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CORRELATION AND REGRESSION

CORRELATION: “ Correlation analysis deals with the association between two or more variables.” “Correlation analysis attempts to determine the ‘degree of relationship’ between variables.” Types of Correlation: Positive or negative Simple and Multiple 3. Linear and Non linear.

“When two variables move in the same direction, that is when one increases the other also increases and when one decreases the other also decreases, such a relation is called positive correlation” Example Typically, in the summer as the temperature increases people are thirstier.

When two variables move in the different directions, that is when one increases the other decreases and when one decreases the other increases, such a relation is called negative correlation.” Example: Demand of commodity And its price

When two variables change in constant proportion, it is called linear correlation When two variables do not change in any constant proportion, the relationship is said to be non-linear correlation Simple Correlation implies the study of relationship between two variables only. Like the relationship between price and demand When the relationship among three or more than three variables is studied simultaneously, it is called multiple correlation.  

Strength of Linear Association

Correlation Coefficient “r” indicates… strength of relationship (strong, weak, or none) direction of relationship positive (direct) – variables move in same direction negative (inverse) – variables move in opposite directions r ranges in value from –1.0 to +1.0 -1.0 0.0 +1.0 Strong Negative No Relation. Strong Positive

Degree Of Correlation: Degree Positive Negative  Perfect +1 -1 High .75 to 1 -.75 to -1  Moderate .25 to .75 -.25 to -.75 Low 0 to +.25 0 to -.25 Zero 0 0

Methods of Studying Correlation: Scatter Diagram Method Graphic Method Karl Pearson’s Coefficient of Correlation Concurrent Deviation Method Karl Pearson’s Coefficient of Correlation: r =  x y / N x y Here x = X – (mean of X series) & y = Y –(mean of Y series) x = Standard deviation of X series y = Standard deviation of Y series N = Number of pairs of observations.

Or r =  x y /   x X  y Here x = X –(mean of X series) y = Y –(mean of Y series) r = the correlation coefficient ( the value lies between-1 r  +1) 2 2

 N  dx – ( dx)  N  dy – (dy) When deviations are taken from an assumed mean r = N  dx dy -  dx  dy  N  dx – ( dx)  N  dy – (dy) 2 2 2 2 Here dx = sum of deviations of X series from assumed mean dy = sum of deviations of Y series from assumed mean dx dy = sum of product of the deviation of X & Y series from their assumed mean dx = sum of squares of the deviation of X series from assumed mean dy = sum of squares of the deviation of Y series from 2 2

Ex Find out Karl Pearson’s co-efficient of correlation Height of Father(inch) Height of Son(inch)   65 67 66 68 67 65 67 68 68 72 69 72 70 69 72 71 72                71

Example: Calculate the Karl Pearson’s coefficient of correlation from the following data. Marks in accountancy : 48 35 17 23 47 Marks in Statistics : 45 20 40 25 45

Spearman’s Rank Correlation Method: Where Ranks are given Where ranks are not given Where ranks are given: R = 1 – 6  D 2 N 3 – N Here D = differences of the two rank

Example: The ranking of 10 students in two subjects A and B are as follows: A B 06 03 05 08 03 04 10 09 02 01 04 06 09 10 07 07 08 05 01 02 Calculate the rank correlation Coefficient.

When ranks are not given: When we are given the actual data and not the ranks, it will be necessary to assign the ranks. Ranks can be assigned By taking either highest value as 1 or the lowest value as 1. But whether we start with the lowest value or the highest value we must follow the same method in both the cases. Example: Calculate the Spearman’s coefficient of correlation between marks assigned to 10 students by judges X and Y in a certain Competitive test as shown below:

Marks by Judge X Marks By Judge Y 52 65 53 68 42 43 60 38 45 77 41 48 37 35 38 30 25 25 27 50

Where values are repeated: R = 1 – 6 ( D 2 + 1/12 ( m1 – m1) + 1/12 (m2 – m2 ) + ………) 3 3 N 3 – N Where,  D 2 = sum of the square of rank difference. N = no. of pairs of items. m1, m2 = no. of items having same rank.

Example: Calculate the rank coefficient correlation: X Y 80 12 78 13 75 14 68 14 67 16 60 15 55 17 50 19 40 20

Probable Error (P.E) = 0.6745 1-r2 N The limits for population coefficient of correlation: = r ± P.E. Also Standard Error (S.E) = 1-r2

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