Correlation Analysis

A measure of association between two or more numerical variables. For examples height & weight relationship price and demand relationship CORRELATION

Correlation Analysis Independent and Dependent Variables Independent variable: The variable that is the basis of estimation is called. Dependent variable: The variable whose value is to be estimated is called dependent variable. The dependent variables are dependent on independent variables.

Correlation Analysis Example StudentHours studied% Marks

Correlation Analysis USE 1. With the help of correlation analysis we can measure in one figure the degree of relationship existing between the variables. 2. Correlation analysis contributes to the economic behavior, aids in locating the critically important variables on which disturbances spread and suggest to him the paths through which stabilizing forces become effective. 3.In business, correlation analysis enables the executive to estimate costs, sales, price and other variables on the basis of some other series with which these costs, sales or prices may be functionally related.

Correlation Analysis Example Independent variable in this example is the number of hours studied. The mark the student obtains is a dependent variable. The mark student obtains depend upon the number of hours he or she will study. Are these two variables related?

Correlation Analysis Types of correlation Correlation Positive and negative Simple, partial and multiple

Correlation Analysis  Height & Weight  Income & Expenditure  Training & performance Positive correlation A positive relationship exists when both variables increase or decrease at the same time.

Correlation Analysis  Strength and age  Demand & Price Negative correlation A negative relationship exist when one variable increases and the other variable decreases or vice versa.

Correlation Analysis Simple, partial and multiple correlation Association between only two variables is Simple correlation. (e.g. Height & Weight) Association among more than two variables is Multiple correlation. (e.g. Capital, Production cost, Advertisement cost & Profit) Incase of multiple correlation the association between two variables is called Partial correlation when effects of other variables remain constant. (e.g. correlation between Capital & Profit when the effects of Production cost & Advertisement cost remain unchanged.)

Correlation Analysis Scatter plots A scatter plot is a chart that shows the relationship between two quantitative variables measured on the same observations. In a scatter plot, one of the variables (usually the independent variable) is plotted along the horizontal or X axis and the other is plotted along the vertical or Y axis.

Correlation Analysis Specific Example Specific Example For seven random summer days, a person recorded the and their, during a three-hour period spent outside. For seven random summer days, a person recorded the temperature and their water consumption, during a three-hour period spent outside. Temperature (F) Water Consumption (ounces)

Correlation Analysis How would you describe the graph?

Correlation Analysis Types of correlations Y X Y X Y Y X X (continued) Perfect positive Perfect negative Strong positive Strong negative

Correlation Analysis No linear correlation x = height y = IQ Height IQ

Correlation Analysis Correlation Coefficient A quantity which measures the direction and the strength of the linear association between two numerical paired variables is called correlation coefficient.

Correlation Analysis Pearson’s correlation coefficient or product moment correlation Pearson’s Correlation coefficient (continued)

Correlation Analysis Example 1 A company has brought out an annual report in which the capital investment and profits were given for the few years. Capital Investment (cores) Profits (lakh)

Correlation Analysis Calculation X= Capital investment Y= Profits

Correlation Analysis Continue… = = = = =

Correlation Analysis Example 2 A departmental store has the following statistics of sales for a period of last one year of 8 salesmen who have varying years of experience. SalesmenYears of exp. Annual sales(tk)

Correlation Analysis Calculation X= years of experiences Y= Annual Sales

Correlation Analysis Continue… salesmen

Correlation Analysis Properties of r r lies between -1 to +1. i.e., The correlation coefficient is a symmetric measure. The r will be negative or positive depending on whether the sign of the numerator of the formula is positive or negative. The correlation coefficient is a dimensionless quantity, implying that it is not expressed in any unit of measurement.

Correlation Analysis Interpretation r=1 indicates a perfect positive correlation or relationship. In this case, all the points in a scatter diagram lie on a straight line that has a upward direction. r=-1 indicates a perfect negative correlation or relationship. In this case, all the points in a scatter diagram lie on a straight line that has a downward direction. r=0 indicates that the variables are not linearly related or no correlation.

Correlation Analysis Interpretation Value of r close to 1 indicates a strong positive correlation or strong positive linear relationship Value of r close to -1 indicates a strong negative correlation or strong negative linear relationship Positive value of r close to 0 indicates a weak positive correlation or weak linear relationship. Negative value of r close to 0 indicates a weak negative correlation or weak negative linear relationship.

Correlation Analysis Perfect negative corr. Perfect positive corr. Zero corr. Weak negative corr. Strong negative corr. Weak positive corr. Strong positive corr Moderate negative corr. Moderate positive corr. Negative correlationPositive correlation

Correlation Analysis Correlation Coefficient Interpretation Coefficient Range Strength of Relationship Very weak weak Moderate Strong Very strong

Correlation Analysis Interpret the following i. r = ii. r = 1 iii. r = 0.5 iv. r = -1 v. r = 0 vi. r=.92

Correlation Analysis Types of correlations Y X Y X Y Y X X (continued) r=1 r=-1 r close to +1 r close to -1

Correlation Analysis Y X Type of correlation r close to zero

Correlation Analysis r = ? Why? 1. r = ? Why? 2. Interpret the following

Correlation Analysis Y X 3. r = ? Why? Interpret the following

Correlation Analysis Characteristics of Correlation Correlation does not tell us anything about causation. To calculate correlation, both variables must be quantitative (not categorical). A positive value for r indicates a positive association between x and y. A negative value for r indicates a negative association between x and y.

Correlation Analysis Regression Analysis Regression analysis is a technique of studying the relationship of one independent variable with one or more dependent variables with a view to estimating or predicting the average value of the dependent variable in terms of the known or fixed values of the independent variables.

Correlation Analysis Objectives of regression Estimate the relationship that exists between the dependent variable and the independent variable. Determine the effect of each of the independent variables on the dependent variables. Prediction the value of the dependent variable for a given value of the independent variable.

Correlation Analysis Regression vs. Correlation The correlation answers the STRENGTH of linear association between paired variables, say X and Y. On the other hand, the regression tells us the FORM of linear association that best predicts Y from the values of X. In case of correlation, it never measure cause and effect relationship whereas regression specially measures this.

Correlation Analysis Regression vs. Correlation Linear regression are not symmetric in terms of X and Y. That is interchanging X and Y will give a different regression value. On the other hand, if you interchange variables X and Y in the calculation of correlation coefficient you will get the same value of this correlation coefficient.

Correlation Analysis Types of regression Linear regression that shows the relationship between one dependent variable and one independent variable. Multiple regression that shows the relationship between one dependent variable and two or more independent variables.

Correlation Analysis Regression model (equation) A model is mathematical equation that describes the relationship between a dependent variable and a set of independent variables. Intercept term slope term Dependent variable Independent variable

Correlation Analysis Interpretation  Y is dependent variable  X is independent variable  a is intercept term, also the expected value of Y for X=0.  b is slope term, also known as regression coefficient. It represents the amount of change in Y for each unit change in X.

Correlation Analysis Estimates of a and b The least squares principle is used to estimate a and b. The equations to determine a and b are

Correlation Analysis Properties of b It lies between Negative value of b indicates the relationship between two variables is negative. Positive value of b indicates the relationship between two variables is positive. b=0 indicates there is no relationship between the two variables.

Correlation Analysis Example 1 Age of trucks years54317 Repair expense last year in hundreds of $

Correlation Analysis Calculation == ==

Correlation Analysis Example 1 Mr. A, president of a financial services, believes that there is a relationship between the no. of client contacts and the dollar amount of sales. To document this assertion, Mr. A gathered the following sample information. No. of contacts Sales ( $ Thousand)

Correlation Analysis Calculation Find the Regression co-efficient and interpret it. Find the regression equation that express the relationship between these two variables. Determine the amount of sales if 40 contacts are made.

Correlation Analysis Problem Given data X Y Draw a scatter diagram 2. Calculate the correlation coefficient and interpret it. 3. Find the regression coefficient and interpret it. 4. Determine the value of Y when X= 11, 18

Correlation Analysis Coefficient of Determination The coefficient of determination (r 2 ) is the proportion of the total variation in the dependent variable (Y) that is explained or accounted for by the variation in the independent variable (X). It is the square of the coefficient of correlation. It ranges from 0 to 1. It does not give any information on the direction of the relationship between the variables.

Correlation Analysis