Calculate the correlation coefficient between X and Y and comment on their relationship: X: -3, -2, -1, 1, 2, 3 Y: 9, 4, 1, 1, 4, 9 (Ans. r = 0)
Calculation Table:
| X | Y | X² | Y² | XY |
|---|---|---|---|---|
| -3 | 9 | 9 | 81 | -27 |
| -2 | 4 | 4 | 16 | -8 |
| -1 | 1 | 1 | 1 | -1 |
| 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 4 | 16 | 8 |
| 3 | 9 | 9 | 81 | 27 |
| ΣX = 0 | ΣY = 28 | ΣX² = 28 | ΣY² = 196 | ΣXY = 0 |
Step 1: Calculate the required sums. N = 6 ΣX = 0, ΣY = 28, ΣX² = 28, ΣY² = 196, ΣXY = 0
Step 2: Apply the formula for correlation coefficient. r = [ΣXY - (ΣX)(ΣY)/N] / [²/N) × ²/N)]
Step 3: Substitute the values. Numerator = 0 - ()/6 = 0 Denominator = × Since the numerator is 0, the value of r is 0.
Comment on Relationship: Since r = 0, the two variables are uncorrelated. There is no linear relation between X and Y. However, other types of relation may be there.
Explanation
The question requires calculating Karl Pearson's coefficient of correlation. The calculation shows that while the numerator (covariance) is zero, indicating no linear association, the variables X and Y have a perfect non-linear relationship (Y = X²). The textbook context explicitly states that 'If r = 0 the two variables are uncorrelated. There is no linear relation between them. However other types of relation may be there.' This distinction is crucial for students to understand that r=0 implies no linear correlation, not necessarily independence.
Solution Steps
Step 1: Construct a table calculating X², Y², and XY for each pair of values.
Step 2: Sum the columns to find ΣX, ΣY, ΣX², ΣY², and ΣXY. Results are ΣX=0, ΣY=28, ΣX²=28, ΣY²=196, ΣXY=0.
Step 3: Substitute these values into the Pearson correlation formula.
Step 4: Observe that the numerator equals 0, resulting in r = 0.
Step 5: Interpret the result using the textbook definition of r = 0: the variables are uncorrelated with no linear relation, though other relations may exist.