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“JUST THE MATHS”
UNIT NUMBER
18.3
STATISTICS 3
(Measures of dispersion (or scatter))
by
A.J.Hobson
18.3.1 Introduction
18.3.2 The mean deviation
18.3.3 Practical calculation of the mean deviation
18.3.4 The root mean square (or standard) deviation
18.3.5 Practical calculation of the standard deviation
18.3.6 Other measures of dispersion
18.3.7 Exercises
18.3.8 Answers to exercises
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View Full Document UNIT 18.3  STATISTICS 3
MEASURES OF DISPERSION (OR SCATTER)
18.3.1 INTRODUCTION
Averages typify a whole collection of values, but they give little information about how the
values are distributed within the whole collection.
For example, 99.9, 100.0, 100.1 is a collection which has an arithmetic mean of 100.0 and so
is 99.0,100.0,101.0; but the second collection is more widely dispersed than the ﬁrst.
It is the purpose of this Unit to examine two types of quantity which typify the distance of
all the values in a collection from their arithmetic mean. They are known as measures of
dispersion (or scatter) and the smaller these quantities are, the more clustered are the values
around the arithmetic mean.
18.3.2 THE MEAN DEVIATION
If the
n
values
x
1
,
x
2
,
x
3
, ......
,
x
n
have an arithmetic mean of
x
, then
x
1

x
,
x
2

x
,
x
3

x
,......
,
x
n

x
are called the
“deviations”
of
x
1
,
x
2
,
x
3
,......
,
x
n
from the arithmetic
mean.
Note:
The deviations add up to zero since
n
X
i
=1
(
x
i

x
) =
n
X
i
=1
x
i

n
X
i
=1
x
=
n
x

n
x
= 0
DEFINITI0N
The
“mean deviation”
(
or, more accurately, the “mean absolute deviation
) is deﬁned by
the formula
M
.
D
.
=
1
n
n
X
i
=1

x
i

x

1
18.3.3 PRACTICAL CALCULATION OF MEAN DEVIATION
In calculating a mean deviation, the following shortcuts usually turn out to be useful,
especially for larger collections of values:
(a) If a constant,
k
, is subtracted from each of the values
x
i
(i = 1,2,3.
..
n
), and also we use
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This note was uploaded on 02/01/2012 for the course ECON 101 taught by Professor Meonk during the Spring '11 term at Abu Dhabi University.
 Spring '11
 Meonk
 Economics

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