Difference Between Rank Coefficient and Karl Pearson’s Coefficient of Correlation
Rank Coefficient and Karl Pearson’s Coefficient of Correlation: In engineering mathematics, coefficients of correlation are statistical measures that quantify the degree and direction of the relationship between two variables. They are used to determine how one variable may predict or relate to another, which is crucial in various engineering applications such as quality control, signal processing, and system optimization.
Table of Content
- Rank Coefficient of Correlation
- Formula
- Example
- Karl Pearson’s Coefficient of Correlation
- Difference between Rank Coefficient and Karl Pearson’s Coefficient of Correlation
Rank Coefficient of Correlation
Rank Coefficient of Correlation is the method of determining 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. 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) |
|---|---|
| 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
(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 | 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
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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 is 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. |
FAQs on Rank Coefficient and Karl Pearson’s Coefficient of Correlation
What is the Rank Correlation Coefficient?
The Rank Correlation Coefficient, also known as Spearman’s Rank Correlation Coefficient, is a measure of the strength and direction of the association between two ranked variables. It is denoted by ρ (row).
How is Karl Pearson’s Coefficient of Correlation calculated?
Karl Pearson’s Coefficient of Correlation, denoted by rrr, is calculated using the formula: [Tex]r = \frac{n(\sum xy) – (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) – (\sum x)^2][n(\sum y^2) – (\sum y)^2]}} [/Tex]
What is the difference between Spearman’s Rank Correlation and Pearson’s Correlation?
The main difference between Spearman’s Rank Correlation and Pearson’s Correlation is the type of data they are used for. Spearman’s Rank Correlation is used for ordinal data and measures the strength and direction of the association between two ranked variables. Pearson’s Correlation is used for interval or ratio data and measures the linear relationship between two continuous variables.
What does a Pearson correlation coefficient of 0.8 indicate?
A Pearson correlation coefficient of 0.8 indicates a strong positive linear relationship between the two variables being measured. This means that as one variable increases, the other variable also tends to increase.