Question

Calculate the arithmetic mean using the shortcut method.

Wages(x)	100	110	120	140
No. of worker(f)	10	20	40	50

A) 126.22
B) 123.66
C) 133.66
D) 143.66

Answer 1

We will start with making the same table and then we will find $f(x)$ in the table. Then we are going to sum up the f and $f(x)$ then we will apply the arithmetic mean method to the table. We will assume a middle point which in this case we will take 30. The method we will use is the Shortcut arithmetic method. This way we will get the answer.

Complete step by step solution:
The middle value of the frequency is A
Here the value of A is 30

Wages(d)	No. of workers(f)	$d = f - A$	$f \times d$
100	10	-20	-200
110	20	-10	-200
120	30	0	0
130	40	10	400
140	50	20	1000
$\sum {f = 150}$		$\sum {f.d = 1000}$

Now we will apply shortcut method
The below is the formula for shortcut arithmetic method
$\bar X = A + \dfrac{{\sum {f.d} }}{{\sum f }}$
Now we will put the values from the above table
$\Rightarrow \bar X = 30 + \dfrac{{1000}}{{150}}$
Hence the arithmetic mean is
$\Rightarrow \bar X = 36.6667$

Note:
Arithmetic mean is a commonly used average to represent data. It is obtained by simply adding all the values and dividing them by the number of items. Arithmetic mean can be a simple arithmetic mean or weighted arithmetic mean. There are many ways we can find the Arithmetic mean of the data. The types are, Direct method and shortcut method. It helps to get the data average.