Methods of computation of correction stat.
- 1. Methods of Computation of Correlation Submitted By: Suthar Yogendra M. Ph.D. (Agri.) Plant Pathology Reg. No.: 1010121030 BACA, AAU, Anand Submitted To: Dr. A. D. Kalola Professor and Head Department of Agricultural Statistics BACA, AAU, Anand AG. STAT. 534 : STATISTICAL METHODS FOR CROP PROTECTION-II 1
- 2. Correlation • Correlation is statistical analysis which measures and analyses the degree or extent to which the two variables fluctuate with reference to each other. • It measures the closeness of the relationship. • Correlation does not indicate cause and effect of relationship. • Price and supply, income and expenditure are correlated. Definitions : • Correlation analysis attempts to determine the degree of relationship between variables - Ya- Kun-Chou • Correlation is an analysis of the co-variation between two or more variables- A. M. Tuttle 2
- 3. Methods of computation of correlation 1. Scatter diagram method 2. Graphic method 3. Algebraic method: Karl Pearson’s coefficient of correlation 4. Spearman’s Rank method 3
- 4. 1. Scatter diagram method 4
- 5. Scatter diagram method • It is the simplest method of studying the relationship between two variables diagrammatically. • One variable is represented along the horizontal axis and the second variable along the vertical axis. For each pair of observations of two variables, we put a dot in the plane. • There are as many dots in the plane as the number of paired observations of two variables. The direction of dots shows the scatter or concentration of various points. This will show the type of correlation. 5
- 6. Scatter diagram method 1. If all the plotted points form a straight line from lower left hand corner to the upper right hand corner then there is Perfect positive correlation. We denote this as r = +1 2. If all the plotted dots lie on a straight line falling from upper left hand corner to lower right hand corner, there is a perfect negative correlation between the two variables. In this case the coefficient of correlation takes the value r = -1. 3. If the plotted points in the plane form a band and they show a rising trend from the lower left hand corner to the upper right hand corner the two variables are highly positively correlated. 4. If the points fall in a narrow band from the upper left hand corner to the lower right hand corner, there will be a high degree of negative correlation. 5. If the plotted points in the plane are spread all over the diagram there is no correlation between the two variables. We denote this as r = 0. 6
- 7. Height Weight No Correlation Moderate Negative Correlation Perfect Negative Correlation High Degree of Positive correlation Perfect Positive relationship 7
- 8. Advantages: • Simple & Non Mathematical method • Not influenced by the size of extreme item • First step in investing the relationship between two variables Disadvantages: • Can not adopt the an exact degree of correlation 8
- 10. Graphic method • This method is used the individual values of the two variables and plotted on the graph paper. • We thus obtain two curves, one for X- variable and another for Y variable. • By examining the direction and closeness of the two curves drawn we can infer whether or not the variables are related. • If both the curves drawn on the graph is moving in the same direction (either upper or downward) correlation is said to be positive. • On the other hand, if the curves are moving in the opposite directions correlation is said to be negative. 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 Graphic Method X Y X 1 2 3 4 5 6 7 8 9 Y 9 8 10 12 11 13 14 16 15 10
- 12. Karl Pearson’s coefficient of correlation • Karl pearson, a great biometrician and statistician, suggested a mathematical method for measuring the magnitude of linear relationship between the two variables. It is most widely used method in practice and it is known as pearsonian coefficient of correlation. It is denoted by ‘r’. • It is a measure of intensity of association between two variables in a bivariate population. • The formula for calculating ‘r’ is 12
- 13. Example : 1 Calculate coefficient of correlation from the following data X 1 2 3 4 5 6 7 8 9 Y 9 8 10 12 11 13 14 16 15 x y 1 9 1 81 9 2 8 4 64 16 3 10 9 100 30 4 12 16 144 48 5 11 25 121 55 6 13 36 169 78 7 14 49 196 98 8 16 64 256 128 9 15 81 225 135 45 108 285 1356 597 n Y X XY xy n X X x 2 2 2 n Y Y y 2 2 2 13
- 14. 4. Spearman’s Rank method 14
- 15. Spearman’s Rank correlation • It is studied when no assumption about the parameters of the population is made. • This method is based on ranks. It is useful to study the qualitative measure of attributes like honesty, colour, beauty, intelligence, character, morality etc. • The individuals in the group can be arranged in order and there on, obtaining for each individual a number showing his/her rank in the group. • This method was developed by Edward Spearman in 1904. It is defined as by the formula Rank correlation, R Where, = squares of difference of ranks n= number of pairs of observations 15
- 16. • In case of a tie, the rank of each tied value is the mean of all positions they occupy. 1 12 1 6 1 2 3 2 n n p P d r i s Where di² = Square of difference of rank n = Number of pairs P = Number of items where ranks are common 16
- 17. Example : 2 In a marketing survey the price of tea and coffee in a town based on quality was found as shown below. Find relation between and tea and coffee price. Price of tea 88 90 95 70 60 75 50 Price of coffee 120 134 150 115 110 140 110 Price of tea Rank Rank d di2 88 3 120 4 1 1 90 2 134 3 1 1 95 1 150 1 0 0 70 5 115 5 0 0 60 6 110 6 0 0 75 4 140 2 2 4 50 7 100 7 0 0 6 R The relation between price of tea and coffee is positive at 0.89. Based on quality the association between price of tea and price of coffee is highly positive. 17
- 18. Test for significance of correlation coefficient • If ‘r’ is the observed correlation coefficient in a sample of ‘n’ pairs of observations from a bivariate normal population, then Prof. Fisher proved that under the null hypothesis, H0: ρ = 0 • The variables, x & y follows bivariate normal distribution, the population correlation coefficient of x and y is denoted by ρ, then it is often of interest to test whether ρ is zero or different from zero, on the basis of observed correlation coefficient ‘r’ • Thus if ‘r’ is the sample correlation coefficient based on a sample of ‘n’ observations, then the appropriate test statistic for testing the null hypothesis H0: ρ = 0, against the alternative hypothesis H1 ρ ≠ 0 • Test statistic, • t follows Student’s t – distribution with (n-2) d.f. • If calculated value of t > table value of t with (n-2) d.f. at specified level of significance, then the null hypothesis is rejected. That is, there may be significant correlation between the two variables. Otherwise, the null hypothesis is accepted 18
- 19. Comparison of sample 'r' with population value Ho: = 0 (both the variables are not linearly associated) Ha: 0 2) - (n ) r - (1 ρ - r t 2 2 2 . r y x xy ρ - r t r of SE 2 ) r - (1 r of SE 2 n 2 1 2 r n r t under Ho : = 0 19
- 20. • If cal. t table t0.05, (n-2) d.f. Ho: rejected • Rejection of Ho : Means there is an association between two variables under study. • If cal. t < table t0.05 (n-2) d.f. Ho: accepted • Acceptance of Ho: indicates that there is no association between two variables in the population. 20
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