From the following frequency distribution, find :

(i) the median

(ii) lower quartile

(iii) upper quartile

(iv) inter quartile range

Variate	Frequency
15	4
18	6
20	8
22	9
25	7
27	8
30	6

The given variates are already in ascending order. We construct the cumulative frequency table as under

Here, n (no. of observations) = 48, which is even.

(i) As n is even,

$\therefore \text{Median} = \dfrac{\dfrac{n}{2} \text{th observation} + \big(\dfrac{n}{2} + 1\big)\text{ th observation}}{2} \\[1em] = \dfrac{\dfrac{48}{2} \text{th observation} + \big(\dfrac{48}{2} + 1\big)\text{ th observation}}{2} \\[1em] = \dfrac{\text{ 24th observation + 25th observation}}{2} \\[1em]$

All observations from 19th to 27th are equal, each = 22.

Hence, median

$= \dfrac{22 + 22}{2} \\[1em] = \dfrac{44}{2} \\[1em] = 22.$

Hence, median = 22.

(ii) As n is even,

$\therefore \text{Lower quartile} (Q_1) = \dfrac{n}{4} \text{th observation} \\[1em] = \dfrac{48}{4} \\[1em] = 12 \text{th observation}$

∴ Lower quartile (Q₁) = 12th observation = 20.

Hence, lower quartile = 20.

(iii) As n is even,

$\therefore \text{Upper quartile} (Q_3) = \dfrac{3n}{4} \text{th observation} \\[1em] = \dfrac{3 × 48}{4} \\[1em] = \dfrac{144}{4} \\[1em] = 36 \text{th observation}$

∴ Upper quartile (Q₃) = 36th observation = 27.

Hence, upper quartile = 27.

(iv) Inter quartile range = Q₃ - Q₁ = 27 - 20 = 7.

Hence, inter quartile range = 7.