The weekly wages (in ₹) of 60 workers in a factory are grouped as follows. Find the mean wage using the direct method, and cross-check your answer using the assumed mean method with $A=325$.

| Wages (₹) | 200–250 | 250–300 | 300–350 | 350–400 | 400–450 |
|---|---|---|---|---|---

Question

The weekly wages (in ₹) of 60 workers in a factory are grouped as follows. Find the mean wage using the direct method, and cross-check your answer using the assumed mean method with $A=325$.

| Wages (₹) | 200–250 | 250–300 | 300–350 | 350–400 | 400–450 |
|---|---|---|---|---|---|
| Number of workers ($f$) | 8 | 16 | 20 | 10 | 6 |

Accepted Answer

**Direct method.** Class marks: $225, 275, 325, 375, 425$.

| Class | $f$ | $x$ | $fx$ |
|---|---|---|---|
| 200–250 | 8 | 225 | 1800 |
| 250–300 | 16 | 275 | 4400 |
| 300–350 | 20 | 325 | 6500 |
| 350–400 | 10 | 375 | 3750 |
| 400–450 | 6 | 425 | 2550 |
| **Total** | **60** | | **19000** |

$\sum f = 8+16+20+10+6=60$. $\sum fx = 1800+4400+6500+3750+2550$: $1800+4400=6200$, $6200+6500=12700$, $12700+3750=16450$, $16450+2550=19000$.
$$\bar{x} = \dfrac{19000}{60} = \dfrac{950}{3} \approx 316.67$$

**Cross-check — assumed mean method with $A=325, h=50$.** $d=x-325$: $-100,-50,0,50,100$; $u=d/50$: $-2,-1,0,1,2$.

| $f$ | $u$ | $fu$ |
|---|---|---|
| 8 | −2 | −16 |
| 16 | −1 | −16 |
| 20 | 0 | 0 | …

Class	$f$	$x$	$fx$
200–250	8	225	1800
250–300	16	275	4400
300–350	20	325	6500
350–400	10	375	3750
400–450	6	425	2550
Total	60		19000

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