When trying to explain rank correlation, first thing that should come in front is the grading or rank given to a particular group of individuals.
For example, if we suppose that based on two characteristics n group of individuals are ranked then the equation will come as:
r = 1 – 6∑di2/ n(n2 – 1)
Here (x, y;), i= 1,2…, n is the ranked individuals
Here, di=Xi–Yi
One thing must be clarified before using – 1≤ r ≤ 1that one rank will not appear more than once. When the ranks are repeated then another correlation factor needs to be added. If we take that ranks are repeated m number of times then the equation will be added asm(m2 – 1) with Σd2
Let us measure the correlation of coefficient of the data mentioned below:
| X | 85 | 74 | 85 | 50 | 65 | 78 | 74 | 60 | 74 | 90 |
| Y | 78 | 91 | 78 | 58 | 60 | 72 | 80 | 55 | 68 | 70 |
Here n = 10, so the solution is:
| Â Â Â Â Â Â Â Â Â Â Â X | Â Â Â Â Â Â Â Â Â Â Y | Rank X (x) | Rank Y (y) | d = x – y | Â Â Â Â Â Â Â Â Â d2 |
| 85
74 85 50 65 78 74 60 74 90 |
78
91 78 58 60 72 80 55 68 70 |
2.5
6 2.5 10 8 4 6 9 6 1 |
3.5
1 3.5 9 8 5 2 10 7 6 |
-1
5 -1 1 0 -1 4 -1 -1 -5 |
1
25 1 1 0 1 16 1 1 25 |
| ∑ | 0 | 72 |
In this solution, you can see that 85 are repeated two times for which the rank became 2.5 where it could be 2 and 3. Then 74 from the X series, is also repeated three times giving it the rank of 6 where it could be 5, 6, 7. Coming next to 78 from Y series which is repeated three times and getting the rank of 3.5 where it could have been 3 and 4.
As mentioned above, for the repeated ranks, a correlation factor will be added for each distribution, these are written below:
2(4 – 1)/12 = 1/2
3(9 -1)/12 = 2
2(4 – 1)/12 = 1/2
After measuring these three correlation factor we come to:
½ + 2 + ½ = 3
Hence we come to the rank correlation:
r = 1 – 6(72 + 3)/ 10 (100 – 1) = 0.545
Links of Previous Main Topic:-
- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Coefficient of correlation
- Regression of equations
Links of Next Statistics Topics:-
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Estimation
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory
