Correlation-Regression

It deals with association between two or more variables

Correlation analysis deals with covariation between two or more variables

Types1. Positive or negativeSimple or multipleLinear or non-linear

Methods of Measuring correlation1. Graphic Method2. Diagramatic Method- Scatter Diagram3. Algebraic methoda. Karl Pearson’s Coefficient of correlationb. Spearman’s Rank Co-efficient Correlationc. Coefficient of Concurrent deviationsd. Least Squares Method

Karl Pearson’s Coefficient of Correlation

Σ dx dy γ ( Gamma) = ------------------------- √ Σ dx2 Σ dy2

Σ dx dy = ------------------------- N σxσy dx = x-xbardy = y- ybardx dy = sum of products of deviations from respective

arithmetic means of both series

Karl Pearson’s Coefficient of Correlation After calculating assumed or working mean Ax & Ay Σ dx dy – (Σ dx) *( Σ dy)γ ( Gamma) = -------------------------------- √ [ NΣ dx2 - (Σ dx)2 * [Σ Ndy2 - (Σ dy)2 ]Σ dx dy = total of products of deviation from assumed

means of x and y seriesΣ dx = total of deviations of x seriesΣ dy = total of deviations of y seriesΣ dx2 = total of squared deviations of x seriesΣ dy2 = total of squared deviations of y seriesN= No. of items ( no. of paired items

Karl Pearson’s Coefficient of Correlation After calculating assumed or working mean Ax &

Ay Σ dx x Σ dy Σ dx dy - ---------------- Nγ ( Gamma) = ------------------------- (Σ dx)2 (Σ dy)2

√ [ Σ dx2 - --------- ] x [ Σ dy2 - ------------] N N

Assumptions of Karl Pearson’s Coefficient of Correlation 1. Linear relationship exists between the variablesProperties of Karl Pearson’s Coefficient of Correlation 1.value lies between +1 & - 12.Zero means no correlation3.γ ( Gamma) = √ bxy X byxWhere bxy X byx are regression coefficicentMerit Convenient for accurate interpretation as it gives degree &

direction of relationship between two variables

Limitations 1. Assumes linear relationship , even though it

may not be2. Method & process of calculation is difficult &

time consuming3. Affected by extreme values in distribution

Probable Error of Karl Pearson’s Coefficient of Correlation

1- γ2

Probable Error of γ ( Gamma) = 0.6745 -------- √ N

Q7.Calculate coefficient of correlation for following data

X65 63 67 64 68 62 70 66 68 67 69 71

Y 68 66 68 65 69 66 68 65 71 67 68 70

Ans Σ dx dy γ ( Gamma) = ------------------------- √ Σ dx2 Σ dy2

Σ dx dy = ------------------- N σxσy

1 2 3 4 5 6 7 8 9 10 11 12SumX Xbar

X 65 63 67 64 68 62 70 66 68 67 69 71 800 66.67

Y 68 66 68 65 69 66 68 65 71 67 68 70 811 67.58

dx=x-xbar -1.67 -3.67 0.33 -2.67 1.33 -4.67 3.33 -0.67 1.33 0.33 2.33 4.33

dx2 2.78 13.44 0.11 7.11 1.78 21.78 11.11 0.44 1.78 0.11 5.44 18.7884.67

dx.dy -0.69 5.81 0.14 6.89 1.89 7.39 1.39 1.72 4.56 -0.19 0.97 10.4740.33

dy=y-ybar 0.42 -1.58 0.42 -2.58 1.42 -1.58 0.42 -2.58 3.42 -0.58 0.42 2.42

dy2 0.17 2.51 0.17 6.67 2.01 2.51 0.17 6.67 11.67 0.34 0.17 5.8438.92

Σ dx dy sum dx2* sumdy2

3294.9

√ Σ dx2 Σ dy2 57.40

coeff of correlation = 0.70

Q8. following information about age of husbands & wives. Find correlation coefficient

Husband 23 27 28 29 30 31 33 35 36 39

Wife 18 22 23 24 25 26 28 29 30 32

γ ( Gamma) =0.99

1 2 3 4 5 6 7 8 9 10SumX Xbar

X 23 27 28 29 30 31 33 35 36 39 311 31.10

Y 18 22 23 24 25 26 28 29 30 32 257 25.70

dx=x-xbar -8.10 -4.10 -3.10 -2.10 -1.10 -0.10 1.90 3.90 4.90 7.90

dx2 65.61 16.81 9.61 4.41 1.21 0.01 3.61 15.21 24.01 62.41202.

9

dx.dy 62.37 15.17 8.37 3.57 0.77 -0.03 4.37 12.87 21.07 49.77178.

3

dy=y-ybar -7.70 -3.70 -2.70 -1.70 -0.70 0.30 2.30 3.30 4.30 6.30

dy2 59.29 13.69 7.29 2.89 0.49 0.09 5.29 10.89 18.49 39.69158.

1

Σ dx dy sum dx2* sumdy232078.4

9

√ Σ dx2 Σ dy2 179.10

coeff of correlation = 1.00

Q9. ten competitors in a cooking competition are ranked by three judges in the following way .by using rank coorelation method find out which pair of judges have nearest approachAns P&Q= -0.21 , Q &R=--0.3 P &R = +0.64

Q9. ten competitors in a cooking competition are ranked by three judges in the following way .by using rank coorelation method find out which pair of judges have nearest approach

P Q R

1 1 3 6

2 6 5 4

3 5 8 9

4 10 4 8

5 3 7 1

6 2 10 2

7 4 2 3

8 9 1 10

9 7 6 5

10 8 9 7

Rank coefficient of correlation 6Σ d2 ρ (rho) = 1 - ------------------- N3-N Σ d2 = total of squared differenceN = number of items

P Q RRp-Rq dpq2

Rq-Rr dqr2

Rp-Rr dpr2

1 1 3 6 -2 4 -3 9 -5 25

2 6 5 4 1 1 1 1 2 4

3 5 8 9 -3 9 -1 1 -4 16

4 10 4 8 6 36 -4 16 2 4

5 3 7 1 -4 16 6 36 2 4

6 2 10 2 -8 64 8 64 0 0

7 4 2 3 2 4 -1 1 1 1

8 9 1 10 8 64 -9 81 -1 1

9 7 6 5 1 1 1 1 2 4

10 8 9 7 -1 1 2 4 1 1

1000 200 214 0 60

6Sigma d2 1200 1284 360

N3-N 990 6Sigma d2/N3-N 1.21 1.297 0.3636

P= -0.21 -0.297 0.636364

Campus Overview

907/A Uvarshad, GandhinagarHighway, Ahmedabad – 382422.

Ahmedabad Kolkata

Infinity Benchmark, 10th Floor, Plot G1,Block EP & GP, Sector V, Salt-Lake, Kolkata – 700091.

Mumbai

Goldline Business Centre Linkway Estate, Next to Chincholi Fire Brigade, Malad (West), Mumbai – 400 064.

Thank You