case,
the mean (
m
) is subtracted from each value and the product (i.e. the outcome) of this
subtraction is squared. h e squared dif erences are added together. Once this is done
the sum is multiplied by
1
/
n
and the square root is taken of this value. In words, the
standard deviation is the square root of the average squared dif erence between
observed values and their mean average. h e squaring is necessary because if the
dif erence between each value and its mean is not squared, then the sum of dif erences
will be zero. Where the square root is not taken the resulting value is called the
variance.
h e mean and standard deviation as dei ned above are population statistics. In rec-
ognition of the fact that usually we have only a sampl
e,
an alternative form of the
standard deviation is computed. h e sample mean,
z
(
z
with a bar on top), is
computed as above (i.e. the population mean in Equation 3.1). h e sample standard
deviation,
s
, is computed in the same way as in Equation 3.2 except that
1
/
n
is replaced
by
1
/
(
n
- 1)
. h e reason for this requires some explanation. h e mean must be computed
before we can compute the variance and a quantity known as the number of degrees
of freedom is
n
minus the number of parameters (such as the mean) estimated, thus
n
- 1 in this case (see O'Sullivan and Unwin (2002) for a further account). Another
way of putting this is that one degree of freedom is used up in estimating the mean and
if we know the mean then we only need
n
- 1 values to calculate the value of the
n
th
sample and thus know all values (i.e. if we have i ve values in total then we need only
four values and the mean to work out the i t h value) (Rogerson, 2006). h e sample
standard deviation is given by:
n
1
(
)
2
Â
(3.3)
s
=
z
-
z
i
n
-
1
i
=
1
Note that population statistics are by convention given by Greek characters (e.g.
s
)
and sample statistics by ordinary lower case letters (e.g.
s
).
Using the same data as before (11, 14, 13, 9, and 6 with a mean average of 10.6),
the sample standard deviation is calculated using:
5
(
)
Â
2
2
2
2
2
2
zz
-
=
(11
-
10.6)
+
(14
-
10.6)
+
(13
-
10.6)
+
(9
-
10.6)
+
(6
-
10.6)
i
i
=
1
=+ +++ =
0.16
11.56
5.76
2.56
21.16
41.20
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