CHAPTER 18. STATISTICS 18.2
Note: To get the deviations, subtract each number from the mean.
X Deviation (X−X ̄) Deviation squared (X−X ̄)^2
57 1 1
53 − 3 9
58 2 4
65 9 81
48 − 8 64
50 − 6 36
66 10 100
51 − 5 25
�
X = 448�
x = 0�
(X−X ̄)^2 = 320
Note: The sum of the deviations of scores about their mean is zero. Thisalways happens; that is
(X−X ̄) = 0, for any set of data. Why is this? Find out.
Calculate the variance (add the squared results together and divide this total by the number of items).
Variance =�
(X−X ̄)^2
n
=320
8
= 40
Standard deviation =√
variance=��
(X−X ̄)^2
n=�
320
8
=
√
40
= 6. 32
Difference between Population Variance and Sample Variance
As with variance, thereis a distinction betweenthe standard deviation, σ, of a whole populationand
the standard deviation, s, of sample extracted from the population.
When dealing with thecomplete population the (population) standarddeviation is a constant,a pa-
rameter which helps to describe the population.When dealing with a sample from the population the
(sample) standard deviation varies from sampleto sample.
In other words, the standard deviation can be calculated as follows:
- Calculate the mean value ̄x.
- For each data value xicalculate the difference xi− ̄x between xiand the mean value ̄x.
- Calculate the squaresof these differences.
- Find the average of the squared differences. This quantity is the variance, σ^2.
- Take the square rootof the variance to obtainthe standard deviation, σ.
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