Question

An aircraft has $120$ passenger seats. The number of seats occupied during $100$ flights is given in the following table :

Number of seats	$100-104$	$104-108$	$108-112$	$112-116$	$116-120$
Frequency	$15$	$20$	$32$	$18$	$15$

Determine the mean number of seats occupied over the flights.

Answer 1

Step 1: Find the assumed mean for the given data:

The assumed mean of the given data is calculated as

$a=\frac{upperclasslimit+lowerclasslimit}{2}$

$\Rightarrow$$a=\frac{120+100}{2}$

$\Rightarrow$$a=110$

Step 2: Create a frequency distribution table based on the given and calculated data:

The deviation $\left(d_i\right)$ for each class interval is calculated as $d_i=x_i-a$, where $x_i$ is the mean of each class interval respectively.

Class Interval	Mean of Class interval $\left(x_i\right)$	Frequency $\left(f_i\right)$	Deviation $(d_i=x_i–a)$	$f_i{d}_i$
$100-104$	$102$	$15$	$-8$	$-120$
$104-108$	$106$	$20$	$-4$	$-80$
$108-112$	$110$	$32$	$0$	$0$
$112-116$	$114$	$18$	$4$	$72$
$116-120$	$118$	$15$	$8$	$120$
		$N=\Sigma f_i=100$		$\Sigma f_i{d}_i=-8$

Step 3: Using the formula calculate the mean:

The mean $\left(\overline{x}\right)$ is given as

$\overline{x}$$=a+\frac{\underset{}{\sum f_i{d}_i}}{\sum _{}{f}_i}$

Substituting the values we get

$\overline{x}$$=110+\left(\frac{-8}{100}\right)$

$\Rightarrow$$\overline{x}$$=109.92$

Since the number of seats cannot be a decimal number we round off to the nearest integer.

$\Rightarrow$$\overline{x}$$=110$

Hence, the mean number of seats occupied during $100$ flights is $110$.