Question

Calculate the mean and the median for the following distribution:

Number	$5$	$10$	$15$	$20$	$25$	$30$	$35$
Frequency	$1$	$2$	$5$	$6$	$3$	$2$	$1$

Answer 1

Step 1: Make the required Cumulative frequency distribution table.

First, we have to prepare a table to find the value of $f_i{x}_i$ and cumulative frequency.

Let the frequency be $\left(f\right)$

The cumulative frequency is obtained by adding the frequency to the cumulative frequency of the predecessors.

$x_i$	$f_i$	$f_i$$x_i$	Cumulative frequency
$5$	$1$	$5$	$1$
$10$	$2$	$20$	$1+2=3$
$15$	$5$	$75$	$3+5=8$
$20$	$6$	$120$	$8+6=14$
$25$	$3$	$75$	$14+3=17$
$30$	$2$	$60$	$17+2=19$
$35$	$1$	$35$	$19+1=20$
	$\sum _{}{f}_i=20$	$\sum _{}{f}_i{x}_i=390$

Step 2:Calculate Mean and Median.

From the table,

$\sum _{}{f}_i{x}_i=390$

$\sum _{}{f}_i=20$

Substitute the values in the arithmetic mean formula,

The Arithmetic mean $=\frac{\sum _{}{f}_i{x}_i}{\sum _{}{f}_i}$

$=\frac{390}{20}$

$=19.5$

Substitute the value of $n=20$ in the arithmetic median formula,

The arithmetic median $=\frac{1}{2}{\left[\frac{n}{2}\right]}^{th}term+{\left[\frac{n}{2}+1\right]}^{th}term$

$$\begin{gathered} =\frac{1}{2}{\left[\frac{20}{2}\right]}^{th}term+{\left[\frac{20}{2}+1\right]}^{th}term \\ =\frac{1}{2}{\left[10\right]}^{th}term+{\left[11\right]}^{th}term \\ ={\left[10\right]}^{th}term=20 \\ ={\left[11\right]}^{th}term=20 \\ =\frac{1}{2}\left[20+20\right] \\ =20 \end{gathered}$$

Therefore, the median is$=20$.

Hence, the arithmetic mean of the given distribution $=19.5$and the arithmetic median$=20$.