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