Lec18p4 - JUST THE MATHS UNIT NUMBER 18.4 STATISTICS 4 (The...

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“JUST THE MATHS” UNIT NUMBER 18.4 STATISTICS 4 (The principle of least squares) by A.J.Hobson 18.4.1 The normal equations 18.4.2 Simplified calculation of regression lines 18.4.3 Exercises 18.4.4 Answers to exercises
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UNIT 18.4 - STATISTICS 4 THE PRINCIPLE OF LEAST SQUARES 18.4.1 THE NORMAL EQUATIONS Suppose two variables, x and y , are known to obey a “straight line law” of the form y = a + bx , where a and b are constants to be found. Suppose also that, in an experiment to test this law, we obtain n pairs of values, ( x i , y i ), where i = 1,2,3,. .., n . If the values x i are assigned values, they are likely to be free from error, whereas the observed values, y i will be subject to experimental error. The principle underlying the straight line of “best fit” is that, in its most likely position, the sum of the squares of the y -deviations, from the line, of all observed points is a minimum. Using partial differentiation, it may be shown that n X i =1 y i = na + b n X i =1 x i --- (1) and n X i =1 x i y i = a n X i =1 x i + b n X i =1 x 2 i --- (2) . The statements (1) and (2) are two simultaneous equations which may be solved for a and b . They are called the “normal equations” .
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Lec18p4 - JUST THE MATHS UNIT NUMBER 18.4 STATISTICS 4 (The...

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