summation 2 - Chapter One The summation sign and the rules...

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Chapter One The summation sign and the rules of summation 1. The summation Sign It is frequently necessary in statistical and psychometric calculations to take the sum of a number of values. The symbol used to indicate this operation of adding up a group of numbers is a capital Greek Sigma - . However, the instruction ‘to take the sum of’ is rather vague without an indication of what it is that is to be summed. It is necessary to have a system of notation to specify precisely which values are to be summed. Let us suppose that we have a set of four scores: 2, 4, 6, 8, and that we let X be a general symbol for any one of these scores. The set of scores now consists of four X’s which are 2, 4, 6 and 8. If we now assign a subscript to each of the X’s , we can assign an X which a given subscript to each score thus: X 1 = 2; X 2 = 4; X 3 = 6; X 4 = 8. In the case of four scores the subscripts will naturally run from 1 to 4, but there are usually more than four scores involved, and so it is desirable to have a generalised notation, so that we can apply the system to any group of scores. A general symbol for the number of scores is N . Any collection of scores will consist of N scores. If we want a general reference to a single score without specifying exactly which we can use the subscript i. Thus X is the i th score. Consider the following set of scores: X 1 = X 2 = 7 X 3 = 8; X 4 = 9; X 5 = 10.
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Philip Ley . Quantitative Aspects of Psychological Assessment © 1972, 2007 Philip Ley Text re-typed for computer by Irene Page 2 What is the value of X i when i = N ? The answer is 10. There are five scores, therefore N = 5. X i when i = N must be X 5 , which is the symbol for the fifth score, which is 10. In similar fashion the value of X 1 where i = 2 is 7; where i = 1 it is 6 and so on. These symbols are often used in connection with the summation sign to indicate exactly which scores are to be summed. For example: X 1 i = 1 N = take the sum of all scores from X 1 toX N i . e . X 1 + X 2 + X 3 + ... X N The symbols above and below the summation sign are called the limits of the summation. The value of i under the summation sign tells you where to start the addition, and the values above the summation sign tells you where to stop.
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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summation 2 - Chapter One The summation sign and the rules...

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