Correlation and Regrretion
- 2. Submitted to : Prof. Tushar Mangukiya Prof. Sneha Patel Submitted by : Desani Parth - 140763106002
- 3. ď Correlation ďĄ Bivariate Distribution ďĄ Scatter Diagram ďĄ Coefficient of Correlation ďĄ Method of Calculation ďĄ Example ď Regression ďĄ Regression Line ďĄ Regression Coefficients ďĄ Rank Correlation Coefficients ďĄ Example
- 5. ď when a distribution has two variables then it is called bivariate. ďś Example: we consider the raw data collected from census considering only two variables, age and sex (26, M), (18, F), (48, F),(32, M), .. ď The first data shows age and second data shows sex of persons in each bracket. Thus it is the case of bivariate distribution. The bivariate raw data is usually presented in the form of a table called âthe bivariate frequency tableâ.
- 6. ď The simplest way to represent bivariate data in a diagram is called scatter diagram. ď In this diagram plot the corresponding pairs of numerical values(x , y) of the two series in the xy- plane. ď It is known as Dot Diagram. If the line goes upward & this upward movement is from left to right it will show positive correlation. ď If the line moves downward & this movement is from left to right, it will show negative correlation. ď The degree of slope will give the degree of correlation. ď If the points are scattered widely , it will show absence of correlation.
- 7. 0 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 Y-Values y values Negative
- 8. 0 2 4 6 8 10 12 0 0.1 0.2 0.3 0.4 0.5 0.6 Y-Values Column1 Positive
- 9. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 Y-Values y-value y-value2 Absence
- 10. As no numerical measure is possible with scatter diagram, Karl Pearson derived a mathematical formula to measure the degree of correlation between two variables, known as Karl Pearson's coefficient of correlation. It is also known as â product moment method.â For a bivariate distribution (x, y) the coefficient of correlation is denoted by and is defined asrxy ď¨ ďŠ ďłďł yx xy yx r ,cov ď˝ ďłďł yx n XYďĽď˝ ďĽ ďĽ ďĽď˝ 22 YX XY
- 11. a) Direct method: ď¨ ďŠď ď ď¨ ďŠ ďşďť ďš ďŞďŤ ďŠ ďď ď ď˝ ď˝ ďĽ ďĽďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ 2222 22 yyxx yxyx r r iiii iiii xy xy NN n YX XY
- 12. A sample of 6 children was selected, data about their age in years and weight in kilograms was recorded as shown in the following table . It is required to find the correlation between age and weight. Weight (Kg)Age (years)serial No 1271 862 1283 1054 1165 1396
- 14. r = 0.759âŚâŚ..........strong direct correlation ď¨ ďŠď ď ď¨ ďŠ ďşďť ďš ďŞďŤ ďŠ ďď ď ď˝ ďĽ ďĽďĽ ďĽ ďĽ ďĽ ďĽ 2222 yyxx yxyx r iiii iiii xy NN n ď¨ ďŠ ď¨ ďŠ ď¨ ďŠď ďď¨ ďŠ ď¨ ďŠď ď22xy 667426.412916 66414616 r ďď ď´ď ď˝
- 15. b) Step deviation method: The above method is simple, but becomes tedious if the mean of the variable is not integers. This can be overcome by the use of assumed mean. Let A and B be the assumed mean of the variables x and y respectively. Let, ď¨ ďŠ ď¨ ďŠB A yd xd iy ix ďď˝ ďď˝ ď¨ ďŠď ď ď¨ ďŠď ďďĽ ďĽďĽ ďĽ ďĽ ďĽ ďĽ ďď ď ď˝ 2222 dddd dddd r yyxx yxyx xy NN N
- 16. E. R.I. R.Student 1011051 1031042 1001023 981014 951005 96996 104987 92968 97939 949210 Find the coefficients of correlation between Intelligence Ratio (I.R.) and Emotional Ration (E.R.) from the following data :
- 17. Let X be intelligence ration And Y be emotional ratio. We construct the following table. Taking A = 100 B = 100 dx = X â A = X â 100 dy = Y â B = Y - 100
- 18. X Y dx = X - 100 dx² dy = Y - 100 dy² dxdy 105 101 5 25 1 1 5 104 103 4 16 3 9 12 102 100 2 4 0 0 0 101 98 1 1 -2 4 -2 100 95 0 0 -5 25 0 99 96 -1 1 -4 16 4 98 104 -2 4 4 16 -8 96 92 -4 16 -8 64 32 93 97 -7 49 -3 9 21 92 94 -8 64 -6 36 48 - - âdx = -10 âdx² = 180 âdy = -20 âdy² = 180 âdxdy = 112
- 19. ď The correlation coefficient between I. R. and E. R. is given by, ď¨ ďŠď ď ď¨ ďŠ ďşďť ďš ďŞďŤ ďŠ ďď ď ď˝ ďĽ ďĽďĽ ďĽ ďĽ ďĽ ďĽ 2222 yyxx yxyx r iiii iiii xy NN n ď ď ď ď22 )20()180(10)10()180(10 )20)(10()112(10 ďďď ďďď ď˝rxy ď ď ď ď4001800)100(1800 2001120 ďď ď ď˝rxy ď ď ď ď14001700 920 ď˝rxy 5963.0 83.1542 920 42.37*23.41 920 ď˝ď˝ď˝rxy
- 20. c) For grouped data: ď¨ ďŠ ďş ďť ďš ďŞ ďŤ ďŠ ďˇ ď¸ ďś ď§ ď¨ ďŚď ďşďť ďš ďŞďŤ ďŠ ď ď ď˝ ďĽ ďĽďĽ ďĽ ďĽ ďĽ ďĽ 2222 dfdfdfdf dfdfddf r ii N ii N ii N yyxx yixyx xy
- 22. Let be the given observations and line to be fitted âŚâŚ(1) Using the method of least squares, we can estimate the values of a and b. That is by using from (1) But, ď¨ ďŠď¨ ďŠ ď¨ ďŠyxyxyx nn ,, 2211 ď bxay ďŤď˝ ďĽ ďĽ ďĽ ďĽ ďĽ ďŤď˝ ďŤď˝ bxaxy xbany ď¨ ďŠď¨ ďŠ ď¨ ďŠ ď¨ ďŠ2 ďĽďĽďĽ ďďŤďď˝ďď xxbxxayyxx ď¨ ďŠ 0ď˝ďďĽ xx
- 23. The line of best fit becomes. ď¨ ďŠxxyy x y xyr ďď˝ď ďł ďł ďł ďł y x xyrď˝ ď¨ ďŠď¨ ďŠ ď¨ ďŠ 222 ďł x n XY X XY xx yyxx b ďĽ ďĽ ďĽ ďĽ ďĽ ď˝ď˝ ď ďď ď˝
- 24. i. When or ii. When , where A and B are assumed values. yYxX yx ii ďď˝ďď˝ , ďĽ ďĽ ďĽ ďĽ ď˝ď˝ 22 , X XY Y XY bb xyxy ď¨ ďŠ ď¨ ďŠďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ď ď ď˝ ď ď ď˝ 2222 , xxn yxxyn yyn yxxyn bb yxxy BA ydxd iyix ďď˝ďď˝ , ď¨ ďŠ ď¨ ďŠďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ď ď ď˝ ď ď ď˝ 22 2 , 2 dd dddd b dd dddd b xx yxyx yx yy yxyx xy n n n n
- 25. iii. For frequency distribution : ď¨ ďŠ ď¨ ďŠďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ďĽ ď ď ď˝ ď ď ď˝ 2222 , dd dddd b dd dddd b xx yxyx yx yy yxyx xy ffn ffn ffn ffN
- 26. ď It is a non-parametric measure of correlation. ď This procedure makes use of the two sets of ranks that may be assigned to the sample values of x and Y. ď Spearman Rank correlation coefficient could be computed in the following cases: ďĄ Both variables are quantitative. ďĄ Both variables are qualitative ordinal. ďĄ One variable is quantitative and the other is qualitative ordinal. 1)n(n (di)6 1r 2 2 s ď ďď˝ ďĽ
- 27. ď In a study of the relationship between level education and income the following data was obtained. Find the relationship between them and comment. Income (Y) level education (X) sample numbers 25Preparatory.A 10Primary.B 8University.C 10secondaryD 15secondaryE 50illiterateF 60University.G