Calculation of Product-moment Co-efficient of Linear Correlation

in

Bivariate Distribution- joint distribution of 2 variables

 

(Scatter Diagram - fill up the frequency)

x-score ↓| y-score → - - - - - - - -
Σx'y'              
                 
                 
(15)
-

 
(14)
-

 
(13)
-

 
(12)
-

 
(11)
-

 
(10)
-

 
(9)
-

 
(8)
-

 
(7)
-

 
(6)
-

 
(5)
-

 
(4)
-

 
(3)
-

 
(2)
-

 
(1)
-

 
                 
                 
                 
                 

(Fill this first)

               
               
Assumed Mean(AM)-   Assumed Mean(AM)-    
  Interval -                        Interval -                         
Score-x Mid-point (x) frequency-x Score-y Mid-point (y) frequency-y
AM Cumulative Frequency AM Cumu. Frequency
SD--  σx Coefficient of Correlation -- r SD--σy    
    Standard error of  r for large N        
    Coefficient of alienation --   k        
    coefficient of forecasting      E efficiency        
    Regression Eqn-I ȳ= (r* σy/σx )x = x Standard Error of ȳ  
    Regression Eqn-II x̄=(r* σx/σy )y = y Standard Error of x̄  
               

Formulae:

Arithmetic Mean = Assumed Mean + ci; where c = fx' / (sum of the frequencies=N ), i= no. of entries in a class interval;

Standard Deviation = i*√ [(Σ fx'2 / N) - c2]

Take Assumed Mean as the mid value corresponding to maximum frequency. In general, any value with the range of values can

 be taken as Assumed Mean.

Co-efficient of correlation : r = ((Σx'y') /N) -((Σfx')*(Σfy')/(N*N')) /  [ ( σx   / interval length)* (σy / interval length) ]

or

r = [((Σx'y') /N) -cxcy] / σ'y σ' x   

where Cx =  fx' / N             and   Cy =  fy' / N     and  σ' =  σ / (class interval)

Standard Error of Y = σy  *  √(1 - r2 )

Standard Error of X = σx  *  √(1 - r2 )

Coefficient of alienation  k = √(1 - r2 )

Coefficient of forecasting efficiency E = 1 - K

Standard Error of r = (1 - r2 ) / √ N    for large N

Example

X-variable - weight in pounds

Weight interval frequency
100-109 (ht interval- 1+1+1) 3
110-119 ( 1+2+5+2) 10
120-129  (0+7+7+9+4+1) 28
130-139(1+1+10+11+11+3) 37
140-149 (0+2+3+8+6+3) 22
150-159 (0+0+0+2+3+4) 9
160-169 (0+0+0+1+2+2) 5
170-179 (0+0+0+0+2+3+1) 6

 

Y-variable - Height  in inches

Height interval frequency
60-61 (ht interval- 1+1+0+1) 3
62-63 ( 1+2+7+1+2) 13
64-65  (1+5+7+10+3) 26
66-67 (0+2+9+11+8+2+1) 33
68-69 (0+0+4+11+6+3+2+2) 28
70-71 (0+0+1+3+3+4+2+3) 16
72-73 (0+0+0+0+0+0+0+1) 1
   

sd of height=2.62 ; sd of weight=15.55 r=0.60