Page 180 - DMGT404 RESEARCH_METHODOLOGY

Research Methodology

Notes \ r = r
XY uv
This shows that correlation between X and Y is equal to correlation between u and v,
where u and v are the variables obtained by change of origin and scale of the variables X
and Y respectively.
This property is very useful in the simplification of computations of correlation. On the
basis of this property, we can write a short-cut formula for the computation of r :
XY
nå u v - (å u )(å v )
i i i i
r = ...(10)
XY nå u - (å u i ) 2 nå v - (å v i ) 2
2
2
i
i
2. The coefficient of correlation lies between – 1 and + 1.
To prove this property, we define
X  X Y Y
x '  i and '  i
y
i i
 
X Y
X i  X  2 Y i  Y  2
2
2
y
\ x '  2 and '  2
i
i
 
X Y
  X  X  2   Y  Y  2
2
2
or  x '  i and  y '  i
i
 2 i  2
X Y
From these summations we can write å ' x  å ' y  n
2
2
i
i
1
  X  X Y  Y 
n i i 1  X i  X   Y i  Y  1
y
Also, r =          x ' ' i
i
  Y n   X    Y  n
X
Consider the sum x  + y . The square of this sum is always a non-negative number,
i i
2
i.e., (x  + y )  0.
i i
Taking sum over all the observations and dividing by n, we get
1 å ( 'x + 2  0 1 å ( 'x + ' y + 2 '  0
2
2
x
n i i ) ' y or n i i i i ) ' y
1 ' x + å ' y + å
2
1
2
2
y
or å i i x ' '  0
i
i
n n n
or 1 + 1 + 2r ³ 0 or 2 + 2r ³ 0 or r ³ – 1 .... (11)
Further, consider the difference x  – y . The square of this difference is also non-negative,
i i
2
i.e., (x  – y )  0.
i i
Taking sum over all the observations and dividing by n, we get
1 å ( 'x - ) ' y 2  0
n i i
1 å 2 2
x
or ( 'x + ' y - 2 ' i i ) ' y  0
i
i
n
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