The number of hours studied ($x$) and marks obtained out of 50 ($y$) by 5 students are given below. Find the covariance between hours studied and marks obtained.

| Student | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Hours studied ($x$) | 5 | 10 | 15 | 20 | 25 |
| Marks obt

Question

The number of hours studied ($x$) and marks obtained out of 50 ($y$) by 5 students are given below. Find the covariance between hours studied and marks obtained.

| Student | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Hours studied ($x$) | 5 | 10 | 15 | 20 | 25 |
| Marks obtained ($y$) | 12 | 18 | 20 | 25 | 30 |

Accepted Answer

**Step 1 — Find the means.** $\bar{x} = \dfrac{5+10+15+20+25}{5} = \dfrac{75}{5}=15$. $\bar{y} = \dfrac{12+18+20+25+30}{5} = \dfrac{105}{5}=21$.

**Step 2 — Tabulate deviations and their products.**

| $x$ | $y$ | $x-\bar x$ | $y-\bar y$ | $(x-\bar x)(y-\bar y)$ |
|---|---|---|---|---|
| 5 | 12 | −10 | −9 | 90 |
| 10 | 18 | −5 | −3 | 15 |
| 15 | 20 | 0 | −1 | 0 |
| 20 | 25 | 5 | 4 | 20 |
| 25 | 30 | 10 | 9 | 90 |
| **Total** | | | | **215** |

$\sum(x-\bar x)(y-\bar y) = 90+15+0+20+90 = 215$.

**Step 3 — Divide by $n$.**
$$	ext{Cov}(x,y) = \dfrac{215}{5} = 43$$

**Independent check (direct method).** $\sum xy = 5(12)+10(18)+15(20)+20(25)+25(30) = 60+180+300+500+750=1790$. Then $\dfrac{1}{5}(1790) - (15)(21) = 358-315=43$ — matches exactly.

> [!ANSWER]
> $	ext{Cov}(x,y)=43$

Student	1	2	3	4	5
Hours studied ( $x$ )	5	10	15	20	25
Marks obtained ( $y$ )	12	18	20	25	30

$x$	$y$	$x-\bar x$	$y-\bar y$	$(x-\bar x)(y-\bar y)$
5	12	−10	−9	90
10	18	−5	−3	15
15	20	0	−1	0
20	25	5	4	20
25	30	10	9	90
Total				215

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