The weekly income (in ₹ hundred) of 45 shopkeepers in a market is grouped as follows. Find the Standard Deviation (a) by the assumed-mean method, and (b) by the step-deviation method (take $A=70$, $h=20$), and verify the two agree.

| Weekly income (₹ hundred) | 20–40 | 40–60 | 6

Question

The weekly income (in ₹ hundred) of 45 shopkeepers in a market is grouped as follows. Find the Standard Deviation (a) by the assumed-mean method, and (b) by the step-deviation method (take $A=70$, $h=20$), and verify the two agree.

| Weekly income (₹ hundred) | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 |
|---|---|---|---|---|---|
| Number of shopkeepers ($f$) | 6 | 10 | 15 | 9 | 5 |

Accepted Answer

**Step 1 — Tabulate class marks and deviations, assumed-mean method, $A=70$.**

| Class | $f$ | Class mark $x$ | $d=x-70$ | $fd$ | $fd^2$ |
|---|---|---|---|---|---|
| 20–40 | 6 | 30 | $-40$ | $-240$ | 9600 |
| 40–60 | 10 | 50 | $-20$ | $-200$ | 4000 |
| 60–80 | 15 | 70 | 0 | 0 | 0 |
| 80–100 | 9 | 90 | 20 | 180 | 3600 |
| 100–120 | 5 | 110 | 40 | 200 | 8000 |
| **Total** | **45** | | | **$-60$** | **25200** |

$$\sigma^2 = \dfrac{25200}{45} - \left(\dfrac{-60}{45}\right)^2 = 560 - 1.7778 = 558.222, \qquad \sigma = \sqrt{558.222} \approx 23.627$$

**Step 2 — Cross-check by the step-deviation method, $A=70$, $h=20$.**

| Class | $f$ | $u=(x-70)/20$ | $fu$ | $fu^2$ |
|---|---|---|---|---|
| 20–40 | 6 | $-2$ | $-12$ | 24 |
| 40–60 | 10 | $-1$ | $-10$ | 10 |
| 60–80 | 15 | 0 | 0 | 0 |
| 80–100 | 9 | 1 | 9 | 9 |
| 100–120 | 5 | 2 | 10 | 20 |
| **Total** | **45** | | **$-3$** | **63** | …

Class	$f$	Class mark $x$	$d=x-70$	$fd$	$fd^2$
20–40	6	30	$-40$	$-240$	9600
40–60	10	50	$-20$	$-200$	4000
60–80	15	70	0	0	0
80–100	9	90	20	180	3600
100–120	5	110	40	200	8000
Total	45			$-60$	25200

Class	$f$	$u=(x-70)/20$	$fu$	$fu^2$
20–40	6	$-2$	$-12$	24
40–60	10	$-1$	$-10$	10
60–80	15	0	0	0
80–100	9	1	9	9
100–120	5	2	10	20
Total	45		$-3$	63

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