Measuring Relationships Prepared by: Bhakti Joshi January 13, 2012

Correlations Attempts to determine the degree of relationship or association between two variables Attempts to determine the most suitable form of relationship

Examples Savings and interest rates Investment and interest rates Rise in prices and demand for food Fall in prices and cost of raw material Rise in oil prices and costs of transportation Rise in oil prices and prices of food

Relationships for discussion Sales and marketing expenses? Capital employed and ROCE? Stock prices and Market Indices?

Correlation Analysis One of the two variables affect the other – Rise in price of tea affects the demand The two variables may act upon each other – Rise in oil prices and cost of exploring and drilling for oil Two variables may be acted upon by the outside influences – Demand for cars and high-end mobiles phones are correlated due to rising incomes (or purchasing power) of consumers High correlation due to sheer coincidence without any identified relationship – High demand for a particular shoe size and higher incomes of people in a particular locality

Assumption of Linearity Merriam Webster defines it as: “…of relating to, resembling or having a graph that is a line and especially a straight line” “…involving a single dimension”

Linearity Examples

Types of Correlation: Pearson’s Pearson’s product-moment correlation coefficient (denoted by r) – Defined as the covariance of two variables (X and Y) divided by the product of their standard deviations r XY = Cov (X,Y) = s X s Y  (x i x) _ (Yi(Yi Y)Y)  _ (x i x) _    (Yi(Yi Y)Y) _ 22

₍ ₎ Types of Correlation: Pearson’s (Contd…) r XY =  xixi YiYi   xixi  YiYi xixi  2 n n xixi  2 ₍ ₎   2 n  2 YiYi YiYi

Example 1. Values of XValues of Y

Interpretation Correlation Coefficient lies between -1 and +1 Negative sign implies correlation in opposite direction Positive sign implies correlation in the same direction If X and Y are independent, they are uncorrelated but the converse is not true

Example 2. Values of XValues of Y ? 48 ?7 Arithmetic means of X and Y are 6 and 8 respectively. Find correlation coefficient -0.92

Example 3. Price (in Rs)Demand (in KGs)

Probable Error (P.E.) of r “The probable error of correlation coefficient is an amount which if added to and subtracted from the mean correlation coefficient, gives limits within which the chances are even that a coefficient of correlation from a series selected at random will fall” This simply implies that whether the calculation of coefficient of correlation falls within particular limits (or the calculations are dependable)

Probable Error (P.E.)of r (contd…) P.E. (r) = X 1 – r 2 Limits of (population) correlation coefficient is defined as r – P.E. (r) <= ρ <= r + P.E. (r) where ρ denotes correlation coefficient in population and r denotes correlation coefficient in sample By convention the rules are – If |r|<6P.E. (r), then correlation is not significant and no correlation between 2 variables – If |r| >6 P.E.(r), then correlation is significant and this implies presence of strong correlation between 2 variables – If correlation coefficient is greater than 0.3 and probable error is relatively small, the correlation coefficient should be considered as significant n

Example 3 Find out correlation between age and playing habit from the following information. HINT: Playing habit can be denoted by ‘Y’ which is (Regular players/No. of students) X 100 AgeNo. of Students Regular Players

Types of Correlation: Spearman’s Rank Correlation or Spearman’s rho ρ = 1 – 6 d i 2 n(n 2 – 1) Where, d i = X i – Y i Used when characteristics are ranked (characteristics in beauty contest) 

Example Competit ors Judge X Judge Y Judge Z A136 B654 C589 D1048 E371 F2 2 G423 H91 I765 J897 Using rank correlation determine which pair of judges have the nearest approach to common tastes to beauty? Calculate d 2 XY and ρ XY Calculate d 2 YZ and ρ YZ Calculate d 2 XZ and ρ XZ

Spearman’s rho - Interpretation ‘rho’ value lies between ‘-1’ and ‘+1’ ‘-1’ is negative rank correlation and ‘+1’ is positive rank correlation Positive rank correlation implies that a high (low) rank of an individual according to one characteristic is accompanied by its high (low) rank according to the other (Consistency in ranking) Negative rank correlation implies that a high (low) rank of an individual according to one characteristic is accompanied by its low (high) rank according to the other (No consistency in ranking but does not mean inconsistency)

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