2. CURVE FITTING AND ITS APPLICATIONS

3. Group no 05  UW-13-EE-Bsc-021  UW-13-EE-Bsc-005  UW-13-EE-Bsc-007  UW-13-EE-Bsc-019  UW-13-EE-Bsc-043  UW-13-EE-Bsc-103

4. Content 1. Definition 2. Scatter Diagram, 3. Karl Pearson Coefficient of Correlation 4. Limits for Correlation Coefficient 5. Applications 6. Definition of Regression 7. Importance of Regression Analysis 8. USE IN ORGANIZATION 9. METHODS OF STUDYING REGRESSION 10. Applications

5. Definition of Correlation Correlation is a statistical tool which studies the relationship between the two variable under study Correlation analysis involves various method and techniques for studying and measuring the extent of the relationship between the two variable If there is any relation between two variables i.e. when one variable changes the other also changes in the same or in the opposite direction, we say that the two variables are correlated. It means the study of existence, magnitude and direction of the relation between two or more variables. The famous astronomist Bravais, Prof. Sir Fanci’s Galton, Karl Pearson Prof. Neiswang and so many others have contributed to this great subject. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

6. Negative Correlation–as x increases, y decreases x = hours of training (horizontal axis) y = number of accidents (vertical axis) Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar Scatter Plots and Types of Correlation 60 50 40 30 20 10 0 02468 1214161820 Hours of Training Accidents

7. Positive Correlation–as x increases, y increases x = SAT score y = GPA GPA Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar Scatter Plots and Types of Correlation 4.00 3.75 3.50 3.00 2.75 2.50 2.25 2.00 1.50 1.75 3.25 300350400450500550600650700750800 Math SAT

8. No linear correlation x = height y = IQ Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar Scatter Plots and Types of Correlation 160 150 140 130 120 110 100 90 80 606468727680 Height IQ

9. Types of Correlation Positive and negative correlation Linear and non-linear correlation Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

10. Positive and negative correlation If two variables change in the same direction, then this is called a positive correlation. For example: Advertising and sales. If two variables change in the opposite direction then the correlation is called a negative correlation. For example: T.V. registrations and cinema attendance. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

11. Linear and non-linear correlation 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. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

12. Degrees of Correlation Through the coefficient of correlation, we can measure the degree or extent of the correlation between two variables. On the basis of the coefficient of correlation we can also determine whether the correlation is positive or negative and also its degree or extent. Perfect correlation: If two variables changes in the same direction and in the same proportion, the correlation between the two is perfect positive Absence of correlation: If two series of two variables exhibit no relations between them or change in variable does not lead to a change in the other variable Limited degrees of correlation: If two variables are not perfectly correlated or is there a perfect absence of correlation, then we term the correlation as Limited correlation

13. Degrees of Correlation High degree, moderate degree or low degree are the three categories of this kind of correlation. The following table reveals the effect of coefficient or correlation. DegreesPositiveNegative Absence of correlation  Zero0 Perfect correlation  + 1 High degree  + 0.75 to + 1- 0.75 to -1 Moderate degree  + 0.25 to + 0.75- 0.25 to - 0.75 Low degree  0 to 0.250 to - 0.25

14. Methods Of Determining Correlation We shall consider the following most commonly used methods. (1) Scatter Plot (2) Kar Pearson’s coefficient of correlation

15. Scatter Plot Method 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. The degree of correlation is denoted by ‘ r ’ and its direction is given by the signs positive and negative. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

16. Scatter Plot Method i) If all points lie on a rising straight line the correlation is perfectly positive and r = +1 (see fig.1 ) ii) If all points lie on a falling straight line the correlation is perfectly negative and r = -1 (see fig.2) iii) If the points lie in narrow strip, rising upwards, the correlation is high degree of positive (see fig.3) iv) If the points lie in a narrow strip, falling downwards, the correlation is high degree of negative (see fig.4) v) If the points are spread widely over a broad strip, rising upwards, the correlation is low degree positive (see fig.5) vi) If the points are spread widely over a broad strip, falling downward, the correlation is low degree negative (see fig.6) vii) If the points are spread (scattered) without any specific pattern, the correlation is absent. i.e. r = 0. (see fig.7) Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

17. Scatter Plot Method i) If all points lie on a rising straight line the correlation is perfectly positive and r = +1 (see fig.1 ) ii) If all points lie on a falling straight line the correlation is perfectly negative and r = -1 (see fig.2) Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

18. Scatter Plot Method iii) If the points lie in narrow strip, rising upwards, the correlation is high degree of positive (see fig.3) iv) If the points lie in a narrow strip, falling downwards, the correlation is high degree of negative (see fig.4) Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

19. Scatter Plot Method v) If the points are spread widely over a broad strip, rising upwards, the correlation is low degree positive (see fig.5) vi) If the points are spread widely over a broad strip, falling downward, the correlation is low degree negative (see fig.6) vii) If the points are spread (scattered) without any specific pattern, the correlation is absent. i.e. r = 0. (see fig.7) Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

20. Scatter Plot Method Though this method is simple and is a rough idea about the existence and the degree of correlation, it is not reliable. As it is not a mathematical method, it cannot measure the degree of correlation. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

21. 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. It is defined as Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

22. Karl Pearson’s coefficient of correlation

23. Note : r is also known as product-moment coefficient of correlation. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

24. Karl Pearson’s coefficient of correlation Example Calculate the coefficient of correlation between the heights of father and his son for the following data. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar Height of father (cm): 165166167168167169170172 Height of son (cm): 167168165172168172169171 Solution: n = 8 ( pairs of observations )

25. Height of father x i Height of son y i x = x i -xy = y i -yxyx2x2 y2y2 165167-3-2694 166168-2241 167165-44116 167168 111 16817203009 16917213319 17016920040 172171428164  x i =1344  y i =135200  xy=24  x 2 =36  y 2 =44 Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

27. Karl Pearson’s coefficient of correlation Example From the following data compute the coefficient of correlation between x and y.. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

28. Limits for Correlation Coefficient Pearsonian correlation coefficient can not exceed 1 numerically. In other words it lies between -1 and +1. Symbolically, -1 <=r<=+1 Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar

29. Applications Architectural engineering Building engineering Building services engineering Chemical engineering Computer engineering Civil engineering Electrical engineering Environmental engineering Industrial engineering Materials engineering Mechanical engineering Safety engineering

30. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar Regression Analysis Regression analysis is a mathematical measure of the averages relationship between two or more variable in terms of the original units of data. Types of Regression (i)Simple Regression (Two Variable at a time) (ii)Multiple Regression (More than two variable at a time) Linear Regression: If the regression curve is a straight line then there is a linear regression between the variables. Non-linear Regression/ Curvilinear Regression: If the regression curve is not a straight line then there is a non-linear regression between the variables.

31. 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 2 or more variables, which is known as multiple regression. It shows the nature of relationship between two or more variable.

32. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar USE IN ORGANIZATION 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.

33. Subject :- Statistics & Numerical Mathematics Created By Prof. Santosh Ashilwar METHODS OF STUDYING REGRESSION: REGRESSION GRAPHICALLY FREE HAND CURVE LESAST SQUARES ALGEBRAICALLYLESAST SQUARES DEVIATION METHOD FROM AIRTHMETIC MEAN DEVIATION METHOD FORM ASSUMED MEAN Or

34. Algebraically 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 Where, 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 )

35. Example1-: From the following data obtain the two regression equations using the method of Least Squares. X32748 Y61859 Solution- : XYXYX2X2 Y2Y2 3618936 21241 78564964 45201625 89726481

36. 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

37. 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 Substituting the values of a and b in the Regression equation X and Y X=0.49+0.74Y

38. Applications Architectural engineering Building engineering Building services engineering Chemical engineering Computer engineering Civil engineering Electrical engineering Environmental engineering Industrial engineering Materials engineering Mechanical engineering Safety engineering