Examples of Arithmetic Mean

Example:

The following data shows distance covered by $$100$$ people to perform their routine jobs.

Distance (Km)
$$0 – 10$$
$$10 – 20$$
$$20 – 30$$
$$30 – 40$$
Number of People
$$10$$
$$20$$
$$40$$
$$30$$

Calculate the arithmetic mean by step-deviation method; also explain why it is better than direct method in this particular case.

 

Solution:

The given distribution is grouped data and the variable involved is distance covered, while the number of people represents frequencies.

Distance Covered in (Km)
Number of People
$$f$$
Mid Points
$$x$$
$$u = \left( {\frac{{x – 5}}{{10}}} \right)$$
$$fu$$
$$0 – 10$$
$$10$$
$$5$$
$$ – 1$$
$$ – 10$$
$$10 – 20$$
$$20$$
$$15$$
$$0$$
$$0$$
$$20 – 30$$
$$40$$
$$25$$
$$ + 1$$
$$40$$
$$30 – 40$$
$$30$$
$$35$$
$$ + 2$$
$$60$$
Total
$$\sum f = 100$$
$$\sum fu = 90$$

Now we will find the arithmetic mean as $$\overline X = A + \frac{{\sum fu}}{{\sum f}} \times h$$
Here
$$A = 15$$,    $$\sum fu = 90$$,    $$\sum f = 100$$   and   $$h = 10$$
$$\overline X = 15 + \frac{{90}}{{100}} \times 10 = 24$$ Km

Explanation:

Here from the mid points ($$x$$) it is very clear that each mid point is a multiple of $$5$$ and there is also a gap of $$10$$ from mid point to midpoint, i.e. class size or interval ($$h$$). Keeping this in mind, we should use the step-deviation method instead of direct method.

 

Example:

The following frequency distribution showing the marks obtained by $$50$$ students in statistics at a certain college. Find the arithmetic mean using (1) direct method (2) short-cut method (3) step-deviation.

Marks
$$20 – 29$$
$$30 – 39$$
$$40 – 49$$
$$50 – 59$$
$$60 – 69$$
$$70 – 79$$
$$80 – 89$$
Frequency
$$1$$
$$5$$
$$12$$
$$15$$
$$9$$
$$6$$
$$2$$

 

Solution:

Direct Method
Short-Cut
Method
Step-Deviation
Method
Marks
$$f$$
$$x$$
$$fx$$
$$D = x – A$$
$$fD$$
$$u = \frac{{x – A}}{h}$$
$$fu$$
$$20 – 29$$
$$1$$
$$24.5$$
$$24.5$$
$$ – 30$$
$$ – 30$$
$$ – 3$$
$$ – 3$$
$$30 – 39$$
$$5$$
$$34.5$$
$$172.5$$
$$ – 20$$
$$ – 100$$
$$ – 2$$
$$ – 10$$
$$40 – 49$$
$$12$$
$$44.5$$
$$534.5$$
$$ – 10$$
$$ – 120$$
$$ – 1$$
$$ – 12$$
$$50 – 59$$
$$15$$
$$54.5$$
$$817.5$$
$$0$$
$$0$$
$$0$$
$$0$$
$$60 – 69$$
$$9$$
$$64.5$$
$$580.5$$
$$10$$
$$90$$
$$1$$
$$9$$
$$70 – 79$$
$$6$$
$$74.5$$
$$447.5$$
$$20$$
$$120$$
$$2$$
$$12$$
$$80 – 89$$
$$2$$
$$84.5$$
$$169.5$$
$$30$$
$$60$$
$$3$$
$$6$$
Total
$$50$$
$$2745$$
$$20$$
$$2$$

 

(1) Direct Method:

$$\overline X = \frac{{\sum fx}}{{\sum f}} = \frac{{2745}}{{50}} = 54.9$$ or $$55$$ Marks

(2) Short-Cut Method:
$$\overline X = A + \frac{{\sum fD}}{{\sum f}}$$       Where $$A = 54.5$$
$$ = 54.5 + \frac{{20}}{{50}} = 54.5 + 0.4 = 54.9$$Marks

(3) Step-Deviation Method:
$$\overline X = A + \frac{{\sum fu}}{{\sum f}} \times h$$   Where $$A = 54.5$$        $$h = 10$$

$$ = 54.5 + \frac{2}{{50}} \times 10$$

$$ = 54.5 + 0.4 = 54.9$$ Marks