# Arithmetic Mean

It is the most commonly used average or measure of the central tendency applicable only in case of quantitative data. Arithmetic mean is also simply called “mean”. Arithmetic mean is defined as:

Arithmetic mean is quotient of sum of the given values and number of the given values”.

The arithmetic mean can be computed for both ungroup data (raw data: a data without any statistical treatment) and grouped data (a data arranged in tabular form containing different groups).

If $X$ is the involved variable, then arithmetic mean of $X$ is abbreviated as $A.M$ of $X$ and denoted by $\overline X$. The arithmetic mean of $X$ can be computed by any of the following methods.

 Method’s Name Nature of Data Ungrouped Data Grouped Data Direct Method $A.M{\text{ of }}X = \overline X = \frac{{\sum x}}{n}$ Indirect or Short-Cut Method $A.M{\text{ of }}X = \overline X = A + \frac{{\sum D}}{n}$ $A.M{\text{ of }}X = \overline X = A + \frac{{\sum fD}}{{\sum f}}$ Method of Step-Deviation $A.M{\text{ of }}X = \overline X = A + \frac{{\sum u}}{n} \times c$ $A.M{\text{ of }}X = \overline X = A + \frac{{\sum fu}}{{\sum f}} \times h$

Where
$x:$ Indicates values of the variable $X$.
$n:$ Indicates number of values of $X$.
$f:$ Indicates frequency of different groups.
$A:$Indicates assumed mean.
$D:$ Indicates deviation from $A$ i.e, $D = (x - A)$
$u = \frac{{x - A}}{{c{\text{ or }}h}}$
$u:$ Step-deviation and $c:$ Indicates common divisor
$h:$ Indicates size of class or class interval in case of grouped data.
$\sum :$ Summation or addition.

Example (1):
The one-sided train fare of five selected BS students is recorded as follows $(\ ){\text{ :}}$ $10$, $5$, $15$, $8$ and $12$. Calculate arithmetic mean of the following data.
Solution:

Let train fare is indicated by $x$, then

 $x{\text{ }}(\ )$ $10$ $5$ $15$ $8$ $12$ $\sum x = 50$

Arithmetic mean of $X = \overline X = \frac{{\sum x}}{n}$, we decide to use above-mentioned formula. Form the given data, we have $\sum x = 50$ and $n = 5$. Placing these two quantities in above formula, we get the arithmetic mean for given data.

Example (2):

Given the following frequency distribution of first year students of a particular college.

 Age (Years) $13$ $14$ $15$ $16$ $17$ Number of Students $2$ $5$ $13$ $7$ $3$

Solution:

The given distribution belongs to a grouped data and the variable involved is ages of first year students. While the number of students Represent frequencies.

 Ages (Years) $x$ Number of Students $f$ $fx$ $13$ $2$ $26$ $14$ $5$ $70$ $15$ $13$ $195$ $16$ $7$ $112$ $17$ $3$ $51$ Total $\sum f = 30$ $\sum fx = 454$

Now we will find the Arithmetic Mean as $\overline X = \frac{{\sum fx}}{{\sum f}} = \frac{{454}}{{30}} = 15.13$ years.

Example (3):

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 $10$ $20$ $40$ $30$

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 (Km) Number of Persons $f$ Mid Points $x$ $fx$ $0 - 10$ $10$ $5$ $50$ $10 - 20$ $20$ $15$ $300$ $20 - 30$ $40$ $25$ $1000$ $30 - 40$ $30$ $35$ $1050$ Total $\sum f = 100$ $\sum fx = 2400$

Now we will find the Arithmetic Mean as $\overline X = \frac{{\sum fx}}{{\sum f}} = \frac{{2400}}{{100}} = 24$ Km.