B.DEVI PRAVALLIKA 8701,MTECH-CIVIL
REPORT ON COEFFICIENT OF CORRELATION (COMPUTATION MATHS) The function between two variables are related related to one another and and the patte atterrn of the the data data poin points ts on the the scat scatte terr plot plot can can illustrate various patterns and relationships, including •
data correlation
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positive or direct relationships between variables
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negative or inverse relationships between variables
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non-linear patterns
As we conned our attention to the analysis of observations on a single variable. There are many phenomenon where one variable are related to the changes in the other variable.
For example the yield of a crop varies with the amount of the rain rainfa fall ll,, the the pr pric ice e of a comm commod odit itiy iy incr increa ease ses s with with the the reduction in its supply. From the above examples we have notied that change in one variable is associated with the change in other variable this relation which exists is known as CORRELATION. uch data connecting two variables is known as !"ivariate population# DEFINITION OF CORRELATION
The change in one variable depends on the change in another variable variable is known as correlation. correlation. it is denoted denoted by !r# !r# and and it is ment mentio ione ned d as -$ % r % $ the valu value e shoul should d lie betw betwee een n thes these e limi limiti ting ng posi positi tion on.. in case case of solv solvin ing g the the
problems the straight line tting is used y= ! "# and the normal e&uations are derived using this straight line e&uation' ( y ) na * b +x
( xy) a+x * b+x correlation than a study using social science data. The quantity r , called the linear correlation coefficient , measures the strength and the direction of a linear relationship between two variables. The linear correlation coefficient is sometimes referred to as the Pearson product moment correlation coefficient in honor of its developer Karl Pearson. The value of r is such that -1 r !1. The ! and " signs are used for positive linear correlations and negative linear correlations, respectively.
Positive correlation: #f x and y have a strong positive linear correlation, r is close to !1. $n r value of e%actly !1 indicates a perfect positive fit. Positive values indicate a relationship between x and y variables such that as values for x increases the value y increases Negative correlation: #f x and y have a strong negative linear correlation, r is close to -1. $n r value of e%actly -1 indicates a perfect negative fit. Negative values indicate a relationship between x and y such that as values for x increase, values y decrease. No correlation: #f there is no linear correlation or a wea& linear correlation, r is close to '. A value near zero means that there is a random, nonlinear relationship between the two variables. (ote that r is a dimension less quantity) that is, it does not depend on the units that are being employed.
$ perfect correlation of * 1 occurs only when the data points all lie e%actly on a straight line. #f r + !1, the slope of this line is positive. #f r + -1, the slope of this line is negative. $ correlation greater than '. is generally described as strong , whereas a correlation less than '. is generally described as weak . These value can vary based upon the type of data that is being e%amined.
f an increase or decrease in the values of one variable corresponds to an increase or decrease in the other ,the correlation is said to be positive. f the increase or decrease in one corresponds to the decrease or increase in the other the correlation is said to be negative. f there is no relationship indicated between the variables they are said to be independent or uncorrelated. To obtain a measure of relationship between two variables we plot their corresponding values on the graph taking one of variables on the x axis and the other along y axis. /et the origin is shifted to x and y where x and y are the new corrdinates. 0ow the points x and y are so distributed along over the four &uardants of xy plane that the product is positive in the rst and third &uardants but negative in second and fourth &uardants. The algebric sum of the products can be taken as describing the trend of dots in all the &uardants
f ( xy is positive , the trend of dots is through the rst and the third &uardants
f ( xy is negative , the trend of dots is through the second and fourth &uardants
f the ( xy is 1ero, the points indicate no trend that is points are evenly distributed through the four &uardants Method of Calculation:
a2 3irect method substituting the value of 4x and 4y in the above formule we get r)+56789+5+62 .
Another form of the formula 9$2 which is &uite handy for calculation is r) n+xy-+x+y 7 8: 9n+x-9+x2 ; x :n+y9+y2;; This formula is used when the means are integers b2 tep deviation method' the direct method becomes very large ,lengthy and tedious if the means of two series are not integers in such cases ,use is made of assumed means if dx and dy are step deviations from the assumed means r) n+ dxdy- +dx +dy78: 9n +dx-9+dx2; x :n+dy9+dy22; where
dx)9x-a27h dy)9y-b27k here, dx)deviation of the central values from the assumed means of x series . dy)deviation of the central values from the assumed means of y-series . f) fre&uency corresponding to the pair 9x,y2 n) total number of fre&uency 9+ f 2 U$%$ &' &%*+& '
measures and describes the strength and direction of the relationship "ivariate techni&ues re&uires two variable scores from the same individuals 9dependent and independent variables2. ?ultivariate when more than two independent variables 9e.g e@ect of advertising and prices on sales2. ariables must be ratio or interval scale C&%%+ &' &%*+& t also called as Bearson product moment correlation coe=cient. The algebraic method of measuring the correlation is called the coe=cient of correlation. T/%% % *y +/%% &%%+$ &' &%*+& Carl BearsonDs
A measure of the strength and direction of a linear relationship between two variables
S*% C&%*+& f there are only two variable under study, the correlation is said to be simple. E#*%: The correlation between price and demand is simple. M2*+*% C&%*+&$ >hen one variable is related to a number of other variables, the correlation is not simple. t is multiple if there is one variable on one side and a set of variables on the other side.
E#*%3 Eelationship between yield with both rainfall and fertili1er together is multiple correlations P+* C&%*+& The correlation is partial if we study the relationship between two variables keeping all other variables constant. E#*%3 The Eelationship between yield and rainfall at a constant temperature is partial correlation. P&%+%$ &' +/% C&%%+ &' &%*+&
