The marks scored by 16 students in a class test are :

3, 6, 8, 13, 15, 5, 21, 23, 17, 10, 9, 1, 20, 21, 18, 12.

Find :

(i) the median

(ii) lower quartile

(iii) upper quartile

(iv) inter quartile range.

On arranging the numbers in ascending order we get,

1, 3, 5, 6, 8, 9, 10, 12, 13, 15, 17, 18, 20, 21, 21, 23.

(i) Here, n (no. of observations) = 16, which is even.

$\therefore \text{Median} = \dfrac{\dfrac{n}{2} \text{th observation} + \big(\dfrac{n}{2} + 1\big)\text{th observation}}{2} \\[1em] = \dfrac{\dfrac{16}{2} \text{th observation} + \big(\dfrac{16}{2} + 1\big)\text{th observation}}{2} \\[1em] = \dfrac{\text{ 8th observation + 9th observation}}{2} \\[1em] = \dfrac{12 + 13}{2} \\[1em] = \dfrac{25}{2} \\[1em] = 12.5$

Hence, the median of following data = 12.5.

(ii) Here, n (no. of observations) = 16, which is even.

$\text{Lower Quartile} = \dfrac{n}{4}\text{th observation} \\[1em] = \dfrac{16}{4} \\[1em] = 4\text{th observation} \\[1em] = 6.$

Hence, lower quartile = 6.

(iii) Here, n (no. of observations) = 16, which is even.

$\text{Upper Quartile} = \dfrac{3n}{4}\text{th observation} \\[1em] = \dfrac{48}{4} \\[1em] = 12\text{th observation} \\[1em] = 18.$

Hence, upper quartile = 18.

(iv) Inter quartile range = Upper quartile - Lower quartile = 18 - 6 = 12.

Hence, inter quartile range = 12.