Pearson's correlation coefficient
- 1. “ Pearson’s Correlation Coefficient” Presented by : Eng. Waleed Alzaghal YouTube Channel : Waleed Alzaghal
- 2. Pearson Correlation Coefficient The correlation coefficient (r) is a measure of the strength and the direction of a linear relationship between two variables x and y.
- 3. Strength and Direction of Correlation
- 4. A single outlier point can significantly change the correlation coefficient r
- 5. r = 0.88 r = - 0.08
- 6. Obs. x i y j 1 x 1 y 1 2 x 2 y 2 3 x 3 y 3 … … … … … … … … … … … … … … … n x n y n Sum
- 8. Obs. x y 1 1 8 2 10 4 3 9 4 4 6 5 5 5 7 6 3 7 7 2 9 Sum 36 44 x 2 1 100 81 36 25 9 4 256 y2 64 16 16 25 49 49 81 300 xy 8 40 36 30 35 21 18 188 • Sum Up x and y • Compute x2 and Sum Up • Compute y2 and Sum Up • Compute xy and Sum Up Apply,
- 9. 𝑟 = 𝑛 𝑥𝑦 − ( 𝑥)( 𝑦) (𝑛 𝑥2 − ( 𝑥)2)(𝑛 𝑦2 − ( 𝑦)2 𝑟 ≈ − 0.94 (Strong negative relationship) Obs. x y x 2 y2 xy 1 1 8 1 64 8 2 10 4 100 16 40 3 9 4 81 16 36 4 6 5 36 25 30 5 5 7 25 49 35 6 3 7 9 49 21 7 2 9 4 81 18 Sum 36 44 256 300 188
- 10. ( x - x̅ ) -4.14 4.86 3.86 0.86 -0.14 -2.14 -3.14 0.02 ( x - x̅ )2 17.14 23.62 14.90 0.74 0.02 4.58 9.86 70.86 ( y - y̅ ) 1.71 -2.29 -2.29 -1.29 0.71 0.71 2.71 -0.03 ( y - y̅ )2 2.92 5.24 5.24 1.66 0.50 0.50 7.34 23.43 ( x - x̅ )( y - y̅ ) -7.08 -11.13 -8.84 -1.11 -0.10 -1.52 -8.51 -38.29 • Compute the Mean for x and y • Compute ( x - x̅ ) and Sum up • Compute ( x - x̅ )2 and Sum up • Compute ( y - y̅ ) and Sum up • Compute ( y - y̅ )2 and Sum up • Compute ( x - x̅ )( y - y̅ ) and Sum up Obs. x y 1 1 8 2 10 4 3 9 4 4 6 5 5 5 7 6 3 7 7 2 9 Sum 36 44 Count 7 7 Mean 5.14 6.29 Apply, r = ( 𝑥 − 𝑥 )( 𝑦 − 𝑦 ) 𝑥 − 𝑥 2 ( 𝑦− 𝑦 )2
- 11. r = ( 𝑥 − 𝑥 )( 𝑦 − 𝑦 ) 𝑥 − 𝑥 2 ( 𝑦− 𝑦 )2 r ≈ − 0.94 (Strong negative relationship)
- 13. Obs. x y x 2 y2 xy 1 8 16 64 256 128 2 9 9 81 81 81 3 10 4 100 16 40 4 11 1 121 1 11 5 12 0 144 0 0 6 13 1 169 1 13 7 14 4 196 16 56 8 15 9 225 81 135 9 16 16 256 256 256 Sum 108 60 1356 708 720 𝑟 = 𝑛 𝑥𝑦 − ( 𝑥)( 𝑦) (𝑛 𝑥2 − ( 𝑥)2). (𝑛 𝑦2 − ( 𝑦)2 𝑟 = 9 720 − (108)(60) ( 9 1356 − 108 2)( 9 708 − 60 2) r = 0.0 (No relationship because of the nonlinearity between the two variables) (r does not describe curved relationships)