The profit (in ₹ lakh) earned by 40 firms in an industry survey is grouped as follows. Find the Variance and Standard Deviation (a) by the assumed-mean method, and (b) by the step-deviation method (take $A=50$, $h=20$), and verify the two agree.

| Profit (₹ lakh) | 0–20 | 20–40

Question

The profit (in ₹ lakh) earned by 40 firms in an industry survey is grouped as follows. Find the Variance and Standard Deviation (a) by the assumed-mean method, and (b) by the step-deviation method (take $A=50$, $h=20$), and verify the two agree.

| Profit (₹ lakh) | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 |
|---|---|---|---|---|---|
| Number of firms ($f$) | 4 | 8 | 16 | 8 | 4 |

Accepted Answer

**Step 1 — Tabulate class marks and deviations, assumed-mean method, $A=50$.**

| Class | $f$ | Class mark $x$ | $d=x-50$ | $fd$ | $fd^2$ |
|---|---|---|---|---|---|
| 0–20 | 4 | 10 | $-40$ | $-160$ | 6400 |
| 20–40 | 8 | 30 | $-20$ | $-160$ | 3200 |
| 40–60 | 16 | 50 | 0 | 0 | 0 |
| 60–80 | 8 | 70 | 20 | 160 | 3200 |
| 80–100 | 4 | 90 | 40 | 160 | 6400 |
| **Total** | **40** | | | **0** | **19200** |

$$\sigma^2 = \dfrac{\sum fd^2}{N} - \left(\dfrac{\sum fd}{N}\right)^2 = \dfrac{19200}{40} - 0^2 = 480, \qquad \sigma = \sqrt{480} \approx 21.909$$

**Step 2 — Cross-check by the step-deviation method, $A=50$, $h=20$.**

| Class | $f$ | $u=(x-50)/20$ | $fu$ | $fu^2$ |
|---|---|---|---|---|
| 0–20 | 4 | $-2$ | $-8$ | 16 |
| 20–40 | 8 | $-1$ | $-8$ | 8 |
| 40–60 | 16 | 0 | 0 | 0 |
| 60–80 | 8 | 1 | 8 | 8 |
| 80–100 | 4 | 2 | 8 | 16 |
| **Total** | **40** | | **0** | **48** | …

Class	$f$	Class mark $x$	$d=x-50$	$fd$	$fd^2$
0–20	4	10	$-40$	$-160$	6400
20–40	8	30	$-20$	$-160$	3200
40–60	16	50	0	0	0
60–80	8	70	20	160	3200
80–100	4	90	40	160	6400
Total	40			0	19200

Class	$f$	$u=(x-50)/20$	$fu$	$fu^2$
0–20	4	$-2$	$-8$	16
20–40	8	$-1$	$-8$	8
40–60	16	0	0	0
60–80	8	1	8	8
80–100	4	2	8	16
Total	40		0	48

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