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R 1 perfect ve correlation between the variables

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Unformatted text preview: 2 ].[ N Σfv 2 − (Σfv) 2 ] 40(−38) − 6 × 9 = [40 × 50 − 62 ].[40 × 47 − 92 ] −1520 − 54 −1574 = = = −0.8373 (2000 − 36) × (1880 − 81) 1964 × 1799 Properties of Correlation: 1. Correlation coefficient lies between –1 and +1 (i.e) –1 ≤ r ≤ +1 x −x y −y Let x’ = ; y’ = σx σy Since (x’ +y’ )2 being sum of squares is always non-negative. (x’ +y’ )2 ≥0 x’ 2 + y’ 2 +2 x’ y’ ≥ 0 2 y− y x− x x− x y− y Σ + 2Σ ≥0 + Σ σy σx σx σy 2 2 2Σ( x − x ) (Y − Y ) Σ( x − x ) Σ( y − y ) + + ≥0 σ x2 σ y2 σ xσ y 2 dividing by ‘ n’ we get 21 11 11 . Σ( x − x) ( y − y ) ≥ . Σ( x − x ) 2 + . Σ( y − y ) 2 + σ x2 n σ y2 n σ xσ y n 0 1 1 2 .σ x 2 + .cov( x, y ) ≥ 0 σ y2 + σ x2 σ y2 σ xσ y 1 + 1 + 2r ≥ 0 2 + 2r ≥ 0 2(1+r) ≥ 0 (1 + r) ≥ 0 –1 ≤ r -------------(1) 206 Similarly, (x’ –y’ )2 ≥ 0 2(l-r) ≥0 l - r ≥0 r ≤ +1 --------------(2) (1) +(2) gives –1 ≤ r ≤ 1 Note: r = +1 perfect +ve correlation. r = −1 perfect –ve correlation between the variables. Property 2: Property 3: Property 4: Property 5: Property 6: ‘ r’ is independent of change of origin and scale. It is a pure number independent of units of measurement. Independent variables are uncorrelated but the converse is not true. Correlation coefficient is the geometric mean of two regression coefficients. The correlation coefficient of x and y is symmetric. rxy = ryx. Limitations: 1. Correlation coefficient assumes linear relationship regardless of the assumption is correct or not. 2. Extreme items of variables are being unduly operated on correlation coefficient. 3. Existence of correlation does not necessarily indicate causeeffect relation. Interpretation: The following rules helps in interpreting the value of ‘ r’ . 1. When r = 1, there is perfect +ve relationship between the variables. 2. When r = -1, there is perfect –ve relationship between the variables. 3. When r = 0, there is no...
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