The price ($x$, in ₹) of a commodity and the quantity demanded ($y$, in units) over 5 weeks are given below. Find Pearson's correlation coefficient and comment on what the scatter diagram of this data would look like.

| Week | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Pric

Question

The price ($x$, in ₹) of a commodity and the quantity demanded ($y$, in units) over 5 weeks are given below. Find Pearson's correlation coefficient and comment on what the scatter diagram of this data would look like.

| Week | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Price ($x$) | 10 | 20 | 30 | 40 | 50 |
| Quantity demanded ($y$) | 40 | 35 | 25 | 20 | 10 |

Accepted Answer

**Step 1 — Find the means.** $\bar{x} = \dfrac{10+20+30+40+50}{5} = \dfrac{150}{5} = 30$. $\bar{y} = \dfrac{40+35+25+20+10}{5} = \dfrac{130}{5} = 26$.

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

| $x$ | $y$ | $x-\bar{x}$ | $y-\bar{y}$ | $(x-\bar{x})(y-\bar{y})$ | $(x-\bar{x})^2$ | $(y-\bar{y})^2$ |
|---|---|---|---|---|---|---|
| 10 | 40 | −20 | 14 | −280 | 400 | 196 |
| 20 | 35 | −10 | 9 | −90 | 100 | 81 |
| 30 | 25 | 0 | −1 | 0 | 0 | 1 |
| 40 | 20 | 10 | −6 | −60 | 100 | 36 |
| 50 | 10 | 20 | −16 | −320 | 400 | 256 |
| **Total** | | | | **−750** | **1000** | **570** |

**Step 3 — Apply the formula.**
$$r = \dfrac{-750}{\sqrt{1000}\sqrt{570}} = \dfrac{-750}{\sqrt{570000}} = \dfrac{-750}{754.98} \approx -0.993$$ …

$x$	$y$	$x-\bar{x}$	$y-\bar{y}$	$(x-\bar{x})(y-\bar{y})$	$(x-\bar{x})^2$	$(y-\bar{y})^2$
10	40	−20	14	−280	400	196
20	35	−10	9	−90	100	81
30	25	0	−1	0	0	1
40	20	10	−6	−60	100	36
50	10	20	−16	−320	400	256
Total				−750	1000	570

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