Page 183 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes Since the Spearman's formula is based upon the assumption of different ranks to different
individuals, therefore, its correction becomes necessary in case of tied ranks. It should be noted
that the means of the ranks will remain unaffected. Further, the changes in the variances are
2
usually small and are neglected. However, it is necessary to correct the term Sd and accordingly
i
2
m m 1
the correction factor , where m denotes the number of observations tied to a particular
12
2 4 1
rank, is added to it for every tie. We note that there will be two correction factors, i.e.,
12
3 9 1
and in the above example.
12

8.2.2 Limits of Rank Correlation

A positive rank correlation implies that a high (low) rank of an individual according to one
characteristic is accompanied by its high (low) rank according to the other. Similarly, a negative
rank correlation implies that a high (low) rank of an individual according to one characteristic
is accompanied by its low (high) rank according to the other. When = +1, there is said to be
perfect consistency in the assignment of ranks, i.e., every individual is assigned the same rank
2
with regard to both the characteristics. Thus, we have d = 0 and hence, = 1.
i
Similarly, when = –1, an individual that has been assigned 1st rank according to one characteristic
must be assigned nth rank according to the other and an individual that has been assigned 2nd
rank according to one characteristic must be assigned (n – 1)th rank according to the other, etc.
Thus, the sum of ranks, assigned to every individual, is equal to (n + 1), i.e., X + Y = n + 1 or
i i
Y = (n + 1) – X , for all i = 1, 2, ...... n.
i i
Further, d = X – Y = X – (n + 1) + X = 2X – (n + 1)
i i i i i i
Squaring both sides, we have

d 2 2X n 1 2 4X 2 n 1 2 4 n 1 X
i i i i
Taking sum over all the observations, we have
2
2 4n n 1 2n 1 2 4n n 1
d i 2 4 X i 2 n n 1 4 n 1 X i 6 n n 1 2

2
2 n n 1 n 1 n n 1
n n 1 2n 1 n 1 2 n 1
3 3 3
Substituting this value in the formula for rank correlation we have
2
6n n 1 1
1 1
3 n n 2 1
Hence, the Spearman's coefficient of correlation lies between - 1 and + 1.

Example: The following table gives the marks obtained by 10 students in commerce and
statistics. Calculate the rank correlation.
Marks in Statistics : 35 90 70 40 95 45 60 85 80 50
Marks in Commerce : 45 70 65 30 90 40 50 75 85 60

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