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Difference Between Rank Coefficient and Karl Pearson’s Coefficient of Correlation

  • Last Updated : 27 Dec, 2021

Rank Coefficient of Correlation is the method of determination of coefficient of correlation. It is also named Spearman’s Coefficient of Correlation. It measures the linear association between ranks assigned to individual items according to their attributes. Attributes are those variables that cannot be numerically measured such as intelligence of people, physical appearance, honesty, etc. 

It is developed by British psychologist Charles Edward Spearman. It is used when the variables cannot be measured meaningfully as in the case of quantitative variables such as price, income, weight, etc. Basically, it is used when values are expressed qualitatively.

Formula:

Rank Coefficient of Correlation (rs)= 1 – 6ΣD2  / (N3–N)

Example:

Rank in Computers (X)Rank in English(Y)
12
24
31
45
53
68
77
86

Solution:

Rank in Computers (X)Rank in Computers (Y)Differences Between Ranks D = (X-Y)D2
12-11
24-24
3124
45-11
5324
68-24
7700
8624
   6ΣD2 = 22

Here, n =  8

(rs)= 1 – 6ΣD2  / (N3–N)
     = 1- 6 * 22 / 504 
     = 1- 132/504
     = 0.74

Rank Coefficient of Correlation (rs)= 0.74

Karl Pearson’s Coefficient of Correlation:

Karl Pearson’s Coefficient of Correlation (or Product moment correlation or simple correlation coefficient or covariance method ) is based upon the arithmetic mean and the standard deviation.

According to Karl Pearson, the correlation coefficient of two variables is obtained by dividing the sum of the products of the corresponding deviations of the various items of two series from the respective means by the product of their standard deviations and the number of pairs of observation. Basically, it is based on the covariance of the concerned variables.

Formula is:

Karl Pearson's Coefficient of Correlation  (r) = NΣXY−ΣX.ΣY / √NΣX2 - (Σx)2 √NΣY2 - (ΣY)2

Example: 

Find the value of Karl Pearson’s coefficient correlation from the following table:

SUBJECT Y
14399
22165
32579
44275
55787
65981

Solution:

SUBJECTXYXYX2Y2
143994257 18499801
221651365  4414225
3257919756256241
4427531501764  5625
557874959 3249 7569
65981 47793481 6561
Σ247  486 204851140940022
 (r) = NΣXY−ΣX.ΣY / √NΣX2 - (Σx)2 √NΣY2 - (ΣY)2
 (r) = 6(20,485) – (247 × 486) / [√[[6(11,409) – (2472)] × [6(40,022) – 4862]]]

Karl Pearson’s Coefficient of Correlation  (r) = 0.5298 

Difference between Rank Coefficient and Karl Pearson’s Coefficient of Correlation

The difference between Rank Coefficient and Karl Pearson’s Coefficient of Correlation is as follows:

Sr. No.Rank Coefficient Karl Pearson’s Coefficient
1.It is suitable when data is given in the qualitative form.It is a suitable method when data is given in the quantitative form.
2.It cannot be applied in the case of bivariate frequency distribution.It is an effective method to determine the correlation in the case of grouped series.
3.It is not possible to determine the combined coefficient of correlation.If coefficients of correlation and number of items of each subgroup os given then one can determine the combined coefficient of correlation items.
4.Changing the actual values in the series does not result in a change in the coefficient of correlation.Changing the actual values in the series results in a change in the coefficient of correlation.
5.The coefficient of correlation is perfectly positive if both the series have equal corresponding ranks i.e. D = 0 for each.The coefficient of correlation is perfectly positive if both the series change uniformly i.e. X and Y series are related linearly correlation.
6.It is difficult to use and understand.It is easier to use and understand.
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