The Product Moment Correlation Coefficient (also called Pearson’s r) is a number that shows how strongly two things (or variables) are related to each other.
For example:
If X is your study hours, and Y is your marks, Pearson’s r tells us: 'When you
study more, do your marks increase?'
Key Points:
• The value of r is always between −1 and +1.
|
r
Value |
Meaning |
|
+1 |
Perfect
positive relationship |
|
0 |
No
relationship |
|
−1 |
Perfect
negative relationship |
• A positive value means both increase
together.
• A negative value means when one increases, the other
decreases.
Formula of
Pearson’s r
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Where:
• n = number of data points
• X, Y = data values
• Σ means 'total sum of'
Let's Calculate
with an Example
Given Data:
|
X (Study
Hours) |
Y
(Marks) |
|
6 |
8 |
|
7 |
9 |
|
10 |
12 |
|
5 |
6 |
|
12 |
14 |
|
11 |
10 |
|
9 |
7 |
|
12 |
11 |
So here, n = 8 (we have 8 data points)
Step
1: Make a Table
|
X |
Y |
X² |
Y² |
XY |
|
6 |
8 |
36 |
64 |
48 |
|
7 |
9 |
49 |
81 |
63 |
|
10 |
12 |
100 |
144 |
120 |
|
5 |
6 |
25 |
36 |
30 |
|
12 |
14 |
144 |
196 |
168 |
|
11 |
10 |
121 |
100 |
110 |
|
9 |
7 |
81 |
49 |
63 |
|
12 |
11 |
144 |
121 |
132 |
Totals: ΣX=72, ΣY=77, ΣX²=700, ΣY²=791,
ΣXY=734
Step
2: Substitute in Formula
Use the formula with:
n = 8, ΣX = 72, ΣY = 77, ΣXY = 734, ΣX² = 700, ΣY² = 791
Step
3: Calculate
Numerator:
8 × 734 = 5872, and 72 × 77 = 5544 → 5872 − 5544 = 328
Denominator:
8 × 700 = 5600, 72² = 5184 → 5600 − 5184 = 416
8 × 791 = 6328, 77² = 5929 → 6328 − 5929 = 399
√(416 × 399) = √166224 ≈ 407.7
Step
4: Final Calculation
r = 328 / 407.7 ≈ 0.8044
Final Answer
r ≈ 0.804
Interpretation
Since r ≈ 0.804, we can say there is a strong
positive relationship.
This means: when X increases (study hours), Y also increases (marks).