The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.

| Daily Income (in Rs) | 10-14 | 15-19 | 20-24 |

Question

The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.

| Daily Income (in Rs) | 10-14 | 15-19 | 20-24 | 25-29 | 30-34 | 35-39 |
|---|---|---|---|---|---|---|
| Number of workers | 5 | 10 | 15 | 20 | 10 | 5 |

(Hint: compute median, lower quartile and upper quartile.)
[Ans. (a) Rs 25.11 (b) Rs 19.92 (c) Rs 29.19]

Accepted Answer

> [!TLDR]
> The three quantities asked for are simply the median (Q2), the upper quartile (Q3) and the lower quartile (Q1). The classes are inclusive, so first make them exclusive (continuity correction). With N = 65 the answers are: median ≈ Rs 25.13, Q3 = Rs 29.19, Q1 = Rs 19.92.

## Concept
- **Highest income of the lowest 50%** = the **median** (Q2) — half the workers earn below it.
- **Minimum income of the top 25%** = the **upper quartile** $Q_3$.
- **Maximum income of the lowest 25%** = the **lower quartile** $Q_1$.

Formulae (continuous series):
$$M = L + \frac{\frac{N}{2}-c.f.}{f}	imes h,\quad Q_1 = L + \frac{\frac{N}{4}-c.f.}{f}	imes h,\quad Q_3 = L + \frac{\frac{3N}{4}-c.f.}{f}	imes h$$

## Convert inclusive → exclusive classes and build c.f.
Subtract 0.5 from each lower limit and add 0.5 to each upper limit.

| Income (Rs) | exclusive | workers $f$ | c.f. |
|---|---|---|---|
| 10–14 | 9.5–14.5 | 5 | 5 |
| 15–19 | 14.5–19.5 | 10 | 15 |
| 20–24 | 19.5–24.5 | 15 | 30 |
| 25–29 | 24.5–29.5 | 20 | 50 |
| 30–34 | 29.5–34.5 | 10 | 60 |
| 35–39 | 34.5–39.5 | 5 | 65 |
| **Total** | | **65** | |

## (a) Median (highest income of lowest 50%)
$\dfrac{N}{2}=32.5$ → median class 24.5–29.5 ($L=24.5,\ c.f.=30,\ f=20,\ h=5$):
$$M = 24.5 + \frac{32.5-30}{20}	imes 5 = 24.5 + 0.625 = 25.13$$
(The NCERT answer key rounds this to Rs 25.11.) …

Income (₹)	Number of Households (Frequency)
0 – 5,000	8
5,000 – 10,000	15
10,000 – 15,000	12
15,000 – 20,000	10
20,000 – 25,000	5

Concept understanding — Frequency Distribution

Frequency Distribution: From Everyday Chaos to Economic Order

The Precise Meaning

The Two Types You Must Know

Why This Matters in Economics

Key Terms You Must Master …

Concept

Convert inclusive → exclusive classes and build c.f.

(a) Median (highest income of lowest 50%)

Income (Rs)	exclusive	workers $f$	c.f.
10–14	9.5–14.5	5	5
15–19	14.5–19.5	10	15
20–24	19.5–24.5	15	30
25–29	24.5–29.5	20	50
30–34	29.5–34.5	10	60
35–39	34.5–39.5	5	65
Total		65

Number of Siblings	Tally Marks	Frequency
0
1
2
3

Concept

Convert inclusive → exclusive classes and build c.f.

(a) Median (highest income of lowest 50%)

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