BUSINESS STATISTICS

N. D. VOHRA

Chapter 13

Correlation Analysis

Classification of Statistical

data

One variable

Univariate

More than one variable

Bivariate (Two Variables)

Multivariate (more than

two variables)

INTRODUCTION

INTRODUCTION

For a study of correlation and regression analysis, we consider bivariate and multivariate data.

Correlation analysis: Related to discovery and measurement of degree of co-variation of the variables involved.

Regression analysis: Analysis of the nature of relationship with a view to make estimates of the values of one variable on the basis of the given values of the other variable(s).

Bivariate Data : When two variables move in sympathy with each other so that changes in one variable are associated with changes in the other variable in the same, or in the opposite direction, they are said to be correlated.

When the variables move in same direction, then the correlation is said to be positive while if they are in the opposite directions, the correlation is said to be negative.

Remember that the direction of movement indicated is in general. It means that it is not necessary that in positive correlation a higher value of one variable shall necessarily be accompanied by a higher value of the other.

CORRELATION ANALYSIS

DIRECTION AND DEGREES OF CORRELATION

D I R E C T I O N

Higher values of one variable are associated with higher values of the other variable & lower values with lower values,

Higher values of one variable are associated with higher values of the other variable & lower values with lower values,

Perfect/Strong Correlation

No Correlation

DEG

REE

Linear and Non-linear RelationshipIn a set of bivariate data, when pairs of values are plotted on a graph then they would fall on, or closely on, a straight line, correlation is linear. If they do not, the correlation is nonlinear.

Simple, Multiple and Partial CorrelationsThe correlation is said to be simple when we deal with bivariate data. In case three or more variables are involved so that we are dealing with multivariate data sets, the correlation between variables is multiple or partial.

CORRELATION

In this case, pairs of values are given. The variables are arbitrarily designated as X and Y

and we seek to determine if the two are correlated. And if they are correlated then what is the degree

and direction of such correlation. An idea about the correlation can be had by

showing the data on a scatter diagram. To draw a scatter diagram, plot the values the two

variables on the two axes of a graph – one on the X-axis and the other on Y-axis.

The various airs of values are shown by means of dots.

SIMPLE CORRELATION

While moving to right on the X-axis, if various dots are found to be lying higher and higher on the graph, the correlation between variables is positive. On the other hand, if they are observed to be lying lower and lower, then the correlation is negative.

If various dots may be joined by a straight line, sloping upward or downward, the correlation is said to be perfect. The correlation is positive or negative accordingly as the line is sloping upward or downward.

If the dots do not fall exactly on a line but are very close to being on a line, then there is a high degree of correlation.

GRAPHIC ANALYSIS OF CORRELATION: SCATTER DIAGRAM

The more scattered are the dots, the smaller is the degree of correlation between the variables.

There is no correlation between the variables when the dots are so scattered that there is no clear

direction of their slope, and the dots are falling on a line that is parallel to the X-

axis or the Y-axis. A line parallel to the X-axis implies that the variable Y is not responsive to changes in X whereas a line parallel to the Y-axis implies that X is not sensitive to changes in Y. Hence there is no correlation in either case.

GRAPHIC ANALYSIS OF CORRELATION: SCATTER DIAGRAM

SOME SELECTED SCATTER DIAGRAMS

At National Company the newly recruited salesmen are given a training which is followed by an aptitude test before they are put on the job. The following data collected by the sales manager of the company shows the scores at the aptitude test and sales made in the first quarter of their employment by a total of 10 salesmen. Plot these data on a graph as a scatter diagram and establish whether correlation exists between the test scores and sales.

EXAMPLE

Salesman: 1 2 3 4 5 6 7 8 9 10

Test scores: 18 20 21 22 27 27 28 29 29 29

Sales (000 Rs):

23 27 29 28 28 31 35 30 36 33

SOLUTION

16 17 18 19 20 21 22 23 24 25 26 27 28 29 3020

22

24

26

28

30

32

34

36

38

40

Test Scores, X

Sale

s (

'000

Rs)

Line through

the points

The Karl Pearson’s coefficient of correlation is also called product-moment correlation coefficient.

The coefficient is defined as the ratio of covariance to the product of individual standard deviations of the two series. Thus,

The covariance between X and Y for n pairs of observations is defined as follows:

KARL PEARSON’S COEFFICIENT OF CORRELATION

It may be noted that when calculation is done, as usually it is, using sample data, we have

In either case, when calculation is done considering whole population data or sample data, the formula for coefficient of correlation simplifies to the following:

KARL PEARSON’S COEFFICIENT OF CORRELATION

This coefficient may assume negative as well as positive values and its value can lie only within ±1.

The negative sign of the correlation coefficient implies negative correlation between the variables and positive sign implies a positive correlation.

Ignoring sign, closer the coefficient to zero, smaller the degree of correlation and closer is the value to 1, higher is the degree of correlation.

However, the correlation coefficient should always be interpreted taking in to account the sample size.

KARL PEARSON’S COEFFICIENT OF CORRELATION

0 1r = 0

No correlation

r = 1Perfect

correlation

0.5

For a given series of paired data, the following information is available:

Covariance between X and Y series = −17.8Standard deviation of X series = 6.6Standard deviation of Y series = 4.2No. of pairs of observations = 20

Calculate the coefficient of correlation.

We have,

Thus, variables are negatively correlated.

AN EXAMPLE

By measuring deviations from mean values: Calculate Measure deviations of X and Y values from their

means and represent them as Multiply different pairs of deviations and add the

products to get Square the deviations and add them up. Apply the formula:

CALCULATION OF COEFFICIENT OF

CORRELATION

At National Company the newly recruited salesmen are given a training which is followed by an aptitude test before they are put on the job. The following data collected by the sales manager of the company shows the scores at the aptitude test and sales made in the first quarter of their employment by a total of 10 salesmen. Calculate coefficient of correlation

AN EXAMPLE

Salesman: 1 2 3 4 5 6 7 8 9 10

Test scores: 18 20 21 22 27 27 28 29 29 29

Sales (000 Rs):

23 27 29 28 28 31 35 30 36 33

ScoresX

SalesY

18 23 −7 −7 49 49 49

20 27 −5 −3 15 25 9

21 29 −4 −1 4 16 1

22 28 −3 −2 6 9 4

27 28 2 −2 −4 4 4

27 31 2 1 2 4 1

28 35 3 5 15 9 25

29 30 4 0 0 16 0

29 36 4 6 24 16 36

29 33 4 3 12 16 9

250 300 0 0 123 164 138

SOLUTION

Here,

Further,

To conclude, there appears to be high degree of positive correlation between the test scores and sales.

SOLUTION

By measuring deviations from assumed mean values:

Take assumed means, AX and AY for the two series. Measure deviations of X values from AX and deviations

of Y values from AY . Label these as dx and dY respectively.

Apply the formula:

This formula is useful where the mean values bear fractions.

CALCULATION OF COEFFICIENT OF CORRELATION

Test ScoresX

SalesY

dx

= X-20

dy

= Y-33dx × dY dx

2 dY 2

18 23 −2 −10 20 4 100

20 27 0 −6 0 0 36

21 29 1 −4 −4 1 16

22 28 2 −5 −10 4 25

27 28 7 −5 −35 49 25

27 31 7 −2 −14 49 4

28 35 8 2 16 64 4

29 30 9 −3 −27 81 9

29 36 9 3 27 81 9

29 33 9 0 0 81 0

TOTAL 50 −30 −27 414 228

SOLUTION USING ASSUMED MEANS

Substituting calculated values in the formula, we get

SOLUTION USING ASSUMED MEANS

Without measuring deviations: In this method, the products of the corresponding

X and Y values are computed along with squares of the X and Y values, and the summations of these all are obtained.

Finally, the following formula is applied:

CALCULATION OF COEFFICIENT OF CORRELATION

Test ScoresX

SalesY

XY X2 Y2

18 23 414 324 529

20 27 540 400 729

21 29 609 441 841

22 28 616 484 784

27 28 756 729 784

27 31 837 729 961

28 35 980 784 1,225

29 30 870 841 900

29 36 1,044 841 1,296

29 33 957 841 1,089

250 300 7,623 6,414 9,138

SOLUTION

Here,

The result, evidently, is same by all three methods.

SOLUTION

Name the two variables as X and Y. Now, find mid-points of the different classes for both the

variables. Take deviations, or step-deviations, from assumed mean

values in respect of each of the variables and label these as dx and dY respectively.

Have three columns headed fdY fdY2, fdY dx and three rows

headed fdx, fdx2 and fdY dx.

Multiply marginal frequencies ( total of cell frequencies) with dY dY

2, and enter these products in appropriate columns. Repeat the process for each of the columns and enter the products in appropriate rows.

Obtain the summations of all.

CORRELATION IN GROUPED DATA

Consider each cell frequency individually and obtain from northward the value of dx and from westward the value of dY.

Multiply all the three to get and place the products in respective cells in their top right hand corners.

These values are then added up across the columns for each row and placed in the column headed fdY dx . Similarly, these are totaled up down each column and put in the row labeled fdY

dx . Now,

CORRELATION IN GROUPED DATA

From the following data relating to advertisement expenditure and sales of 40 comparable firms, calculate coefficient of correlation between these two variables.

AN EXAMPLE

Sales Revenue

Advertisement Expenditure (‘000 Rs)Total

(‘000 Rs) 5 – 15 15 – 25 25 – 35 35 – 45

75 – 125 4 1 5

125 – 175 7 6 2 1 16

175 – 225 1 3 4 2 10

225 – 275 1 1 3 4 9

Total 13 11 9 7 40

SOLUTION

Using various inputs calculated,

Notice that no adjustment is required for taking step-deviations instead of deviations.

SOLUTION

Linear Relationship: The product-moment coefficient of correlation assumes essentially that the relationship between the variables is linear in nature.

Normality: A further assumption is that a large number of independent factors operate on each of the variables being correlated in such a way that each of them is normally distributed.

ASSUMPTIONS OF THE COEFFICIENT OF CORRELATION, r

The Karl Pearson’s coefficient of correlation is a pure number and is divorced of the units in which the original data are expressed.

As indicated earlier, the value of the coefficient of correlation varies between ±1.

The coefficient of correlation is independent of the change of origin and scale of the data. Thus, if a constant is added to/subtracted from one or both variable values or if all values are multiplied or divided by a constant, it will have no effect on the value of the coefficient.

PROPERTIES OF THE COEFFICIENT OF CORRELATION, r

Null hypothesis, H0: ρ = 0 (Correlation in the population is zero)

Alternate hypothesis, H1: ρ ≠ 0 (Correlation in the population is other than zero)

Level of significance, α = 0.05 (say) Test statistic:

TESTING THE SIGNIFICANCE OF CORRELATION COEFFICIENT

The data of 10 sales manager of the National Company showed the correlation between test scores and the sales made by salesmen to be equal to 0.818. This suggested a strong correlation between the two variables. Test the significance of correlation coefficient.

AN EXAMPLE

Null hypothesis, H0: ρ = Alternate hypothesis, H1: ρ ≠ 0 Level of significance, α = 0.05 (say) Test statistic: t Decision Rule: If , reject the null hypothesis Computations:

Conclusion: The null hypothesis is rejected at 0.05 level of significance meaning thereby that the correlation in the population is not zero. From the practical standpoint, it indicates for the sales manager that there is correlation in the population of salespersons with respect to their test scores and sales made by them.

SOLUTION:

Sometimes probable error is used in interpreting a correlation coefficient, r. The probable error, PE, is defined as follows:

The correlation coefficient is considered to be significant when it exceeds 6 times the probable error.

It may be noted that the value of probable error is related inversely to the value of n so that smaller the value of n greater is the probable error for a given value of r.

PROBABLE ERROR

It measures how much variation in one variable is explained by variation in the other variable.

It is numerically equal to the square of the coefficient of correlation, r2.

An r2 equal to 0.64 implies that 64 percent of the variation in one variable is due to variation in the other variable.

In the context of a situation where the variables are perfectly correlated so that r = 1 (or −1). In such a case, r2 = 1 implies that all changes in one variable are explained by changes in the other variable.

COEFFICIENT OF DETERMINATION

First, too much importance may not be given to coefficients of correlation obtained from small data sets as they may lead to erroneous conclusions.

In any case, it is always advisable to interpret the value of a given correlation coefficient using the probable error.

Secondly, it should be clearly understood that while a cause-and-effect relationship between two variables would result in a correlation between them the reverse is not true.

Further, sometimes a high correlation may be found between the variables due to chance alone.

COEFFICIENT OF CORRELATION AND ITS LIMITATIONS

Rank correlation is calculated essentially where the variables under consideration cannot quantified being measured on ordinal scale.

However, it can be calculated even where the variables are objectively quantifiable.

This is done by ranking the given data on the basis of the values involved.

The rank correlation coefficient also varies between ±1.

The presence of extreme observations in the data does not distort the value of rank correlation coefficient.

RANK CORRELATION

Let there be n pairs of values of two ranked variables, or two rankings of a variable.

The ranks may already be given or else they may be obtained by ranking the given values as 1, 2,, … , n in ascending or descending order.

Now, find the difference, d, between different pairs of ranks and obtain their squares, d2.

Finally, obtain the summation of the squared differences, and apply the formula:

SPEARMAN’S COEFFICIENT OF RANK CORRELATION, rs

Eight countries were ranked by two directors of a company seeking to expand its activities in the foreign markets in terms of their sales potential. Determine to what extent is the assessment of the two directors agreed.

AN EXAMPLE

Country: A B C D E F G H

Ranking by

Director 1:

7 5 1 8 2 4 3 6

Director 2:

4 6 3 5 2 7 1 8

SOLUTION

CountryRanks

d = R1 – R2 d2

R1 R2A 7 4 3 9B 5 6 −1 1C 1 3 −2 4D 8 5 3 9E 2 2 0 0F 4 7 −3 9G 3 1 2 4H 6 8 −2 4

Total 40

Thus, there is moderate degree of agreement among the directors.

While ranking, it may sometimes not be possible to distinguish clearly between adjacent units.

The ranks are said to be tied in such a case. Similarly, in quantitatively expressed data, tied

ranks are experienced when equal values appear in a given series.

The problem is resolved by assigning the average of the ranks involved to each of them.

TIED RANKS

If there are m items with common ranks, then a value equal to (m3-m)/12 is added to sum of square of difference as a correction factor for calculating coefficient of correlation.

If there is more than one such group of items with common ranks, the correction factor is added as many times as the number of groups.

The coefficient of rank correlation is given by:

TIED RANKS

When the data involve two variables, the correlation between the variables is called simple correlation

When they involve more than two variables, then we study multiple and partial correlations.

In such data, there are two or more independent variables which affect a dependent variable.

Multiple correlation is used to study the joint or cumulative effect of all the given independent variables on the dependent variable.

The partial correlation involves a study of correlation between one independent variable and the dependent variable holding the other independent variable(s) constant statistically.

MULTIPLE AND PARTIAL CORRELATION

If we designate the given three variables as 1, 2 and 3, we can calculate three coefficients of multiple correlation.

COEFFICIENT OF MULTIPLE CORRELATION

If data on three variables are given, we can calculate a total of three partial correlation coefficients.

COEFFICIENTS OF PARTIAL CORRELATION

In a trivariate distribution, it is found that . Obtain

AN EXAMPLE

END OF CHAPTER 13