Lec18p3 - JUST THE MATHS UNIT NUMBER 18.3 STATISTICS 3...

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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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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
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18.3.3 PRACTICAL CALCULATION OF MEAN DEVIATION In calculating a mean deviation, the following short-cuts 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.

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Lec18p3 - JUST THE MATHS UNIT NUMBER 18.3 STATISTICS 3...

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