The alternative formula for obtaining the sum of squares is:
9
Sum of squares =
2
x

2
n
x
The two new expressions are
2
x
and
2
x
(i)
2
x
is called the
sum of the squares of x
.
Using again the 10
measurements of
Example 3
it is derived as follows:
x
:
81
79
82
83
80
78
80
87
82
82
2
x
= 81
2
+ 79
2
+ 82
2
+ 83
2
+ 80
2
+ 78
2
+ 80
2
+ 872 + 82
2
+
82
2
=
66316
(ii)
2
x
is called the
square of the sum of
x
and is calculated
as
follows:
2
x
= (81 + 79 + 82 + 83 + 80 + 78 + 80 + 87 + 82 + 82)
2
=
662596
Substituting in the formula for the sum of squares:
Sum of squares =
4
.
56
10
662596
66316
(iii)
The variance is obtained by dividing the sum of squares by (
1
n
):
27
.
6
)
1
10
(
4
.
56
2
s
(iv)
The standard deviation is the square root of the variance:
mm
s
504
.
2
27
.
6
1.7Degrees of freedom
In obtaining a sample standard deviation to estimate a population
standard deviation, reference was made to the number of degrees of
10
freedom.
Because the concept of degrees of freedom is involved in
many statistical techniques, it now needs a fuller explanation.
Suppose that we are told that a sample of n = 5 observations has a
mean
x
of 50 and we are then asked to ‘invent’ the values of the
observations.
We know that
x
= (
x
x
n
) = 250. If the sum of the
observations is 250, we have freedom of choice only for the first four
observations because the 5
th
must be a number (perhaps a negative
number) that brings the sum to 250.
By way of example, if the first four
numbers are arbitrarily chosen as, say, 40, 25, 18, 130, then in order to
make
x
= 250, we have no further freedom of choice; the fifth
number must be 37.
Degrees of freedom (df) in our present example is
one less than the number of observations.
That is, df =
)
1
(
n
Sometimes the formula for estimating a population parameter contains
a value which is itself an estimate. Thus, to estimate a standard
deviation, knowledge of the mean is required.
Because the value of
the mean is itself estimated from a sample, this ‘costs’ a degree of
freedom (this cost
is Elliott’s ‘tax’ referred to in Section 1.4).
The degrees of freedom are not always
)
1
(
n
; they depend on the
particular estimation in hand.
We explain the rule for deciding the
degrees of freedom for each technique as it arises.
Degrees of
freedom are symbolized by
v
(nu).
1.8The coefficient of variation (CV)
The standard deviation is a measure of the degree of variability in a
sample which estimates the corresponding parameter of a population.
However, it is of limited value for comparing the variability of samples
whose means are appreciably different.
The standard deviation of the
mass of a population of elephants is hundreds of kilograms whilst that
of a population of harvest mice is a few grams.
When comparing
11
variability in samples from populations with different means, the
coefficient of variation (CV)
is used.
This is the ratio of the standard
deviation to the mean, usually expressed as a percentage by
multiplying by 100.
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 Fall '18
 F. TAILOKA
 Standard Deviation, Mean, 1.8 g, 0.54 g, 1.7 degrees, 0.26 g