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MODULE 4 - 18 MAT 41 183, , COMPLEX ANALYSIS, PROBABILITY, &, STATISTICAL METHODS, , , , , , MODULE - 4, , , , , , , , ‘Statistics is a science of facts and figures which may be readily |, available or obtained through the process of direct enquiry or, enumeration. It deals with the methods of collecting, classifying, and analysing the data so as to draw some valid conclusions. !, , , , In this module we discuss two topics namely Correlation and |, Regression followed by Curve Fitting., , ‘Correlation’ can be understood as co-relation. Co-variation of |, fwo independent magnitudes is known as correlation. ‘Regression’, is an estimation of one independent variable interms of the other., , ‘Curve Fitting’ isa method of finding a specific relation connecting, the dependent and independent variables for a given data so as to, Satisfy the data as accurately as possible. Such a curve is called, , the curve of best fit, achieved by the method of least squares. ;, BY curve of best fi y of 1 J
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Complex Analysis, Probability & Statistical Methods MODULE 4- 18.MATAI 185, 184 . ., , STATISTICAL METHODS, , , , , , , , , , , , , , , , , , , , , , [41] Basic Definitions and Formulae in Statistics, , Classification: The significance of a large mass of statistical data known as, raw data cannot be understood unless it is arranged in some definite manner., The process of arrangement in different groups is called classification., Frequency table: It is a tabular arrangement consisting of various classes of, uniform size known as class intervals and the number in each class known, as frequency., , The difference between two consecutive vertical entries of the classes is known, as the width of the class usually denoted by h. The average of the left and, right end points of the class interval is known as the midpoint of the class, usually denoted by x., , Variate : It is a quantity which can have different numerical values., Example : Price, marks, height, weight, age etc., , | Mean ( Arithmetic mean), , If x,,%,,%,,-+, x, bea set of n values of variate x, the mean denoted by ¥, is defined as follows., , i Foi dx or fe, , n m4 n, , For a grouped data in the form of a frequency distribution,, , n, , , , , , where fe "sare the frequency of the classes having corresponding midpoint x,., Variance ( V ) and Standard Deviation ( SD ), , Ifa variate x take values X17 %,,+1, x, ,the variance ( V) is defined as follows., , Lee?, , n, , Vv or V=, , D(e-* ), n
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Complex Analysis, Probability & Statistical Methods MODULE 4 - 18 MAT 41 187, 186 2 ,, xz - O+84+745444943 42, Also for a grouped data, ie, Fo OTe — = 2. 6, | DF @-7F v Ve-#) Thus, mean (X) = 6, iat _ aa, yet oy oF Df 7 = 7(6-6) 48-67 47-6) (5-67 + (4-67 9-6) 4 0-6), 7 ote” 2 4 or SD(o) =2, Standard deviation (SD), o = VV or o? = V ;, 2, ii . 2 a nif SND, ‘ i 2 Ailter: o? = “ (x), Alternative expression for ¢, 2 2 2 2 2 2: 2, 2_1 ~# _ +8 +7 SSP Ae ogy, Consider, © = Ca, 280, of = 2 [at 288+ (2) = 36 =4, 2 x2 Thus, SD =o =2, = 2” _o(zy.g4 0), n n, Illustrative Example - 2, , , , , , , 4 . =y\2, He be x and (X)’ being a constant added n times gives n(X), ere, —— =, n, , Let us find the mean and standard deviation for the following 8rouped data, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 2, i of ie 5 (zy + (2) Class | 1-10 | 11-20 | 21-30 | 31-40 | at-50 | Sisco, 4 Frequency | 3 16 26 31 16 8 |, of = LE)! :, B Class f x fx (x-x) | f(x-¥), ession will be of the form, For a grouped data the expr 1-10 3 55 16.5 702.25 | 2106.75, obit (zy? 11-20 16 15.5 248.0 272.25 | 4356.00, Df 21-30 26 25.5 663.0 42.25 1098.50, s 1 ., eda eal learen and standard deviation for a set of observations : 6,8,7,5,4,9;3 31-40 31 35.5 1100.5 12.25 379.75, We shall fin dat 41-50 16 45.5 728.0 182.25 2916.00, 5. i ata,, This:being: @ raw zy 51-60 8 55.5 444.0 552.25 | 4418.00, gu we = Lok Totals 100 3200 15275, n
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er, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Complex Analysis, Probability & Statistical Methods, , , , o = 152.75 = 12.36, , Correlation and Regression, This topic deals with data concerning independent observations., , Examples :, , 1. Marks of individuals in two subjects., , 2. Height and weight of individuals., , 3, Amount involved in advertising a product and sale of product etc.,, We discuss the aspect of inter-relation between the independent variables., , Correlation and Correlation Coefficient, , Co-variation of two independent magnitudes is known, variables x and y are related in such a way that increase or decrease in one, of them corresponds to increase or decrease in the other, we say that the, variables are positively correlated. ‘Also if increase or decrease in one of them, to decrease or increase in the other, the variables are said to be, , , , as correlation. If two, , corresponds, , negatively correlated., ¢ correlation between two variables x and y is known, , The numerical measure 0}, as Karl Pearson's coefficient of correlation usually denoted by r and is defined, , as follows., , E(x-7y-T), , r= —, no o, x, , y, , This can be put in an alternative form as follows., , If X=x-x,Y =y-y we can write,, , Lek? DE, gg Ew 2Y, 4 n - n, , - (ex EY or no, 6, = Jax Joy, , , , , , xy, , , , , , MODULE 4 - 18 MAT 41, , Hence (1) becomes,, , ree, Property :, , The coefficient of correlation numerically does not exceed unit:, Proof : We have to show that -1 <7 < +1 ,, , 1 xX yy, Le, S= Sg |e i cl x yy, a [2 x} ands= 3D: “=, * y, , o o, x y, , hi =X, y., where X=x-X,Y =y-y. Obviously both S, S’ are > 0, , 1 xX? y?, Now, ee Car ara, , , , o, , a, S; s, oo, , , , , , , , , , 1, pliti+2r] 20, , 3 1, te. =, 2 [2+2r] 20 or (1+r) > 0 or r>-1, , That is, -Il<r, Similarly we can obtain,, , 1, So ==f23l2 2r]20 or (1-r)>0 or 124, That is,, , reel, , (1), , 2), , »}, , 189, , cs)
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190 Complex Analysis, Probability & Statistical Methods, , Combining (1) and (2) we obtain -1 <r < +1, , Note : If r = £1 we say that x and y are perfectly correlated and if r = 0 we say, , , , , , , , , , , , , , , that x and y are non correlated., , [4.22 | Formula for the correlation coefficient r, , z 2 2, , ae 2, , 6, 19,5,, r=—, , y, , 20, °,, Let 2=x-y, , 2a _ ee By 7, , or Z=xX-Y¥, n n, , Proof :, , Hence, (2-2Z) = (x-y)-(¥-Y), or (2-2) =(x-x)-(y-¥) _, Squaring both sides, taking summations and dividing by n we have,, , De-z! _ De zP | Dy-g! _ 2D-F)(y-7), — n, , nH, n Wn, , L(x-¥)y-9), Q 2 2_%F4rO 6 since, f=, ie, 9, = 9, 4°, 276, oy, ‘ GO. o,, , , , , , , , , , , , =o +6? -2ro 9,, x y x y, , , , ie., o., , x-y, , 2 2 at, _ +9, Gecy, Thus, Veo, , 2 S. 9,, , Note : In general if z = ax + by we can obtain as before,, , , , , Zh, o =a or +b o| +2rabo.o,, , , , , , , , , , , 2 =070°+0?o°+2raboo, s =7 0 +b’?o, ie, Srey 8 Oy y wo, , Regression, , i i ther. If, Regression is an estimation of one independent varie le in terms of tre ot a, t fitting straight line in the least sqi, x and yare correlated, the bes: : ., reasonably a good relation between x and y. [Refer the following article 4.3], , , , MODULE 4 - 18 MAT 41 191, , The best fitting Straight line of the form ¥ = ax+b (x being the independent, , variable) is called the regression line of y, , on x and x = ay+b(y being the, independent variable) is called the regressi, , on line of x on y., ¢ Derivation of the equation of the regression lines, , Let y = ax +b be the equation of the regression line of y on x for a given set, , of 1 values (x, y). The associated normal equations are [Refer article 4.3], , Ly =ayxend ».. (2), Dry = ayer soy x ewes), , Dividing (1) by 1 we have,, , dy, Le b, , ——+n— or ¥ =ax+b., n n n ., , This shows that the regression line passes through (x,7)., , We also know that the equation of, , a straight line passing through (4 -¥,), having slope im is,, , , , , , , Y~y, = m(x-x,), , , , , Hence if (x, “Y,) = (%,7) we get,, , yi = m(x-%), , , , , , , , , , , , , , or Y=mX where X = x-¥,Y =y-y., , m X to find the parameter i will be,, , DxY = my X*? or m= a, , xX?, , , , , The normal equation for fitting Y =, , , , , , , & xy = Ure. s,, , , , , , , , , , , , , , , Y xX?, Also we have, o® = &* 6 YX? = ne?, : A §, ro 6 °,, Hence, mt = 2" = ptt, no o
