This document discusses Pearson's Correlation Coefficient (r), which is a measure of the strength and direction of a linear relationship between two variables. It provides an example calculation of r using sample data points. While the example initially shows a strong negative correlation, adding additional non-linear data points results in r equaling 0, indicating no relationship due to the non-linearity between the variables. Pearson's r only describes linear relationships.

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)