Rank Correlation

Ranks are the orders or priorities that are given in accordance with their importance or position. When the data are statistically measured, Karl Pearson's correlation coefficient is particularly helpful. Some characteristics, such as beauty, wisdom, intelligence, honesty, etc., cannot be accurately quantified. Rankings or other types of ratings can be used to quantify these aspects. The degree of association between the two sets of ranks is then referred to as the rank correlation. There is a method that may be used in these circumstances to study the link between these features. The rank correlation coefficient is the name of it. Charles Edward Sparkman, a British psychologist, created this technique in 1904.

Rank correlation is the degree of association between two variables when data are arranged in order or in ranks.

Sparkman Rank-correlation Coefficient:

The Sparkman rank correlation coefficient is a measure of rank correlation created by Sparkman. Rs is typically used to indicate it. The formula for the Sparkman's rank correlation coefficient, which measures the strength of the association between different ranks or grades of two characteristics, is as follows:

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

where,

d= R1 – R2 = difference of rank between paired items

R1= rank of first attribute

R2= rank of second attribute

n= number of paired observations.

Properties of Rank Correlation Coefficient

• The rank correlation coefficient's value varies from -1 to 1.
• When R = +1, the sequence of ranks between the two traits is fully agreed upon.
• When R = -1, there is total disagreement over the rankings.

For example,

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

rs = 1 - \{\frac{6.0}{5(25 – 1)}\}

=1-0

=1

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

=1 - \{\frac{6.40}{5(25 – 1)}\}

=1-2

=-1

When determining Sparkman's rank correlation coefficient, there are three scenarios to consider:

• When a ranking is given,
• When rankings are not provided
• Once the ranks are cycled

Case(I): when actual ranks are given

The following measures must be taken after the actual ranks are announced:

• Compute the difference of ranks (R1 – R2) which is denoted by d.
• Compute d2 to get ∑d2.
• Substitute these values in the formula of rank correlation coefficient i.e.

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

Example:

Following is a ranking of the top ten industries in one state based on profit and working capital for the year:

Solution:

Ranks are assigned here. We can calculate rank correlation coefficient using Sparkman's rank correlation coefficient formula, which is as follows:

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

Computation of Rank Correlation Coefficient

The rank correlation coefficient is calculated using:

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

= 1 - \{\frac{6.36}{10(100 – 1)}\}

=1- \{\frac{216}{990}\}

=1- 0.2182

=0.782

Case(II): when ranks are not given

It will be necessary to assign the ranks when the real data for the variables but not the rankings are provided. The formula for computing the rank correlation coefficient, which is given by Sparkman, can be used to assign ranks by taking either the highest value as 1, the second highest value as 2, and so on, or the lowest value as 1, the lowest value as 2, and so on.

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

Example:

Using the data below, determine the Sparkman's rank correlation coefficient between the cost of advertising (in thousand rupees) and sales (in lakh rupees).

Solution:

The problem does not assign ranks. In all scenarios, we must give ranks beginning with the smaller or larger amount. Let's order from highest value to lowest. Let X and Y represent, respectively, the advertising expense (in thousands of rupees) and the sales (in lakhs of rupees).

Table for calculation of rank correlation coefficient

Now ,

rs = 1 - \{\frac{6.∑d2}{n(n2 – 1)}\}

= 1 - \{\frac{6.30}{10(100 – 1)}\}

=0.82

Case(III): when the ranks are repeated(tied)

A common rank is assigned to all such repeated items when more than one item in a series has the same value for r. The next item will receive the rank immediately after them. These common ranks are the arithmetic means of the ranks that these objects would have received if they were different from one another. For instance, if the third and fourth items are equal in value, the third and fourth items' ranks are each 3.5, and the next item's rank is 5. By adding a tie correlation factor, frac(m(m2-1)12), to d2, the general formula (described above) for computing rank correlation coefficient is modified for this situation. When many items have the same value, the formula for computing the Sparkman rank correlation coefficient is as follows:

rs=1- \{\frac{6[∑d2 + \{\frac{m1(m12-1)}{12}\} + \{\frac{m2(m22-1)}{12}\} +………}{n(n2-1}\}

Where, m1,m2,……etc. be the number of times that an item is repeated.

Example:

Ten candidates were examined by a company. based on the applicants' scores in the statistics and accounting exams. Do the rank correlation coefficient calculation.

Solution:

Let R1 stand for the statistics grade rank and R2 for the accounting grade rank.

Computation of rank correlation coefficient

Here, n= 10, m1= 3, m2=2, m3=2

Rank correlation coefficient

rs=1- \{\frac{6[∑d2 + \{\frac{m1(m12-1)}{12}\} + \{\frac{m2(m22-1)}{12}\} +………}{n(n2-1}\}

=1- \{\frac{6[83 + \{\frac{3(32-1)}{12}\} + \{\frac{2(22-1)}{12}\} + \{\frac{3(32-1)}{12}\} }{10(102-1}\}

=1- \{\frac{6*30}{990}\}

=1-0.1818

=0.82

Refrence

Chaudary, A.K. (2061).Business statistics. kathmandu:Bhundipuran Prakshan

Dhakal Bashanta (2014).Business Statistics,Buddha academic publisher

Sthapit, Azaya Bikram(2006),Business Statistics,Asmita publication