Question

Calculate mode of the following by using empirical formula.

$x$	$18$	$20$	$25$	$30$	$34$	$38$	$40$
$f$	$6$	$7$	$3$	$7$	$7$	$5$	$5$

Answer 1

Step 1: Find the mean

$x$	$f$	$fx$	C
$18$	$6$	$108$	$6$
$20$	$7$	$140$	$13$
$25$	$3$	$75$	$16$
$30$	$7$	$210$	$23$
$34$	$7$	$238$	$30$
$38$	$5$	$190$	$35$
$40$	$5$	$200$	$40$
	$\sum _{}f=40$	$\sum _{}fx=1161$

For the grouped data, mean $=\frac{\sum _{}fx}{{\displaystyle \sum _{}}f}$

$$\begin{gathered} =\frac{1161}{40} \\ =29.025 \end{gathered}$$

Step 2: Find the median

Here, $n$ is even i.e., $40$. Therefore,

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

$=\frac{1}{2}\left({20}^{th}term+{21}^{st}term\right)$

$=\frac{1}{2}\left(30+30\right)$

$=30$

Step 3: Find the mode

By empirical formula, Mode $=3Median-2Mean$

$$\begin{gathered} =3\times 30-2\times 29.025 \\ =90-58.05 \\ =31.95 \end{gathered}$$

Hence, the mode is $31.95$