Question

The daily wages (in rupees) of $18$ workers are:$41,21,38,31,45,23,26,29,30,28,25,35,42,47,53,29,31,$ and $35.$

Find :

(i) The median

(ii) Lower quartile

(iii) Upper quartile

(iv) Interquartile range.

Answer 1

Step 1: Use the Formula for calculating median, lower quartile, upper quartile and interquartile range when $n$ is even.

Median$=\frac{1}{2}\left({\left(\frac{n}{2}\right)}^{th}term+{\left(\frac{n}{2}+1\right)}^{th}term\right)$

Lower quartile$\left(Q_1\right)$$={\left(\frac{n}{4}\right)}^{th}$term

Upper quartile$\left(Q_3\right)$$={\left(\frac{3n}{4}\right)}^{th}$term

Interquartile range$=Q_{3-}{Q}_1$

Where, $n=$total number of workers.

Step 2: Arrange the wages in ascending order.

Arranging the given in ascending order, we get,

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

Here, $n=18$ which is even.

Step 3: (i) Calculate the median.

Median$=\frac{1}{2}\left({\left(\frac{n}{2}\right)}^{th}term+{\left(\frac{n}{2}+1\right)}^{th}term\right)$

We know that, $n=18$

Thus, the median becomes,

Median$=\frac{1}{2}\left({\left(\frac{18}{2}\right)}^{th}term+{\left(\frac{18}{2}+1\right)}^{th}term\right)$

$=\frac{1}{2}\left({\left(9\right)}^{th}term+{\left(9+1\right)}^{th}term\right)$

$=\frac{1}{2}\left({\left(9\right)}^{th}term+{\left(10\right)}^{th}term\right)$

Here, $9^{th}$ term $=31$and ${10}^{th}$ term $=31$,

Median$=\frac{1}{2}(31+31)$

$=\frac{1}{2}\left(62\right)$

$=31$.

Step 4: (ii) Calculate the Lower quartile range.

Lower quartile $\left(Q_1\right)={\left(\frac{n}{4}\right)}^{th}$ term

Lower quartile $\left(Q_1\right)={\left(\frac{18}{4}\right)}^{th}$term

$={\left(4.5\right)}^{th}$term

$={\left(5\right)}^{th}$term

$=28$.

Step 5: (iii) Calculate the Upper quartile range.

Lower quartile $\left(Q_3\right)={\left(\frac{3n}{4}\right)}^{th}$term

Lower quartile $\left(Q_3\right)={\left(\frac{3\times 18}{4}\right)}^{th}$term

$={13.5}^{th}$ term

$={14}^{th}$ term

$=41$.

Step 6: (iv) Calculate the Interquartile range.

Interquartile range $=Q_3-Q_1$

Interquartile range $=41-28$

$=13$.

Final Answer:

Therefore, for the given grouped frequency

(i) Median is 31.

(ii) Lower quartile range is 28.

(iii) Upper quartile range is 41.

(iv) Interquartile range is 13.