Correlation & Regresion

Correlation And Regression: 

Correlation And Regression

Correlation : 

Correlation Is defined as the degree of relationship between two or more variables. Two variables are said to be correlated if the change in one variable results in a corresponding change in the other variable.

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Degree of correlation between two variables is called Single correlation. Degree of correlation between one variable and several variables is called Multiple correlation.

Linear Correlation: 

L inear Correlation Points scattered around a straight line is linear correlation.

Non-Linear Correlation: 

Non-Linear Correlation Points scattered around a curve is non-linear correlation.

Doubtful Correlation: 

Doubtful Correlation Points scattered all around without forming a pattern there may not be a correlation existing.

Types of correlation: 

Types of correlation Positive Correlation : if two variables tend to change together i.e. increase or decrease.

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Negative Correlation : if two variables tend to change in opposite direction i.e. one increases the other decreases.

Correlation Coefficient : 

Correlation Coefficient R = covariance of x and y (S.D. of x)(S.D. of y)

Example: 

Example X values Y values 60 3.1 61 3.6 62 3.8 63 4 65 4.1

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Step 1: Count the number of values. N = 5 Step 2: Find XY, X 2 , Y 2 See the below table

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X Value Y Value X*Y X*X Y*Y 60 3.1 60 * 3.1 = 186 60 * 60 = 3600 3.1 * 3.1 = 9.61 61 3.6 61 * 3.6 = 219.6 61 * 61 = 3721 3.6 * 3.6 = 12.96 62 3.8 62 * 3.8 = 235.6 62 * 62 = 3844 3.8 * 3.8 = 14.44 63 4 63 * 4 = 252 63 * 63 = 3969 4 * 4 = 16 65 4.1 65 * 4.1 = 266.5 65 * 65 = 4225 4.1 * 4.1 = 16.81

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Step 3: Find ΣX, ΣY, ΣXY, ΣX 2 , ΣY 2 . ΣX = 311 ΣY = 18.6 ΣXY = 1159.7 ΣX 2 = 19359 ΣY 2 = 69.82

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Step 4: Now, Substitute in the above formula given. Correlation(r) =[ NΣXY - (ΣX)(ΣY) / Sqrt ([NΣX 2 - (ΣX) 2 ][NΣY 2 - (ΣY) 2 ])] = ((5)*(1159.7)-(311)*(18.6))/ sqrt ([(5)*(19359)-(311) 2 ]*[(5)*(69.82)-(18.6) 2 ]) = (5798.5 - 5784.6)/ sqrt ([96795 - 96721]*[349.1 - 345.96]) = 13.9/ sqrt (74*3.14) = 13.9/ sqrt (232.36) = 13.9/15.24336 = 0.9119

Applications: 

Applications Quality control reviews the factors involved in manufacturing and production; it can make use of statistical sampling of product items to aid decisions in process control or in accepting deliveries. Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in the insurance and finance industries.

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Biostatistics is a branch of biology that studies biological phenomena and observations by means of statistical analysis, and includes medical statistics . Econometrics is a branch of economics that applies statistical methods to the empirical study of economic theories and relationships.

Regression Analysis: 

Regression Analysis A regression is a statistical analysis assessing the association between two variables. It is used to find the relationship between two variables.

Regression Formula:: 

Regression Formula: Regression Equation(y) = a + bx Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX 2 - (ΣX) 2 ) Intercept(a) = (ΣY - b(ΣX)) / N

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where x and y are the variables. b = The slope of the regression line a = The intercept point of the regression line and the y axis. N = Number of values or elements X = First Score Y = Second Score ΣXY = Sum of the product of first and Second Scores ΣX = Sum of First Scores ΣY = Sum of Second Scores ΣX 2 = Sum of square First Scores

Example: 

X values Y values 60 3.1 61 3.6 62 3.8 63 4 65 4.1 Example

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To find regression equation, we will first find slope, intercept and use it to form regression equation.. Step 1: Count the number of values. N = 5 Step 2: Find XY, X 2 See the below data X Value Y Value X*Y X*X 60 3.1 60 * 3.1 = 186 60 * 60 = 3600 61 3.6 61 * 3.6 = 219.6 61 * 61 = 3721 62 3.8 62 * 3.8 = 235.6 62 * 62 = 3844 63 4 63 * 4 = 252 63 * 63 = 3969 65 4.1 65 * 4.1 = 266.5 65 * 65 = 4225

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Step 3: Find ΣX, ΣY, ΣXY, ΣX 2 . ΣX = 311 ΣY = 18.6 ΣXY = 1159.7 ΣX 2 = 19359

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Step 4: Substitute in the above slope formula given. Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX 2 - (ΣX) 2 ) = ((5)*(1159.7)-(311)*(18.6))/((5)*(19359)-(311) 2 ) = (5798.5 - 5784.6)/(96795 - 96721) = 13.9/74 = 0.19 Step 5: Now, again substitute in the above intercept formula given. Intercept(a) = (ΣY - b(ΣX)) / N = (18.6 - 0.19(311))/5 = (18.6 - 59.09)/5 = -40.49/5 = -8.098

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Step 6: Then substitute these values in regression equation formula Regression Equation(y) = a + bx = -8.098 + 0.19x. Suppose if we want to know the approximate y value for the variable x = 64. Then we can substitute the value in the above equation. Regression Equation(y) = a + bx = -8.098 + 0.19(64). = -8.098 + 12.16 = 4.06

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Trend line A trend line represents a trend, the long-term movement in t ime series data after other components have been accounted for. It tells whether a particular data set (say GDP, oil prices or stock prices) have increased or decreased over the period of time. Epidemiology Early evidence relating tobacco smoking to mortality and morbidity came from observationa l studies employing regression analysis. Applications:

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Finance The capital asset pricing model uses linear regression as well as the concept of Beta for analyzing and quantifying the systematic risk of an investment. Economics Linear regression is the predominant empirical tool in economics . For example, it is used to predict consumption spending,fixed investment spending, inventory investment, purchases of a country's exports, spending on imports, the demand to hold liquid assets, labor demand, and labor supply.

Case Study: 

Case Study A survey done by the FAS Research Agency for Chemical Industries in Delhi with following Observation

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Sr. no. Firms Sales ('000 units) Expenses (Rs. '000) 1 Patel Chemical 50 11 2 Shah Fertilizers 50 13 3 J K Pesticides 55 14 4 G R Pvt Ltd 60 16 5 R P Pesticides 65 16 6 Y G Pvt Ltd 65 15 7 RAMIFS Ltd 65 15 8 H B Pvt Ltd 60 14 9 A J Chemicals 60 13 10 S K Fertilizers 50 13

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With the help of Karl Pearson’s coefficient Of Correlation between sales and expenses of the Above firms

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Firms x y u = (x-65)/5 v = y-13 u*u v*v u*v 1 50 11 -3 -2 9 4 6 2 50 13 -3 0 9 0 0 3 55 14 -2 1 4 1 -2 4 60 16 -1 3 1 9 -3 5 65 16 0 3 0 9 0 6 65 15 0 2 0 4 0 7 65 15 0 2 0 4 0 8 60 14 -1 1 1 1 -1 9 60 13 -1 0 1 0 0 10 50 13 -3 0 9 0 0 Total 580 140 -14 10 34 32 0

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R = covariance of x and y (S.D. of x)(S.D. of y) = 0.786

Conclusion: 

Conclusion

Thank You: 

Thank You