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

Last Updated : 21 Jun, 2022

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 (r_s)= 1 – 6ΣD²/ (N³–N)

Example:

Rank in Computers (X)	Rank in English(Y)
1	2
2	4
3	1
4	5
5	3
6	8
7	7
8	6

Solution:

Rank in Computers (X)	Rank in Computers (Y)	Differences Between Ranks D = (X-Y)	D²
1	2	-1	1
2	4	-2	4
3	1	2	4
4	5	-1	1
5	3	2	4
6	8	-2	4
7	7	0	0
8	6	2	4
			6ΣD² = 22

Here, n = 8

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

Rank Coefficient of Correlation (r_s)= 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	X	Y
1	43	99
2	21	65
3	25	79
4	42	75
5	57	87
6	59	81

Solution:

SUBJECT	X	Y	XY	X²	Y²
1	43	99	4257	1849	9801
2	21	65	1365	441	4225
3	25	79	1975	625	6241
4	42	75	3150	1764	5625
5	57	87	4959	3249	7569
6	59	81	4779	3481	6561
Σ	247	486	20485	11409	40022

 (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