The daily output of 19 workers is :

41, 21, 38, 27, 31, 45, 23, 26, 29, 30, 28, 25, 35, 42, 47, 53, 29, 31, 35.

Find :

(i) the median

(ii) lower quartile

(iii) upper quartile

(iv) inter quartile range

On arranging the given wages in ascending order we get,

21, 23, 25, 26, 27, 28, 29, 29, 30, 31, 31, 35, 35, 38, 41, 42, 45, 47, 53.

Here, n (no. of observations) = 19, which is odd.

(i) As n is odd,

$\therefore \text{Median} = \dfrac{n + 1}{2} \text{th observation} \\[1em] = \dfrac{19 + 1}{2} \\[1em] = \dfrac{20}{2} \\[1em] = 10 \text{th observation}$

∴ Median = 10th observation = 31.

Hence, median = 31.

(ii) As n is odd,

$\therefore \text{Lower quartile} (Q_1) = \dfrac{n + 1}{4} \text{th observation} \\[1em] = \dfrac{19 + 1}{4} \\[1em] = \dfrac{20}{4} \\[1em] = 5 \text{th observation}$

∴ Lower quartile (Q₁) = 5th observation = 27.

Hence, lower quartile = 27.

(iii) As n is odd,

$\therefore \text{Upper quartile} (Q_3) = \dfrac{3(n + 1)}{4} \text{th observation} \\[1em] = \dfrac{3(19 + 1)}{4} \\[1em] = \dfrac{60}{4} \\[1em] = 15 \text{th observation}$

∴ Upper quartile (Q₃) = 15th observation = 41.

Hence, upper quartile = 41.

(iv) Inter quartile range = Q₃ - Q₁ = 41 - 27 = 14.

Hence, inter quartile range = 14.