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.