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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 nonnegative.
(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(lr) ≥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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 Winter '08
 Moshiri
 Business

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