REGRESSION ANALYSIS.

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Presentation transcript:

REGRESSION ANALYSIS

MEANING OF REGRESSION: The dictionary meaning of the word Regression is ‘Stepping back’ or ‘Going back’. Regression is the measures of the average relationship between two or more variables in terms of the original units of the data. It also attempts to establish the nature of the relationship between variables that is to study the functional relationship between the variables and thereby provide a mechanism for prediction, or forecasting.

Importance of Regression Analysis Regression analysis helps in three important ways :- It provides estimate of values of dependent variables from values of independent variables. It can be extended to 2or more variables, which is known as multiple regression. It shows the nature of relationship between two or more variable.

USES OF REGRESSION In the field of business regression is widely used. Businessman are interested in predicting future production, consumption, investment, prices, profits, sales etc. So the success of a businessman depends on the correctness of the various estimates that he is required to make. It is also use in sociological study and economic planning to find the projections of population, birth rates. death rates etc.

METHODS OF STUDYING REGRESSION: GRAPHICALLY FREE HAND CURVE LEAST SQUARES ALGEBRAICALLY DEVIATION METHOD FROM AIRTHMETIC MEAN DEVIATION METHOD FORM ASSUMED MEAN Or

Algebraic method-: 1.Least Square Method-: The regression equation of X on Y is : X= a+bY Where, X=Dependent variable Y=Independent variable The regression equation of Y on X is: Y = a+bX Y=Dependent variable X=Independent variable And the values of a and b in the above equations are found by the method of least of Squares-reference . The values of a and b are found with the help of normal equations given below: (I ) (II )

Example1-:From the following data obtain the two regression equations using the method of Least Squares. X 3 2 7 4 8 Y 6 1 5 9 Solution-: X Y XY X2 Y2 3 6 18 9 36 2 1 4 7 8 56 49 64 5 20 16 25 72 81

Substitution the values from the table we get 29=5a+24b…………………(i) 168=24a+142b 84=12a+71b………………..(ii) Multiplying equation (i ) by 12 and (ii) by 5 348=60a+288b………………(iii) 420=60a+355b………………(iv) By solving equation(iii)and (iv) we get a=0.66 and b=1.07

By putting the value of a and b in the Regression equation Y on X we get Y=0.66+1.07X Now to find the regression equation of X on Y , The two normal equation are Substituting the values in the equations we get 24=5a+29b………………………(i) 168=29a+207b…………………..(ii) Multiplying equation (i)by 29 and in (ii) by 5 we get a=0.49 and b=0.74

2.Deviation from the Arithmetic mean method: Substituting the values of a and b in the Regression equation X and Y X=0.49+0.74Y 2.Deviation from the Arithmetic mean method: The calculation by the least squares method are quit cumbersome when the values of X and Y are large. So the work can be simplified by using this method. The formula for the calculation of Regression Equations by this method: Regression Equation of X on Y- Regression Equation of Y on X- Where, and = Regression Coefficient and

Taking deviations from the actual means of X and Y series. Example2-: from the previous data obtain the regression equations by Taking deviations from the actual means of X and Y series. X 3 2 7 4 8 Y 6 1 5 9 Solution-: X Y x2 y2 xy 3 6 -1.8 0.2 3.24 0.04 -0.36 2 1 -2.8 -4.8 7.84 23.04 13.44 7 8 2.2 4.84 4 5 -0.8 0.64 9 3.2 10.24

Regression Equation of X on Y is Regression Equation of Y on X is ………….(II)

3.Deviation from Assumed mean method-: It would be observed that these regression equations are same as those obtained by the direct method . 3.Deviation from Assumed mean method-: When actual mean of X and Y variables are in fractions ,the calculations can be simplified by taking the deviations from the assumed mean. The Regression Equation of X on Y-: The Regression Equation of Y on X-: But , here the values of and will be calculated by following formula:

Dev. From assu. Mean 7 (dx)=X-7 Dev. From assu. Mean 6 (dy)=Y-6 Example-: From the data given in previous example calculate regression equations by assuming 7 as the mean of X series and 6 as the mean of Y series. Solution-: X Y Dev. From assu. Mean 7 (dx)=X-7 Dev. From assu. Mean 6 (dy)=Y-6 dxdy 3 6 -4 16 2 1 -5 25 +25 7 8 4 5 -3 9 -1 +3

The Regression Coefficient of X on Y-: The Regression equation of X on Y-:

It would be observed the these regression equations are same as those The Regression coefficient of Y on X-: The Regression Equation of Y on X-: It would be observed the these regression equations are same as those obtained by the least squares method and deviation from arithmetic mean .