Standard Deviation ()
This is the measure of dispersion of all values in a data set from the arithmetic mean. It is the most
common method for showing dispersion
1.
2.
Calculate the mean ( x ) of all the values
Measure how much each value differs from it by subtracting the mean from each value ( x  x ) values higher (+) and values lower (-)]
3.
4.
Each difference is squared ( x  x )2 (to get rid of the negatives…)
Add together all the squared deviations using the formula
(x  x )
2
and this figure is divided by
the number (n) of values. This is the ‘variance’ of distribution
5.
The standard deviation (S) is the square root of the variance:
S
 (x  x )
2
n
The standard deviation is the best method for showing the extent to which the values cluster around the
mean value. A low standard deviation, for example, will indicate that the values are clustered around the
mean and there is a small spread of data. A high standard deviation will indicate that the values are widely
spread around the mean and, therefore, dispersion is large.
However, the degree of dispersion will vary with the mean value itself. If two data sets have the same
standard deviation but different means the dispersion will be greater for the lower value.
In a normal distribution which is symmetrical:

68% of the values will lie less than ±1 standard deviation from the mean

95% of the values will lie less than ±2 standard deviations from the mean

99% of the values will lie less than ±3 standard deviations from the mean
Calculate the standard deviation of the population data below.
Country
Egypt
Nigeria
Ethiopia
Uganda
Mexico
Bangladesh
India
Pakistan
China
Brazil
Bolivia
Chile
Puerto Rico
 x = 3376
population
(millions) (x)
75
134
74
27
108
146
1121
165
1311
186
9
16
4
xx
( x  x )2
-184.69
-125.69
34114.39
15797.97
-232.69
54144.63
-113.69
861.31
-94.69
12925.41
741854.92
8966.19
-250.69
62845.47
-255.69
65377.37
x = 259.69
(x  x )
2
=
n = number of values
S
 (x  x )
n
Standard Deviation =
1SD = ±
2
(closer to 0 = less deviation)
2SD = ±
(6 marks)
2)
Comment on the standard deviation figure.
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(4 marks)
(Total marks = 10)