Correlation Analysis

Correlation is not concerned with the relationship between variables; instead it gives an estimate as to the degree of association between the variables. Correlation analysis tests for interdependence of the variables.

Regression attempts to describe the dependence of a variable on one (or more) explanatory variables; it implicitly assumes that there is a one-way causal effect from the explanatory variable(s) to the response variable.

TYPES OF CORRELATION

i. Positive and Negativeii. Simple, partial and multipleiii.Linear and non-linear

Positive Correlation

• If the higher scores on X are generally paired with the higher scores on Y, and the lower scores on X are generally paired with the lower scores on Y, then the direction of the correlation between two variables is positive

Negative Correlation

If the higher scores on X are generally paired with the lower scores on Y, and the lower scores on X are generally paired with the higher scores on Y, then the direction of the correlation between two variables is negative.

MULTIPLE & PARTIAL CORRELATION• X1 - Yield of rice• X2 - Amount of Rainfall• X3 - Amount of fertilizers• X4 - Type of soil• X5 - Advanced technologies used.

• Correlation analysis of X1,X2,X3,X4 and X5 is an example of Multiple Correlation whereas if we only study the relation between X1 and X2 it would be an example of Partial Correlation

Linear and Non-linear

• The nature of the graph gives us the idea of the linear type of correlation between two variables. If the graph is in a straight line, the correlation is called a "linear correlation" and if the graph is not in a straight line, the correlation is non-linear or curvi-linear.

Methods Of Determining Correlation

Scatter Plot Karl Pearson’s coefficient of

correlation Spearman’s Rank-correlation

coefficient.

Scatter Plot ( Scatter diagram or dot diagram )

• In this method the values of the two variables are plotted on a graph paper. One is taken along the horizontal ( (x-axis) and the other along the vertical (y-axis). By plotting the data, we get points (dots) on the graph which are generally scattered and hence the name ‘Scatter Plot’.

• The manner in which these points are scattered, suggest the degree and the direction of correlation.

Karl Pearson’s coefficient of correlation

It gives the numerical expression for the measure of correlation. it is noted by ‘ r ’. The value of ‘ r ’ gives the magnitude of correlation and sign denotes its direction.

Formula for coefficient of correlation :

Karl Pearson’s coefficient of correlation

n Σdx.dy - ( Σ dx)(Σ dy)

√ n Σ d²x –(Σ dx)² x √ n Σd²y –(Σ dy)²

Where dx = x – A

dy = y- B (deviations of x and y from their assumed means)

r =

E.G. ( CALCULATION OF KARL PEARSON’S COEFFICIENT OF CORRELATION)

• CALCULATE THE KARL PEARSON’S COEFFICIENT OF CORRELATION FROM THE FOLLOWING DATA

X 100 200 300 400 500 600 700

y 30 50 60 80 100 110 130

x dx d²x y dy d²y dx.dy

100 30

200 50

300 60

400 80

500 100

600 110

700 130

x dx d²x y dy d²y dx.dy

100 -300 90000 30 -50 2500 15000

200 -200 40000 50 -30 900 6000

300 -100 10000 60 -20 400 2000

400 (A)

0 0 80(B)

0 0 0

500 100 10000 100 20 400 2000

600 200 40000 110 30 900 6000

700 300 90000 130 50 2500 15000

2800 0 280000 560 0 7600 46000Σ

Calculations

7 x 46000 - (0x0) √ [7 x 280000 – 0 ] √ [ 7x7600 – 0 ]

= 46000 √ 1960000 x √ 53200 = 0.997

r =

E.G. (KARL PEARSON’S COEFFICIENT OF CORRELATION)

• A COMPANY GIVES ON THE JOB TRANING TO ITS SALESMEN,FOLLOWED BY A TEST.

• IT IS CONSIDERING WHETHER IT SHOULD TERMINATE THE SERVICES OF ANY SALESMAN WHO DOES NOT DO WELL IN THE TEST.

• FOLLOWING DATA GIVE THE TEST SCORES AND SALES (IN 1000 Rs ) MADE BY NINE SALES MEN DURING LAST ONE YEAR.

• COMPUTE COEFFICIENT OF CORRELATION BETWEEN TEST SCORES & SALES.

• DOES IT INDICATE TERMINATION OF SERVICES OF SALESMEN WITH LOW SCORES IS JUSTIFIED.

TEST SCORES (X)

dx d²x SALES( in 1000 Rs) (Y)

dy d²y dx.dy

14 3119 3624 4821 3726 5022 4515 3320 4119 39

TEST SCORES (X)

dx d²x SALES( in 1000 Rs) (Y)

dy d²y dx.dy

14 -5 25 31 -8 6419 0 0 36 -3 924 5 25 48 9 8121 2 4 37 -2 426 7 49 50 11 12122 3 9 45 6 3615 -4 16 33 -6 5620 1 1 41 2 419 (A) 0 0 39 (B) 0 0

Σ 9 129 9 355 202

Calculation

9 x 202 - (9x9) √ (9x129) – (9)² x √ (9 x 355) - (9)² 1818 - 81 √ (1161-81) x √ 3195 - 81 1737 0.9473 √1080 x √ 3114

r =

=

= =

Thank You