For the following frequency distribution, find :

(i) the median

(ii) lower quartile

(iii) upper quartile

Variate	Frequency
25	3
31	8
34	10
40	15
45	10
48	9
50	6
60	2

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

Here, n (no. of observations) = 63, which is odd.

(i) As n is odd,

$\therefore \text{Median} = \dfrac{n + 1}{2} \text{th observation} \\[1em] = \dfrac{63 + 1}{2} \\[1em] = \dfrac{64}{2} \\[1em] = 32 \text{nd observation}$

∴ Median = 32nd observation = 40.

Hence, median = 40.

(ii) As n is odd,

$\therefore \text{Lower quartile} (Q_1) = \dfrac{n + 1}{4} \text{th observation} \\[1em] = \dfrac{63 + 1}{4} \\[1em] = \dfrac{64}{4} \\[1em] = 16 \text{th observation}$

∴ Lower quartile (Q₁) = 16th observation = 34.

Hence, lower quartile = 34.

(iii) As n is odd,

$\therefore \text{Upper quartile} (Q_3) = \dfrac{3(n + 1)}{4} \text{th observation} \\[1em] = \dfrac{3(63 + 1)}{4} \\[1em] = \dfrac{3 \times 64}{4} \\[1em] = 48 \text{th observation}$

∴ Upper quartile (Q₃) = 48th observation = 48.

Hence, upper quartile = 48.