Below is given distribution of profit in Rs. per day of a shop in certain town: Calculate median profit of a shop.
| Profit (in Rs.) | 500 - 900 | 1000 - 1400 | 1500 - 1900 | 2000 - 2400 | 2500 - 2900 | 3000 - 3400 | 3500 - 3900 |
|---|---|---|---|---|---|---|---|
| No. of shops | 8 | 18 | 27 | 21 | 20 | 18 | 8 |
Solution:
Class
(Profit in Rs.)Continuous
ClassesFrequency ($f_i$)
(No. of shops)Cumulative frequency
less than type500 - 900 450 - 950 8 8 1000 - 1400 950 - 1450 18 26 1500 - 1900 1450 - 1950 27 53 → $c.f.$ 2000 - 2400 1950 - 2450 21 → $f$ 74 2500 - 2900 2450 - 2950 20 94 3000 - 3400 2950 - 3450 18 112 3500 - 3900 3450 - 3950 8 120 Total 120 → $N$ Here total frequency $= \sum f_i = N = 120$
$\frac{N}{2} = \frac{120}{2} = 60$
Cumulative frequency (less than type) which is just greater than $60$ is $74$. Therefore corresponding class $1950 - 2450$ is median class.
$L = 1950$, $N = 120$, $c.f. = 53$, $f = 21$, $h = 500$
$$ \begin{aligned} \text{Median} &= L + \left( \frac{\frac{N}{2} - c.f.}{f} \right) h \quad \text{(or equivalently)} \quad L + \left( \frac{N}{2} - c.f. \right) \frac{h}{f} \\[8pt] &= 1950 + \left( \frac{120}{2} - 53 \right) \frac{500}{21} \\[8pt] &= 1950 + (60 - 53) \frac{500}{21} \\[8pt] &= 1950 + (7) \times \frac{500}{21} \\[8pt] &= 1950 + \frac{500}{3} \\[8pt] &= 1950 + 166.67 \\[8pt] &= 2116.67 \end{aligned} $$Median of profit is Rs. 2116.67.