ACCA Paper F2 free course note | Correlation & Regression | Types of Correlation

Correlation is described or classified in several different ways. Three of the most important are

(1) From the viewpoint of inclusion of variables

v  Simple Correlation: Association between only two variables is simple correlation. When only two variables are studied it is a problem of simple correlation. For example, relationship between profit and capital.

v  Multiple Correlation: Association among more than two variables is multiple correlation. In a multiple correlation three or more variables are studied simultaneously. For example, when we study the relationship of profit and capital, production cost and advertisement cost.

v  Partial Correlation: Incase of multiple correlation the association between two variables is called partial correlation when effects of other variables remain constant. In partial correlation we recognize more than two variables. But consider only two variables to be influencing each other, the effect of other influencing variable being kept constant. For example, correlation between capital and profit when the effects of production cost and advertisement cost remain unchanged.

(2) From the view point of direction of variables

v  Positive Correlation: If the change of one variable is associated with the change of other variable is the same direction, then the correlation existing between variable is called positive correlation. For example, if one variable (i,e. investment) is increasing the other (i.e. profit) on an average  is also increasing or, if one variable (i,e. investment) is decreasing the other (i.e. profit) on an average  is also decreasing, then the correlation is said to be positive. 

Investment (X)	Profit (Y)
5	2
10	3
15	4
20	5
25	6

v  Negative Correlation: If the change of one variable is associated with the change of other variable is the opposite direction, then the correlation existing between variable is called negative correlation. For example, if one variable (i,e. supply) is increasing the other (i.e. demand) on an average  is decreasing or, if one variable (i,e. supply) is decreasing the other (i.e. demand) on an average  is  increasing, then the correlation is said to be negative. 

Supply (X)	Demand (Y)
5	25
10	20
15	15
20	10
25	5

v  No correlation: If the change of one variable is no way associated with the change of other variable, then that indicates no relation.

Supply (X)	Demand (Y)
5	5
10	20
15	3
20	50

(3) Linear and Non-Linear

v  Linear Correlation: When the ratio of change in both variables is constant then it is called linear correlation. If the amount of change in one variable tends to bear a constant ratio to the amount of change in other variable then the correlation is said to be linear.

Investment (X)	Profit (Y)
10	70
20	140
30	210
40	280
50	350

It is clear that the ratio of change between two variables is the same. If such variables are plotted on a graph paper, all plotted points would fall on straight line.

v  Non-linear Correlation: When the ratio of change in both variables does not give constant result then it is called non linear correlation. If the amount of change in one variable does not bear a constant ratio to the amount of change in other variable then the correlation is said to be non-linear. For example, if we double the amount of investment, the profit would not necessarily be doubled.

Investment (X)	Profit (Y)
10	70
20	100
30	120
40	150
50	350

ACCA Paper F2 free course note | Correlation & Regression | Correlation

Correlation analysis is the study of the relationship between two or more than two variables. It is the statistical tool we can use to describe the relationships between two or more than two variables. It is also defined as group of techniques to measure the association between two or more than two variables.

To explain, suppose the sales manager of Square Pharmaceuticals Ltd. wants to determine whether there is a relationship between the medicine sold in a month and advertisement.

Correlation analyses are based on the relationship between two (or more) variables. The known variable (or variables) is called the independent variable(s). The variable we are trying to predict is the dependent variable. Let’s take an example. Bankers might base their predictions of customer satisfaction on the interest rate. Thus, the interest rate is the independent variable and the customer satisfaction is the dependent variable. Bankers, for example, may add a second independent variable, environment, to improve their estimate of the customer satisfaction.

Coefficient of Correlation

The measure of correlation called the coefficient of correlation (denoted by the symbol r) summarizes in the figure the direction and degree of correlation.

Coefficient of Correlation is a measure of the strength of the linear relationship between two variables. It requires interval or ratio-scaled data.

Features of Correlation Coefficient

l  It can range from -1 to 1

l  Values of -1 or 1 indicate perfect and strong correlation.

l  The closer to -1, the stronger the negative linear relationship

l  The closer to 1, the stronger the positive linear relationship

l  Values close to 0 indicate weak correlation.

l  Negative values indicate an inverse relationship and positive values indicate a direct relationship.

ACCA Paper F2 free course note | Correlation & Regression | Regression Equation

Regression equation is an equation that expresses the linear relationship between two variables. It is an algebraic expression of the regression line. Since there are two regression lines, there are two regression equations—the regression equation of X on Y is used to describe the variations in the values of X for given changes in Y and the regression equation of Y on X is used to describe the variations in the values of Y for given changes in X.

Regression equation of Y on X

The regression equation of Y on X is expressed as follows:

                Y= a + bX

Where Y is the dependent variable to be estimated and X is the independent variable. The parameter ‘a’ determines the level of the fitted line. The parameter ‘b’ determines the slope of the line, i.e., the change in Y for unit change in X.

                b =

               

                a = - b

Regression equation of X on Y

The regression equation of X on Y is expressed as follows:

                X= a+ bY

 The parameter ‘a’ determines the level of the fitted line. The parameter ‘b’ determines the slope of the line, i.e., the change in X for unit change in Y.

                b =

               

                a = - b

Illustration: Calculate the regression equation of Profit on Capital and Capital on Profit from the following data.

Capital (X)	Profit (Y)
5	2
10	3
15	5
20	8
25	12

ACCA Paper F2 free course note | Correlation & Regression | What is Regression

The dictionary meaning of the term “regression” is the act of returning or going back. The term “regression” was first used in 1877 by Francis Galton while studying the relationship between the height of fathers and sons.

Regression analysis is the technique used to develop the equation and provide the estimate. In regression, we develop an equation to express the linear relationship between two variables. In addition, we are able to estimate the value of the dependent variable Y based on a selected value of the independent variable X.

The statistical tool with the help of which we are in a position to estimate (or predict) the unknown values of one variable from known values of another variable is called regression.

With the help of regression analysis we are in a position to find out the average probable change in one variable given a certain amount of change in another variable.

For example, if we know that advertisement and sales are correlated, we may find out the expected amount of sales for a given advertisement expenditure or the required amount of expenditure for achieving a fixed sales target.

Differences between correlation and regression analysis

There are two important points of difference between correlation and regression analysis. These are

v  Correlation coefficient measures the degree of association between two or more than two variables, whereas regression analysis measures the nature of relationship between the variables.

v  In case of correlation analysis, it never measure cause and effect relationship whereas regression analysis specially measures this.

Regression Line

Regression line is the graphical presentation of equation. It is a line drawn through a scatter plot of two variables If we take the case of two variables X and Y, we shall have two regression lines as the regression line of X on Y and the regression line of Y on X.