GATE 2021 BT – Question 53

Correct answer: 0.49 to 0.51

Explanation

**Pearson's correlation coefficient:**
$$r=\frac{\sum(x-\bar x)(y-\bar y)}{\sqrt{\sum(x-\bar x)^2\sum(y-\bar y)^2}} .$$

**Means.** $\bar x=\dfrac{121+242+363}{3}=242$, $\bar y=\dfrac{360+364+362}{3}=362$.

**Deviations.**
- $x-\bar x=-121,\ 0,\ 121$
- $y-\bar y=-2,\ 2,\ 0$

**Sums.**
- $\sum(x-\bar x)(y-\bar y)=(-121)(-2)+0+0=242$
- $\sum(x-\bar x)^2=121^2+0+121^2=29\,282$
- $\sum(y-\bar y)^2=4+4+0=8$

$$r=\frac{242}{\sqrt{29\,282\times8}}=\frac{242}{484}=\mathbf{0.5}.$$