Correlation

Covariance, Correlation, Scatter Diagram, Karl Pearson's correlation coefficient, Autocorrelation.

1. Correlation and types of correlation. Joshua Rodrick and manas Pradeep

2. Introduction Many a times we come across situations where two variables are interrelated . For example: Marks and intelligence quotient of students. Demand and price of a certain commodity. Rainfall and agricultural production. Income and expenditure of a family. In these situations we may be interested in examining the relation between the two variables. Such interrelated variables are called as correlated variables.

3. Definition of Correlation and Bivariate Data. Correlation is a statistical tool to measure the extent of linear relation between two variables. Bivariate Data: In order to determine correlation, we require data regarding two concerned variables. These data are called bivariate data. Whenever the variables X and Y are mentioned in the same item, they are likely to be correlated.

4. Positive Correlation In some cases, ↑Increase in value of one variable is associated with ↑Increase in value of other variable or ↓Decrease in value of one variable is associated with ↓Decrease in value of other variable. Correlation between these variables is said to be POSITIVE.

5. Negative Correlation I n some situations ↑Increase in value of one variable is accompanied by ↓Decrease in value of other variable and ↓Decrease in value of one variable is accompanied by ↑Increase in value of the other Variable. Correlation between these two variables is said to be NEGATIVE.

6. Examples of Positive and Negative Correlation Positive Correlation Negative Correlation Relationship between Sale of Cold Drinks and Temperature. Relationship between Alcohol Consumption and Driving Ability. Relationship Between DUI’s and Accidents. Relationship between Supply and price of commodity.

7. No Correlation In some cases change in one variable is not related to change in other variable, In these cases there is said to be No Correlation or Zero Correlation between the two variables. For example, There is no relationship between the amount of tea drunk and level of intelligence. There is no relationship between height of students and grades scored in examinations.

8. Measures of Correlation There are several measure of correlation some of which are: Scatter Diagram Karl Pearson’s Coefficient of Correlation Rank Correlation

9. Scatter Diagram Scatter Diagram is a graph of observed potted points where each points represents the value of X & Y as a coordinate. It portrays the relationship between these two variables graphically.

10. Perfect Positive Correlation A perfect positive correlation is given the value of 1.

11. Perfect Negative Correlation A perfect negative correlation is given the value of -1.

12. Positive Correlation If the data points make a straight line going from the origin out to high x- and y-values, then the variables are said to have a positive correlation.

13. Negative Correlation If the line goes from a high-value on the y-axis down to a high-value on the x-axis, the variables have a negative correlation .

14. Non-Linear Correlation Sometimes when we look at a plot of data there is an obvious nonlinear relationship. In other words, the plotted data have an obvious curved appearance. They are known as Non-Linear Correlation

15. Merits and Demerits of Scatter Diagram Merits Demerits Scatter diagram is the simplest method of studying correlation. It does not give a numerical measure of correlation. It is easy to understand. It is a subjective method. It is not influenced by extreme values. It cannot be applied to qualitative data.

16. Karl Pearson’s Coefficient of correlation The Karl Pearson’s correlation coefficient method is quantitative and offers numerical value to establish the intensity of linear relationship between X and Y. Karl Pearson’s coefficient correlation is represented by ‘r’. The Pearson’s correlation measures the direction and degree of linear relationship between two variables.

17. Formula

18. Merits and Demerits of Karl Pearson’s Coefficient of Correlation Merits Demerits Karl Pearson’s coefficient of correlation determines a single values which summarizes the extent of linear relationship. It also indicates types of correlation. It cannot be computed for qualitative data such as honesty and intelligence, beauty and intelligence. It depends upon all observations. It is unduly affected by extreme values. It measures only linear relationship.

19. Applications of Karl Pearson’s Coefficient of correlation The Pearson correlation coefficient can be used to summarize the strength of linear relationship between two data samples. The Pearson’s correlation coefficient is calculated as the covariance of two variables divided by the standard deviation of each data sample.

20. Auto Correlation Sometimes the observations X 1 ,X 2 ….., X n are interrelated among themselves. In other words X i ’s are dependent. To measure such dependence, we compute correlation among the observation such a correlation is called auto correlation.

21. Examples of Autocorrelation During Monsoon, rainfall on n th day depends on rainfall on (n-1) th Day or (n-2) th Day . Price of share on n th day depends upon what happened on earlier day or a few earlier days. In order to analyse the data under this situation we make use of autocorrelation. It has applications in the analysis of time series data.

22. Rank Correlation Karl Pearson’s coefficient of correlation is the best measure of correlation, however it poses difficulty in measuring the correlation between qualitative characteristics. If the qualitative characteristics under study are recorded using ordinal scale, we can arrange the items in ascending or in descending order according to the merit that they possess.

23. Rank Correlation Ranking : Ordered arrangement if items according to merit that they possess is called ranking . Rank : The Number indicating the position in ranking is called as rank . Tie : Tie is said to occur in ranking if two or more items have same merit. In this case we allot common rank to these items. This rank is the average of ranks which would have been allotted if the respective items would differ in merit slightly.

24. Rank Correlation The product moment correlation between ranks is called as Spearman’s rank correlation coefficient. It is denoted by R. R is a Karl Pearson’s coefficient of correlation for bivariate data.

25. Rank Correlation Spearman’s rank correlation is simple to compute as compared t Karl Pearson’s coefficient of correlation. However there is a loss of accuracy, whenever we compute it for quantitative data. Spearman’s rank correlation is most commonly used to measure the correlation between the different traits such as intelligence and mathematical aptitude. Since R is Karl Pearson’s coefficient of correlation between the ranks, it lies between -1 and 1.

26. Rank Correlation With Ties When ranking the data, ties(two or more subjects having exactly the same value of a variable) are likely to occur. In case of ties, the tied observations receive the same average rank.

27. Formula

28. Rank Correlation With Ties If two or more items have same merit or quality, then common rank is allotted to each of such items. This rank is arithmetic mean of ranks which would have been given. In case the corresponding items would differ slightly in quality or merit. The number of items getting same rank called as length of the tie.

29. Rank Correlation With Ties We denote it by m. For example, suppose scores of the student are to be ranked. Let the scores be arranged in increasing order. Supposed 3rd and 4th student have same scores, then we give common rank to both. It will be an arithmetic mean of 3 and 4. Hence, we give rank 3.5 to both the students. In this case length of the tie is m = 2.

30. Thankyou. Joshua Rodrick and manas Pradeep