Question

The mean of the following frequency distribution is $350$, find the missing frequency.

Class	$100 - 200$	$200 - 300$	$300 - 400$	$400 - 500$	$500 - 600$	$600 - 700$
Frequency	$5$	$3$	$3$	$-$	$2$	$1$

Answer 1

Take the missing frequency as a variable. Add all the frequencies and equate it to the frequency representing variable in terms of the considered variable. Now find the mid-term of the class. Substitute the values in the formula of the mean and form a linear equation to find the variable.

Complete step by step solution:
Let us consider the missing frequency as the variable $x$
Number of total pupils $=$ $350$
Add the total number of pupils and equate it to frequency $f$
$\Rightarrow 5 + 3 + 3 + x + 2 + 1 = f$
Add the individual terms which simplifies as follows;
$\Rightarrow f = 14 + x$
Let us take the above equation as equation $1$
We have mean as given
Mean $= 350$
To find $f\left( x \right)$, we have to find the mid-value of all the class intervals. The mid-values of the given class intervals are as given in the below table:

Class	$100 - 200$	$200 - 300$	$300 - 400$	$400 - 500$	$500 - 600$	$600 - 700$
Frequency$f$	$5$	$3$	$3$	$x$	$2$	$1$
Mid-values $f\left( x \right)$	150	250	350	450	550	650

$\Rightarrow \dfrac{{\sum {f(x)f} }}{{\sum f }} = 350$
\Rightarrow \dfrac{{150 \times 5 + 250 \times 3 + 350 \times 3 + 450 \times x + 550 \times 2 + 650 \times 1}}{{14 + x}}$$$$ = 350
Simplifying the equation, we get;
$\Rightarrow \dfrac{{750 + 750 + 1050 + 450x + 1100 + 650}}{{14 + x}} = 350$
Now, adding the terms and taking the denominator to the right-hand side, we get;
$\Rightarrow 450x + 4300 = 350 \times \left( {14 + x} \right)$
Multiplying and rearranging the terms, we get;
$\Rightarrow 450x + 4300 = 4900 + 350x$
Rearranging the terms, we get;
$\Rightarrow 450x - 350x = 4900 - 4300$
Now, subtracting the left-hand side terms and the right-hand side terms, we get;
$\Rightarrow 100x = 600$
Now, cancelling out the common zeros on both the sides and simplifying the equation, we get;
$x = 6$

$\therefore$ The missing frequency = $$$$x = 6

Note: The total number of fillings/observations multiplied by the frequency respectively divided by the total frequency gives you the mean. The mean is the average of the given data, used to conclude the average aspect.