Introduction of Correlation Analysis

Correlation Analysis is a statistical technique used to indicate the nature and degree of relationship existing between one variable and the other(s). It is also used along with regression analysis to measure how well the regression line explains the variations of the dependent variable with the independent variable.

The commonly used methods for studying linear relationship between two variables involve both graphic and algebraic methods. Some of the widely used methods include:

1. Scatter Diagram
2. Correlation Graph
3. Pearson’s Coefficient of Correlation
4. Spearman’s Rank Correlation
5. Concurrent Deviation Method

1. SCATTER DIAGRAM 

This method is also known as Dotogram or Dot diagram. Scatter diagram is one of the simplest methods of diagrammatic representation of a bivariate distribution. Under this method, both the variables are plotted on the graph paper by putting dots. The diagram so obtained is called "Scatter Diagram". By studying diagram, we can have rough idea about the nature and degree of relationship between two variables. The term scatter refers to the spreading of dots on the graph. We should keep the following points in mind while interpreting correlation:

• if the plotted points are very close to each other, it indicates high degree of If the plotted points are away from each other, it indicates low degree of correlation.

Figure: Scatter Diagrams

 

• if the points on the diagram reveal any trend (either upward or downward), the variables are said to be correlated and if no trend is revealed, the variables are
• if there is an upward trend rising from lower left hand corner and going upward to the upper right hand corner, the correlation is positive since this reveals that the values of the two variables move in the same direction. If, on the other hand, the points depict a downward trend from the upper left hand corner to the lower right hand corner, the correlation is negative since in this case the values of the two variables move in the opposite
• in particular, if all the points lie on a straight line starting from the left bottom and going up towards the right top, the correlation is perfect and positive, and if all the points like on a straight line starting from left top and coming down to right bottom, the correlation is perfect and negative.

The various diagrams of the scattered data in Figure depict different forms of correlation.

Example

Given the following data on sales (in thousand units) and expenses (in thousand rupees) of a firm for 10 month:

 

Month :	J	F	M	A	M	J	J	A	S	O
Sales:	50	50	55	60	62	65	68	60	60	50
Expenses:	11	13	14	16	16	15	15	14	13	13

1. Make a Scatter Diagram
2. Do you think that there is a correlation between sales and expenses of the firm? Is it positive or negative? Is it high or low?

Solution:(a) The Scatter Diagram of the given data is shown in Figure

 

Scatter Diagram

Figure shows that the plotted points are close to each other and reveal an upward So there is a high degree of positive correlation between sales and expenses of the firm.

2. CORRELATION GRAPH

This method, also known as Correlogram is very simple. The data pertaining to two series are plotted on a graph sheet. We can find out the correlation by examining the direction and closeness of two curves. If both the curves drawn on the graph are moving in the same direction, it is a case of positive correlation. On the other hand, if both the curves are moving in opposite direction, correlation is said to be negative. If the graph does not show any definite pattern on account of erratic fluctuations in the curves, then it shows an absence of correlation.

Find out graphically, if there is any correlation between price yield per plot (qtls); denoted by Y and quantity of fertilizer used (kg); denote by X.

 

Plot No.:	1	2	3	4	5	6	7	8	9	10
Y:	3.5	4.3	5.2	5.8	6.4	7.3	7.2	7.5	7.8	8.3
X:	6	8	9	12	10	15	17	20	18	24

Solution: The Correlogram of the given data is shown in Figure 4-3

 

Correlation Graph

Figure 4-3 shows that the two curves move in the same direction and, moreover, they are very close to each other, suggesting a close relationship between price yield per plot (qtls) and quantity of fertilizer used (kg)

Remark:        Both the Graphic methods - scatter diagram and correlation graph provide a ‘feel for’ of the data – by providing visual representation of the association between the variables. These are readily comprehensible and enable us to form a fairly good, though rough idea of the nature and degree of the relationship between the two variables. However, these methods are unable to quantify the relationship between them. To quantify the extent of correlation, we make use of algebraic methods - which calculate correlation coefficient.

3. PEARSON’S COEFFICIENT OF CORRELATION 

---

A mathematical method for measuring the intensity or the magnitude of linear relationship between two variables was suggested by Karl Pearson (1867-1936), a great British Biometrician and Statistician and, it is by far the most widely used method in practice.

Karl Pearson’s measure, known as Pearsonian correlation coefficient between two variables X and Y, usually denoted by r(X,Y) or r_xy or simply r is a numerical measure of linear relationship between them and is defined as the ratio of the covariance between X and Y, to the product of the standard deviations of X and Y.

Symbolically

 

Remark: Eq. (4.3) or Eq. (4.3a) is quite convenient to apply if the means X and Y come out to be integers. If X or/and Y is (are) fractional then the Eq. (4.3) or Eq. (4.3a) is quite cumbersome to apply, since the computations of å( X - X )² , å(Y - Y )² and å( X - X )(Y - Y ) are quite time consuming and tedious. In such a case Eq. (4.6) may be used provided the values of X or/ and Y are small. But if X and Y assume large values, the calculation of å X ² , åY ² and å XY is again quite time consuming.

Thus if (i) X and Y are fractional and (ii) X and Y assume large values, the Eq. (4.3) and Eq. (4.6) are not generally used for numerical problems. In such cases, the step deviation method where we take the deviations of the variables X and Y from any arbitrary points is used. We will discuss this method in the properties of correlation coefficient.

Properties of Pearsonian Correlation Coefficient

---

The following are important properties of Pearsonian correlation coefficient:

Pearsonian correlation coefficient cannot exceed 1 numerically. In other words it lies between –1 and +1.

Symbolically,

-1 ≤ r ≤1

Remarks:   (i) This property provides us a check on our calculations. If in any problem, the obtained value of r lies outside the limits + 1, this implies that there is some mistake in our calculations.

• The sign of r indicate the nature of the correlation. Positive value of r indicates positive correlation, whereas negative value indicates negative correlation. r = 0 indicate absence of
• The following table sums up the degrees of correlation corresponding to various values of r:

Value of r	Degree of correlation
±1	perfect correlation
±0.90 or more	very high degree of correlation
±0.75 to ±0.90	sufficiently high degree of correlation
±0.60 to ±0.75	moderate degree of correlation
±0.30 to ±0.60	only the possibility of a correlation
less than ±0.30	possibly no correlation
0	absence of correlation

 

Where A, B, h and k are constants and h > 0, k > 0; then the correlation coefficient between X and Y is same as the correlation coefficient between U and V i.e.,

r(X,Y) = r(U, V) => r_xy = r_uv

Remark: This is one of the very important properties of the correlation coefficient and is extremely helpful in numerical computation of r. We had already stated that Eq. (4.3) and Eq.(4.6) become quite tedious to use in numerical problems if X and/or Y are in fractions or if X and Y are large. In such cases we can conveniently change the origin and scale (if possible) in X or/and Y to get new variables U and V and compute the correlation between U and V by the Eq. (4.7)

3. Two independent variables are uncorrelated but the converse is not true

If X and Y are independent variables then r_xy = 0

However, the converse of the theorem is not true i.e., uncorrelated variables need not necessarily be independent. As an illustration consider the following bivariate distribution.

X          :	1	2	3	-3	-2	-1
Y          :	1	4	9	9	4	1

For this distribution, value of r will be 0.

Hence in the above example the variable X and Y are uncorrelated. But if we examine the data carefully we find that X and Y are not independent but are connected by the relation Y = X². The above example illustrates that uncorrelated variables need not be independent.

Remarks: One should not be confused with the words uncorrelation and independence. r_xy = 0 i.e., uncorrelation between the variables X and Y simply implies the absence of any linear (straight line) relationship between them. They may, however, be related in some other form other than straight line e.g., quadratic (as we have seen in the above example), logarithmic or trigonometric form.

4. Pearsonian coefficient of correlation is the geometric mean of the two regression coefficients, i.e.

The signs of both the regression coefficients are the same, and so the value of r will also have the same sign.

This property will be dealt with in detail in the next lesson on Regression Analysis.

5. The square of Pearsonian correlation coefficient is known as the coefficient of determination.

Coefficient of determination, which measures the percentage variation in the dependent variable that is accounted for by the independent variable, is a much better and useful measure for interpreting the value of r. This property will also be dealt with in detail in the next lesson.

Probable Error of Correlation Coefficient 

The correlation coefficient establishes the relationship of the two variables. After ascertaining this level of relationship, we may be interested to find the extent upto which this coefficient is dependable. Probable error of the correlation coefficient is such a measure of testing the reliability of the observed value of the correlation coefficient, when we consider it as satisfying the conditions of the random sampling.

If r is the observed value of the correlation coefficient in a sample of N pairs of observations for the two variables under consideration, then the Probable Error, denoted by PE (r) is expressed as

There are two main functions of probable error:

1. Determination of limits:

The limits of population correlation coefficient are r ± PE(r), implying that if we take another random sample of the size N from the same population, then the observed value of the correlation coefficient in the second sample can be expected to lie within the limits given above, with 0.5 probability. When sample size N is small, the concept or value of PE may lead to wrong conclusions. Hence to use the concept of PE effectively, sample size N it should be fairly large.

2. Interpretation of 'r': The interpretation of 'r' based on PE is as under:

• If r < PE(r), there is no evidence of correlation, e. a case of insignificant correlation.
• If r > 6 PE(r), correlation is If r < 6 PE(r), it is insignificant.
• If the probable error is small, correlation exist where r > 5

Example 4-3

Find the Pearsonian correlation coefficient between sales (in thousand units) and expenses (in thousand rupees) of the following 10 firms:

Firm:	1	2	3	4	5	6	7	8	9	10
Sales:	50	50	55	60	65	65	65	60	60	50
Expenses:	11	13	14	16	16	15	15	14	13	13

 

Solution: Let sales of a firm be denoted by X and expenses be denoted by Y

Calculations for Coefficient of Correlation

{Using Eq. (4.3) or (4.3a)}

 

Firm	X	Y	dx =X-X	dy =Y-Y	d 2 x	d 2 y	dx.dy
1	50	11	-8	-3	64	9	24
2	50	13	-8	-1	64	1	8
3	55	14	-3	0	9	0	0
4	60	16	2	2	4	4	4
5	65	16	7	2	49	4	14
6	65	15	7	1	49	1	7
7	65	15	7	1	49	1	7
8	60	14	2	0	4	0	0
9	60	13	2	-1	4	1	-2
10	50	13	-8	-1	64	1	8

	= 580	=140		=360	=22	=70

The value of rxy = 0.78 , indicate a high degree of positive correlation between sales and expenses.

Example 4-4

The data on price and quantity purchased relating to a commodity for 5 months is given below:

Month :            January           February          March              April                May Prices(Rs):            10                    10                   11                   12                   12

Quantity(Kg):       5                      6                     4                      3                    3

 

Find the Pearsonian correlation coefficient between prices and quantity and comment on its sign and magnitude.

Solution: Let price of the commodity be denoted by X and quantity be denoted by Y 

Calculations for Coefficient of Correlation

{Using Eq. (4.6)}

 

Month	X	Y	X2	Y2	XY
1	10	5	100	25	50
2	10	6	100	36	60
3	11	4	121	16	44
4	12	3	144	9	36
5	12	3	144	9	36

 

å X =55

åY =21

å X 2 = 609

åY 2 = 95

å XY = 226

 

 

The negative sign of r indicate negative correlation and its large magnitude indicate a very high degree of correlation. So there is a high degree of negative correlation between prices and quantity demanded.

Example 4-5   

Find the Pearsonian correlation coefficient from the following series of marks obtained by 10 students in a class test in mathematics (X) and in Statistics (Y):

X :	45	70	65	30	90	40	50	75	85	60
Y :	35	90	70	40	95	40	60	80	80	50

 

Also calculate the Probable Error.

Solution:

Calculations for Coefficient of Correlation

{Using Eq. (4.7)}

 

X	Y	U	V	U2	V2	UV
45	35	-3	-6	9	36	18
70	90	2	5	4	25	10
65	70	1	1	1	1	1

 

30	40	-6	-5	36	25	30
90	95	6	6	36	36	36
40	40	-4	-5	16	25	20
50	60	-2	-1	4	1	2
75	80	3	3	9	9	9
85	80	5	3	25	9	15
60	50	0	-3	0	9	0
		åU = 2	åV = -2	åU 2 = 140	åV 2 = 176	åUV	= 141

 

PE(r) = 0.0405

So the value of r is highly significant.

SPEARMAN’S RANK CORRELATION

---

Sometimes we come across statistical series in which the variables under consideration are not capable of quantitative measurement but can be arranged in serial order. This happens when we are dealing with qualitative characteristics (attributes) such as honesty, beauty, character, morality, etc., which cannot be measured quantitatively but can be arranged serially. In such situations Karl Pearson’s coefficient of correlation cannot be used as such. Charles Edward Spearman, a British Psychologist, developed a formula in 1904, which consists in obtaining the correlation coefficient between the ranks of N individuals in the two attributes under study.

Suppose we want to find if two characteristics A, say, intelligence and B, say, beauty are related or not. Both the characteristics are incapable of quantitative measurements but we can arrange a group of N individuals in order of merit (ranks) w.r.t. proficiency in the two characteristics. Let the random variables X and Y denote the ranks of the individuals in the characteristics A and B respectively. If we assume that there is no tie, i.e., if no two individuals get the same rank in a characteristic then, obviously, X and Y assume numerical values ranging from 1 to N.

The Pearsonian correlation coefficient between the ranks X and Y is called the rank correlation coefficient between the characteristics A and B for the group of individuals.

Spearman’s rank correlation coefficient, usually denoted by ρ(Rho) is given by the equation

Where d is the difference between the pair of ranks of the same individual in the two characteristics and N is the number of pairs.

Example 4-6

Ten entries are submitted for a competition. Three judges study each entry and list the ten in rank order. Their rankings are as follows:

Entry:	A	B	C	D	E	F	G	H	I	J
Judge J1:	9	3	7	5	1	6	2	4	10	8
Judge J2:	9	1	10	4	3	8	5	2	7	6
Judge J3:	6	3	8	7	2	4	1	5	9	10

 

Calculate the appropriate rank correlation to help you answer the following questions:

• Which pair of judges agrees the most?
• Which pair of judges disagrees the most?

Solution:

Calculations for Coefficient of Rank Correlation

{Using Eq.(4.8)}

 

Entry	Rank by Judges			Difference in Ranks
	J1	J2	J3	d(J1&J2)	d2	d(J1&J3)	d2	d(J2&J3)	d2
A	9	9	6	0	0	+3	9	+3	9
B	3	1	3	+2	4	0	0	-2	4
C	7	10	8	-3	9	-1	1	+2	4
D	5	4	7	+1	1	-2	4	-3	9
E	1	3	2	-2	4	-1	1	+1	1
F	6	8	4	-2	4	+2	4	+4	16
G	2	5	1	-3	9	+1	1	+4	16
H	4	2	5	+2	4	-1	1	-3	9
I	10	7	9	+3	9	+1	1	-2	4
J	8	6	10	+2	4	-2	4	-4	16
					åd2 =48		åd2 =26		åd2 =88

 

 

Spearman’s rank correlation Eq.(4.8) can also be used even if we are dealing with variables, which are measured quantitatively, i.e. when the actual data but not the ranks relating to two variables are given. In such a case we shall have to convert the data into ranks. The highest (or the smallest) observation is given the rank 1. The next highest (or the next lowest) observation is given rank 2 and so on. It is immaterial in which way (descending or ascending) the ranks are assigned. However, the same approach should be followed for all the variables under consideration.

Example 4-7

Calculate the rank coefficient of correlation from the following data:

 

X:	75	88	95	70	60	80	81	50
Y:	120	134	150	115	110	140	142	100

 

Repeated Ranks

In case of attributes if there is a tie i.e., if any two or more individuals are placed together in any classification w.r.t. an attribute or if in case of variable data there is more than one item with the same value in either or both the series then Spearman’s Eq.(4.8) for calculating the rank correlation coefficient breaks down, since in this case the variables X [the ranks of

individuals in characteristic A (1^st series)] and Y [the ranks of individuals in characteristic B (2^nd series)] do not take the values from 1 to N.

In this case common ranks are assigned to the repeated items. These common ranks are the arithmetic mean of the ranks, which these items would have got if they were different from each other and the next item will get the rank next to the rank used in computing the common rank. For example, suppose an item is repeated at rank 4. Then the common rank to be assigned to each item is (4+5)/2, i.e., 4.5 which is the average of 4 and 5, the ranks which these observations would have assumed if they were different. The next item will be assigned the rank 6. If an item is repeated thrice at rank 7, then the common rank to be assigned to each value will be (7+8+9)/3, i.e., 8 which is the arithmetic mean of 7,8 and 9 viz., the ranks these observations would have got if they were different from each other. The next rank to be assigned will be 10.

If only a small proportion of the ranks are tied, this technique may be applied together with Eq.(4.8). If a large proportion of ranks are tied, it is advisable to apply an adjustment or a correction factor to Eq.(4.8)as explained below:

“In the Eq.(4.8) add the factor

 

Hence, there is a very low degree of positive correlation, probably no correlation, between preference share prices and debenture prices.

Remarks on Spearman’s Rank Correlation Coefficient

1. We always have  , which provides a check for numerical
2. Since Spearman’s rank correlation coefficient, r, is nothing but Karl Pearson’s correlation coefficient, r, between the ranks, it can be interpreted in the same way as the Karl Pearson’s correlation
3. Karl Pearson’s correlation coefficient assumes that the parent population from which sample observations are drawn is normal. If this assumption is violated then we need a measure, which is distribution free (or non-parametric). Spearman’s r is such a distribution free measure, since no strict assumption are made about the from of the population from which sample observations are
4. Spearman’s formula is easy to understand and apply as compared to Karl Pearson’s formula. The values obtained by the two formulae, viz Pearsonian r and Spearman’s r are generally different. The difference arises due to the fact that when ranking is used instead of full set of observations, there is always some loss of Unless many ties exist, the coefficient of rank correlation should be only slightly lower than the Pearsonian coefficient.
5. Spearman’s formula is the only formula to be used for finding correlation coefficient if we are dealing with qualitative characteristics, which cannot be measured quantitatively but can be arranged serially. It can also be used where actual data are given. In case of extreme observations, Spearman’s formula is preferred to Pearson’s
6. Spearman’s formula has its limitations also. It is not practicable in the case of bivariate frequency distribution. For N >30, this formula should not be used unless the ranks are

CONCURRENT DEVIATION METHOD

This is a casual method of determining the correlation between two series when we are not very serious about its precision. This is based on the signs of the deviations                        (i.e. the direction of the change) of the values of the variable from its preceding value and does not take into account the exact magnitude of the values of the variables. Thus we put a plus (+) sign, minus (-) sign or equality (=) sign for the deviation if the value of the variable is greater than, less than or equal to the preceding value respectively. The deviations in the values of two variables are said to be concurrent if they have the same sign (either both deviations are positive or both are negative or both are equal). The formula used for computing correlation coefficient rc by this method is given by

Where c is the number of pairs of concurrent deviations and N is the number of pairs of deviations. If (2c-N) is positive, we take positive sign in and outside the square root in Eq. (4.9) and if (2c-N) is negative, we take negative sign in and outside the square root in Eq. (4.9).

Remarks:   (i)   It should be clearly noted that here N is not the number of pairs of observations but it is the number of pairs of deviations and as such it is one less than the number of pairs of observations.

(ii) Coefficient of concurrent deviations is primarily based on the following principle:

“If the short time fluctuations of the time series are positively correlated or in other words, if their deviations are concurrent, their curves would move in the same direction and would indicate positive correlation between them”

Example 4-9   

Calculate coefficient of correlation by the concurrent deviation method

Supply:	112	125	126	118	118	121	125	125	131	135
Price:	106	102	102	104	98	96	97	97	95	90

Solution:

Calculations for Coefficient of Concurrent Deviations

{Using Eq. (4.9)}

 

Supply (X)	Sign of deviation from preceding value (X)	Price (Y)	Sign of deviation preceding value (Y)	Concurrent deviations
112		106
125	+	102	-
126	+	102	=
118	-	104	+
118	=	98	-
121	+	96	-
125	+	97	+	+(c)
125	=	97	=	= (c)
131	+	95	-
135	+	90	-

 

We have

Number of pairs of deviations, N =10 – 1 = 9

c = Number of concurrent deviations

= Number of deviations having like signs

= 2

Coefficient of correlation by the method of concurrent deviations is given by:

             

Hence there is a fairly good degree of negative correlation between supply and price.