RANK CORRELATION TOPIC UNDER STATISTICS.pptx

There are two types of rank correlation pearson correlation
Spearman rank correlation

1. RANK CORRELATION

2. RANK CORRELATION: The degree of association between two rankings. A measure of the strength of association among and between variables. TYPES OF RANK CORRELATION: Spearman rank correlation Pearson correlation

3. SPEARMAN RANK CORRELATION: Spearman's rank correlation coefficient is a statistical measure to show the strength of a relationship between two variables. It basically gives the measure of monotonicity of the relation between two variables i.e., how well the relationship between two variables could be represented using a monotonic function. This method is finding out the lack of it between two variables was developed by the british Psychologist Charles Edward Spearman in 1904. Ten is the minimum number needed in a sample for the spearman's rank test to be valid. CLASSIFICATION OF SPEARMAN RANK CORRELATION: Equal Rank Unique Rank :

4. Equal Rank: R = 1 - 6[ Σ d²+CF] (if rank is repeated) ___________ N³-N CF=Correction Factor=1/12(m³-m) R = 1 - 6[ Σ d²+1/12(m³-m)+1/12(m³-m)+…] _______________________________________ N³-N where d = R1-R2 m= Repetitive occurrence of each rank. n = No. of times the data is repeated. Unique Rank: R = 1 - 6 Σ d² ______ N³-N where d = R1-R2 n = No. of times the data is repeated.

5. PEARSON CORRELATION : The Pearson correlation measures the strength of the linear relationship between two variables. It has a value between -1 to 1 -1 a total negative linear correlation 0 being no correlation and +1 a total positive correlation. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s, and for which the mathematical formula was derived and published by Auguste Bravais in 1844. r = Σ (x-x) (y-y) ▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔▔ √ Σ (x-x)² √ Σ (y-y)²

6. Equal Rank: Marks in Statistic 52 55 62 48 61 56 57 49 Marks in Maths 48 52 59 37 35 46 31 48 Marks in Statistics Marks in Maths R1 R2 d=R1-R2 d ² 52 48 3 5.5 -2.5 6.25 55 52 4 7 -3 9 62 59 8 8 48 37 1 3 -2 4 61 35 7 2 5 25 56 46 5 4 1 1 57 31 6 1 5 25 49 48 2 5.5 -3.5 12.25 Σ d ²=82.5

7. R = 1 - 6[ Σ d²+1/12(m³-m)+1/12(m³-m)+…] _____________________________ N³-N R = 1 - 6[82.5+1/12(2³-2)] = 1 - 6[82.5+1/12(8-2)] = 1 - 6[82.5+1/12(6)] _______________ ______________ _____________ 8³-8 512-8 504 = 1 - 6[82.5+6/12] __________ 504 = 1 - 6[82.5+0.5] __________ 504 = 1- 6[83] _____ 504 = 1 - 498 _____ 504 = 1- 0.99 = 0.01

8. Unique Rank: R = 1 - 6 Σ d² = 1 - 6*8 = 1 - 48 = 1 - 48 _____ _____ ______ ____ N³-N 5³-5 125-5 120 = 1-0.4 = 0.6 X 12 17 22 27 31 y 113 119 117 115 121 x y R1 R2 d=R1-R2 d ² 12 113 1 1 17 119 2 4 -2 4 22 117 3 3 27 115 4 2 2 4 31 121 5 5 Σ d ²=8

9. Which pair of judges have nearest approach to common test of beauty? First Judge 1 6 5 10 3 2 4 9 7 8 Second Judge 3 5 8 4 7 10 2 1 6 9 Third Judge 6 4 9 8 1 2 3 10 5 7 IJ (x) IIJ (y) IIIJ (z) R1 R2 R3 dxy =R1-R2 d²xy dyz =R2-R3 d²yz dxz =R1-R3 d²xz 1 3 6 1 3 6 -2 4 -3 9 -5 25 6 5 4 6 5 4 1 1 1 1 2 4 5 8 9 5 8 9 -3 9 -1 1 4 16 10 4 8 10 4 8 6 36 -4 16 2 4 3 7 1 3 7 1 -4 16 6 36 2 4 2 10 2 2 10 2 -8 64 8 64 4 2 3 4 2 3 2 4 -1 1 1 1 9 1 10 9 1 10 8 64 -9 81 -1 1 7 6 5 7 6 5 1 1 1 1 2 4 8 9 7 8 9 7 -1 1 2 4 1 1 Σ d ²xy=200 Σ d ²yz=214 Σ d ²xz=60

10. Rxy = 1- 6 Σ d² = 1 - (6*200) = 1 - 1200 = 1 - 1200 = 1-1.212 = -0.212 ____ ______ _______ ____ N³-N 10³-10 1000-10 990 Ryz = 1- 6 Σ d² = 1 - (6*214) = 1 - 1284 = 1 - 1284 = 1-1.297 = -0.297 ____ ______ _______ ____ N³-N 10³-10 1000-10 990 Rxz = 1- 6 Σ d² = 1 - (6*60) = 1 - 360 = 1 - 360 = 1-0.363 = 0.636 ____ ______ _______ _____ N³-N 10³-10 1000-10 990 First judge and third judge have nearest approach to common test of beauty

11. r = Σ (x-x) (y-y) = 142 = 142 = 142 = 142 ▔▔▔▔▔▔▔▔▔▔▔ ▔▔▔▔▔▔ ▔▔▔▔▔▔▔▔ ▔▔▔▔▔▔▔▔▔▔ ▔▔▔▔▔▔▔ √ Σ (x-x)² √ Σ (y-y)² √ 28 *√1864 2√7 *2√466 (2*2.645)(2*21.587) 5.29*43.174 = 142 ▔▔▔▔▔▔▔▔▔ 228.39046 = 0.62 x y (x-x) (y-y) (x-x) (y-y) (x-x)² (y-y)² 2 58 -2 -2 4 4 4 4 32 -28 784 5 63 1 3 3 1 9 7 87 3 27 81 9 729 3 67 -1 7 -7 1 49 1 45 -3 -15 45 9 225 6 68 2 8 16 4 64 =28/7=4 y =420/7=60 Σ (x-  )(y-y)=142 Σ (x-  )²=28 Σ (y-y)²=1864 PEARSON CORRELATION :