Question

From the following frequency distribution, find :

(i) the median

(ii) lower quartile

(iii) upper quartile

(iv) inter quartile range

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

Answer 1

Step 1: Use 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: Calculate the cumulative frequency distribution table.

Since the given variable is already in ascending order, thus finding the cumulative frequency alone we get,

Variable	Frequency	Cumulative frequency
$15$	$4$	$4$
$18$	$6$	$4+6=10$
$20$	$8$	$10+8=18$
$22$	$9$	$18+9=27$
$25$	$7$	$27+7=34$
$27$	$8$	$34+8=42$
$30$	$6$	$42+6=48$

Here, total number of observations $n=48$ which is even.

Step 3: (i) Find 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, $n=48$

Thus, the median becomes,

Median$=\frac{1}{2}\left({\left(\frac{48}{2}\right)}^{th}term+{\left(\frac{48}{2}+1\right)}^{th}term\right)$

$=\frac{1}{2}\left({\left(24\right)}^{th}term+{\left(24+1\right)}^{th}term\right)$

$=\frac{1}{2}\left({\left(24\right)}^{th}term+{\left(25\right)}^{th}term\right)$.

Here, the cumulative frequency $24$ and $25$ corresponds to the variable $22.$

Median$=\frac{1}{2}(22+22)$

$=\frac{1}{2}\left(44\right)$

$=22$.

Step 4: (ii) Find the Lower quartile range:

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

Lower quartile $\left(Q_1\right)={\left(\frac{48}{4}\right)}^{th}$ term

$={12}^{th}$ term.

Here, the cumulative frequency $12$ corresponds to the variable $20.$

Then, Lower quartile range$=20$.

Step 5: (iii) Find the Upper quartile range:

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

Upper quartile $\left(Q_3\right)={\left(\frac{3\times 48}{4}\right)}^{th}$term

$={36}^{th}$term.

Here, the cumulative frequency $36$corresponds to the variable $27.$

Then, Upper quartile range $=27$.

Step 6: (iv) Find the Interquartile range:

Interquartile range $=Q_3-Q_1$

Interquartile range $=27-20$

$=7$.

Final Answer:

Therefore, for the given grouped frequency,

(i) Median is $22.$

(ii) Lower quartile range is $20.$

(iii) Upper quartile range is $27.$

(iv) Interquartile range is $7.$