# Examples of Arithmetic Mean

Example (4):

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

 Distance (Km) $0 - 10$ $10 - 20$ $20 - 30$ $30 - 40$ Number of Persons $10$ $20$ $40$ $30$

Calculate Arithmetic Mean by Step-Deviation Method; also explain why it is better than direct method in this particular case.

Solution:

The given distribution belongs to a grouped data and the variable involved is ages of “distance covered”. While the “number of persons” Represent frequencies.

 Distance Covered in (Km) Number of Persons $f$ 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 $\overline X = A + \frac{{\sum fu}}{{\sum f}} \times h$
Where
$A = 15$,    $\sum fu = 90$,    $\sum f = 100$   and   $h = 10$
$\overline X = 15 + \frac{{90}}{{100}} \times 10 = 24$ Km

Explanation:

Here from the mid points ($x$) it is very much clear that each mid point is multiple of $5$ and there is also a gap of $10$ from mid point to midpoint i.e. class size or interval ($h$). Keeping in view this, we should prefer to take method of Step-Deviation instead of Direct Method.

Example (5):

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