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# Correlation analysis

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### Correlation analysis

1. 1. Divyanshu Singh, Yalin Wang MBA, NGO-Management Business StatisticsHand Out: Date: 16/17 December 2011 Correlation AnalysisIntroduction to Correlation AnalysisWhy is correlation analysis done?Methods and ToolsSPSSApplication of Correlation Analysis in Business Management Correlation AnalysisIntroduction to Correlation Analysis:Correlation analysis is a statistical technique to quantify the dependence of two or morevariables. This dependence or degree of correlation is given by correlation coefficient ‘r’ or‘ρ’. Dependence of 2 variables states that values of both the variables increasesimultaneously. The correlation coefficient value lies between +1 and -1. Any value closeto1.0 or more than 0.5 shows positive correlation which means values of both the variablesincrease simultaneously showing linear dependence. A value less that 0.5 and close to -1.0shows negative correlation which means increase in the value of 1 variable shows decrease inthe value of another. There are several techniques for calculating correlation coefficient.Many Types of Correlation coefficient exist and choice of these depends in the type of data tobe analysed.Methods and Tools:Different types of correlation functions exist for different types of data analysis namely, 1) Pearson’s coefficient (r) 2) Spearman’s coefficient (ρ) 3) Point biserial coefficient etc…Degree of linear relationship between the 2 variables (r) is calculated by the ratio of thecovariance of 2 variables to the product of their standard deviations.
2. 2. Divyanshu Singh, Yalin Wang MBA, NGO-Management Business Statistics Date: 16/17 December 2011An r value of zero indicates that there is no relationship between the two variables. Note thatthe correlation coefficient is only intended to detect linear relationships between variablesthat are normally distributed.How are the results of Correlation Analysis interpreted? How to interpret the results from SPSS? (a) is a perfect linear correlation with r = 1 and (d) has a positive linear correlation with 0 <r< 1. Example (b) is a perfect linear correlation with r = -1 and (e) has a negative linear correlation with -1 <r< 0. Example (c) is not correlated with r= 0 and (f) has a non-linear correlation which causes r to be close to zero.
3. 3. Divyanshu Singh, Yalin Wang MBA, NGO-Management Business Statistics Date: 16/17 December 2011Interpretation of results of Correlation Analysis:Linear correlation is measured by calculating the Pearson correlation coefficient. Thiscoefficient is symbolized by r for a sample of data values and by the Greek letter ρ for apopulation. It is common practice to simply refer to this as the correlation coefficient.The correlation coefficient varies between -1.00 and +1.00. An r value of 1 indicates a perfectpositive linear correlation. This happens when the values of both variables increase togetherand their coordinates on a scatter plot form a straight line. An r value of -1 indicates a perfectnegative linear correlation. This happens when the values of one variable increases while theother variable decreases and their coordinates on a scatter plot form a straight line. Values ofr that are not zero show decreasing significance as they approach zero. The scatter plot ofvariables with r values not equal to 1 or -1 does not form a straight line.What is the purpose of Correlation Analysis?Correlation analysis shows the extent to which two quantitative variables vary together,including the strength and direction of their relationship. The strength of the relationshiprefers to the extent to which one variable predicts the other. For example, in a study ofconsumers, you might find that the amount of money spent weekly on groceries variesdirectly with the size of the household; youd expect it to be a strong positive relationship.However, youd expect to find a weak correlation (or none at all) between amount spentweekly on groceries and scores on a customer satisfaction survey. The direction of therelationship shows whether the two variables vary together directly or inversely. In a directrelationship, the two variables increase together. In an inverse relationship, one variable tendsto decrease as the other increases.Correlation analysis can be used to make inferences about one variable which cannot beeasily measured based on on which can be. For example, we cannot measure sales of aproduct which hasnt yet been produced or marketed. Correlation analysis of similar productsmay show us the variables which affect sales. The snack food industry might do a correlationanalysis of sales of snack foods with salt content, discovering that the more salt in potatochips, the higher the sales. This might lead to a business decision to produce snack foods withincreasing amounts of salt, with the goal of driving sales.
4. 4. Divyanshu Singh, Yalin Wang MBA, NGO-Management Business Statistics Date: 16/17 December 2011SPSS software enables such statistical calculations and analysis. Detail on thisfollows:SPSSSteps involved: 1) Create data for analysis or upload an existing data file. 2) Select the data required to analyse. 3) Correlate the data using Bivariate option.Using SPSS:  The Bivariate Correlations procedure computes the pairwise associations for a set of variables and displays the results in a matrix. It is useful for determining the strength and direction of the association between two scale or ordinal variables.  For quantitative, normally distributed variables, choose the Pearson correlation coefficient. If your data are not normally distributed or have ordered categories, choose Kendalls tau-b or Spearman, which measure the association between rank orders.  To check if the result is not by chance. When your research hypothesis states the direction of the difference or relationship, then you use a one-tailed probability. While it is generally safest to use a two-tailed test, there are situations where a one-tailed test seems more appropriate.  Flag significant correlations. Correlation coefficients significant at the 0.05 level are identified with a single asterisk, and those significant at the 0.01 level are identified with two asterisks.
5. 5. Divyanshu Singh, Yalin Wang MBA, NGO-Management Business Statistics Date: 16/17 December 2011  How to interpret the results from SPSS?Pearsons correlation coefficient (0.891) is significant at the 0.000 level which means thatthey are positively correlated.  How to interpret the results from SPSS?The plot shows that efficiency increases with automation and the number of working hoursdecreases due to less intake of human resources.Interpretation of Case Study  Increase in the use of Technology and Efficient service by the bank increase simultaneously  Computerized techniques and ATMs made banking easier and faster.  Automated procedures reduced errors caused manually.  The same applies to other variables as well.  But there is no causation between these variables.