Q.53 If the values of two random variables (X, Y) are (121, 360), (242, 364)
and (363, 362), the value of correlation coefficient between X and Y
(rounded off to one decimal place) is _________.
Problem Statement
If the values of two random variables (X, Y) are (121, 360), (242, 364) and (363, 362), the value of correlation coefficient between X and Y (rounded off to one decimal place) is _________.
Correct Answer: -1.0
Data Summary
- Sum of X (ΣX) = 121 + 242 + 363 = 726
- Sum of Y (ΣY) = 360 + 364 + 362 = 1086
- Sum of XY (ΣXY) = (121×360) + (242×364) + (363×362) = 43560 + 88168 + 131406 = 263134
- Sum of X² (ΣX²) = 121² + 242² + 363² = 14641 + 58564 + 131769 = 204974
- Sum of Y² (ΣY²) = 360² + 364² + 362² = 129600 + 132496 + 131044 = 393140
Pearson Correlation Formula
r = [nΣXY - (ΣX)(ΣY)] / √{[nΣX² - (ΣX)²][nΣY² - (ΣY)²]}
Step-by-Step Calculation
- Numerator: 3(263134) – (726)(1086) = 789402 – 788916 = 486
- Denominator: √{[3(204974) – 726²][3(393140) – 1086²]}
- X Variance Term: 3(204974) – 726² = 614922 – 527076 = 87846
- Y Variance Term: 3(393140) – 1086² = 1179420 – 1180196 = -776
- Final r ≈ -0.9967 → Rounded to -1.0
Key Insights
- X increases sharply: 121 → 242 → 363 (242% increase)
- Y remains nearly constant: 360 → 364 → 362 (hovering around 362)
- Perfect negative correlation: As X rises dramatically, Y stays flat, creating r ≈ -1.0
- Fill-in-the-blank: No multiple choice options provided
Common Errors to Avoid
| Error | Why Wrong | Correct Value |
|---|---|---|
| +0.9 or +1.0 | Wrong sign – ignores inverse relationship | -1.0 |
| 0.0 | Ignores clear pattern in data | -1.0 |
| -0.9 | Rounding error – should round to -1.0 | -1.0 |
