# Arithmetic Mean

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

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

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

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

 Method’s Name Nature of Data Ungrouped Data Grouped Data Direct Method $A.M{\text{ of }}X = \overline X = \frac{{\sum x}}{n}$ $A.M{\text{ of }}X = \overline X = \frac{{\sum fx}}{{\sum f}}$ 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 $A.M{\text{ of }}X = \overline X = A + \frac{{\sum u}}{n} \times c$ $A.M{\text{ of }}X = \overline X = A + \frac{{\sum fu}}{{\sum f}} \times h$

Here
$x:$ indicates value 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, $D = (x – A)$
$u = \frac{{x – A}}{{c{\text{ or }}h}}$
$u:$ indicates step-deviation and $c:$ indicates common divisor
$h:$ indicates size of class or class interval in case of grouped data
$\sum :$ indicates summation or addition

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

Solution:

Let train fare be indicated by $x$, then

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

The arithmetic mean of $X = \overline X = \frac{{\sum x}}{n}$, so we decide to use the above-mentioned formula. From the given data, we have $\sum x = 50$ and $n = 5$. Placing these two quantities in the above formula, we get the arithmetic mean for the given data.
$\overline X = \frac{{50}}{5} = \ 10$

Example:

Provide the given distribution of 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 is grouped data and the variable involved is ages of first year students, while the number of students represents 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:

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

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

Solution:

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

 Distance (Km) Number of People $f$ 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.