Definition

“Correlation is an analysis in which we study the degree of closeness of relationship between the variables.”

Explanation

  • Where the values of two variables vary in such a way that the movements i.e. increase or decrease, in one variable is connected with the movements of another variable, such variables are said to be as correlated. Therefore, correlation is the degree of simultaneous variations between the variables. In some cases, the movements of the variables are in the same direction i.e. an increase in one variable results in another variable, and vice versa. In such case, when movements of variables are in same direction, the correlation is said to be positive. For example, an increase in heights of children is normally accompanied by increase of their weights. Similarly, if the movements of the variables are in opposite direction, such variables are said to be as negative or inverse. For example, an increase in the temperature during the winter is accompanied by a decrease in the sale of warm clothes.
  • It is important to note that in correlation both the variables are assumed to be random. Correlation is the name of sympathy of movements in the variables. It is not the name of movements or change in one variable causes change in another. The changes in variables might be occurs due to a common cause. For example, when we say that heights and weights of children are correlated, it does not mean that an increase in heights also causes an increase in weight. The increase in both the variable is due to a common cause, such as age. Therefore, as a child grows in age, his height and weight also increases.
  • If both the variables tend to move in the same direction, such as, Y tends to increase as X increases, or Y tends to decrease as X decreases, the correlation is called positive or direct correlation.
  • On the other hand, if the variables tend to move in the opposite direction, such as, Y tends to decrease as X increases or vice versa, the correlation is said to be as negative or inverse correlation.

The Coefficient of Correlation

For the purpose of measuring the degree of variability, we employ help from coefficient of correlation. We need to measure the degree of relationship between two variables, which is an abstract coefficient, which is free from specific units used to measure the relationship between the two variables. Such a measure is referred as coefficient of correlation. Coefficient of correlation is represented by the sign “r”.

Properties of Coefficient of Correlation

The important properties of coefficient of correlation are as follows

i. The coefficient of correlation is symmetrical with respect to X and Y i.e. clip_image002[5].

ii. The coefficient of correlation is the geometric mean of the two regression coefficients i.e. clip_image004[4]

iii. The coefficient of correlation is a pure number and does not depend upon the units employed. For example, if the coefficient of correlation between the heights and weights of children of students is computed is 0.98, it will be expressed simply 0.98. This means that it is neither 0.98 inches nor 0.98 pounds.

iv. The correlation coefficient is independent of origin and scale of measurement. This employs that if we take deviations of X and Y from some suitable origins or transform X and Y into u and v respectively, it will not at all affects the correlation coefficient.

v. The coefficient of correlation lies between -1 and 1. I.e. it cannot be less than -1 and greater than +1. Symbolically, it is written as clip_image006[4] If there is perfect positive correlation i.e. r = +1 and if there is perfect negative correlation i.e. r = -1. Lesser degree of correlation results in smaller positive correlation values of r. When r = 0, there is no correlation.

The interpretation of coefficient of correlation as a measure of strength of the linear relationship between two variables is purely a mathematical interpretation and is completely devoid of any cause or effect implications. The fact that two variables tend to increase or decrease together does imply that one has any direct or indirect effect on one another. Both may be influence by other variables in such a manner as to give rise to a strong mathematical relationship. The high correlation merely reflects the common effect of the upward trend of the two variables.

The linear coefficient of correlation between n pairs of variables clip_image008[4] is mathematically represented as follows, which is used for calculating the coefficient of correlation between the two variables,

clip_image010[4]

Example

Find the coefficient of linear correlation between the following values;
X 1 2 4 4 5 7 8 9
Y 1 3 4 6 8 9 11 14

clip_image012[4]

clip_image014[4]

Computations are needed to find coefficients of correlation which are given in the following Table 22;

Table 22

X Y clip_image016[6] clip_image018[6] clip_image020[6] clip_image022[6] clip_image024[6]
1 1 -4 16 -6 36 24
2 3 -3 9 -4 16 12
4 4 -1 1 -3 9 3
4 6 -1 1 -1 1 1
5 8 0 0 1 1 0
7 9 2 4 2 4 4
8 11 3 9 4 16 12
9 14 4 16 7 49 28
∑X = 40 ∑Y = 56 clip_image026[6] clip_image028[4] clip_image030[6] clip_image032[4] clip_image034[4]

By using the formula;

clip_image036[6]

clip_image038[4]

clip_image040[4]

clip_image042[4]

Example

Compute the coefficient of linear correlation between the price (X) and the quantity demanded (Y) for the following pair of values;

X 3 4 5 6 7 8 9 10 11 12
Y 17 16 15 13 11 10 8 9 7 6

clip_image044[4]

clip_image046[4]

Computations for the coefficient of correlation are outlined in the following Table 23;

Table 23

X Y clip_image016[7] clip_image018[7] clip_image020[7] clip_image022[7] clip_image024[7]
3 17 -4.5 20.25 5.8 33.64 -26.1
4 16 -3.5 12.25 4.8 23.04 -16.8
5 15 -2.5 6.25 3.8 14.44 -9.5
6 13 -1.5 2.25 1.8 3.24 -2.7
7 11 -0.5 0.25 -0.2 0.04 0.1
8 10 0.5 0.25 -1.2 1.44 -0.6
9 8 1.5 2.25 -3.2 10.24 -4.8
10 9 2.5 6.25 -2.2 4.84 -5.5
11 7 3.5 12.25 -4.2 17.64 -14.7
12 6 4.5 20.25 -5.2 27.04 -23.4
∑X = 75 ∑Y = 112 clip_image048[4] clip_image050[4] clip_image052[4] clip_image054[4] clip_image056[4]

By using the formula;

clip_image036[7]

clip_image058[4]

clip_image060[4]

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Short Computational Formula

There is another short formula for the computation of coefficient of correlation. It involves fewer computations as compared to previous formula and it is easy to calculate the required answer. The short computational formula for coefficient is as follows;

clip_image064[4]

Example

Find the coefficient of linear correlation, by using the short computational formula, using the following pairs of values;

X

1

2

3

4

5

6

7

8

Y

3

4

6

8

10

12

14

15

Computations for calculating coefficient of correlation are provided in following Table 24;

Table 24

X Y X2 Y2 XY
1 3 1 9 3
2 4 4 16 8
3 6 9 36 18
4 8 16 64 32
5 10 25 100 50
6 12 36 144 72
7 14 49 196 98
8 15 64 225 120
∑X = 36 ∑Y = 72 ∑X2 = 204 ∑Y2 = 790 ∑XY =401

clip_image066[4]

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clip_image072[5]

clip_image074

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