Arithmetic Mean

Arithmetic Mean is simply the mean or average for a set of data or a collection of numbers. In Mathematics, we deal with different types of means, such as arithmetic mean, arithmetic harmonic mean, geometric mean and geometric harmonic mean.

The term arithmetic mean is used to differentiate it from other “means”, such as harmonic and geometric mean. The arithmetic mean is a good average, so it is also known as the average. But, it cannot be used in some cases, like the distribution has open-end classes, the distribution is highly skewed, averages are taken for ratios and percentages, etc.

Table of Contents

Arithmetic Mean Formula
Arithmetic Mean for Ungrouped Data
Arithmetic Mean between Two Numbers
How to Find the Arithmetic Mean of a Series
Weighted Arithmetic Mean
Solved Problems

Arithmetic Mean Formula

The arithmetic mean is the simplest measure of central tendency and is the ratio of the sum of the items to the number of items. It is denoted by x̄. It is considered as the best measure of central tendency. The choice of the average depends upon the distribution of the data and the purpose for which it is used.

If a₁, a₂,……, a_n are the values of variable “a”, then the mean of a is calculated as below:

\(\begin{array}{l}\text{Arithmetic Mean}=\frac{a_{1} + a_{2} + a_{3} +… + a_{n}}{n}\end{array} \)

Where, “n” is the total number of elements in the set. Read More.

Arithmetic Mean for Ungrouped Data

The most popular and commonly used measure of central tendency is the arithmetic mean or simply mean. It is defined as the ratio of aggregate values to the total number of values.

Let us understand, with the help of an example, how to find the AM for ungrouped data. For example, calculate the arithmetic mean of the ages of 10 players: 20, 30, 40, 78, 65, 98, 77, 79, 63, 64.

Solution:

Sum of terms = 20 + 30 + 40 + 78 + 65 + 98 + 77 + 79 + 63 + 64 = 614

Number of terms = 10

Arithmetic mean = 614/10 = 61.4

Therefore, the arithmetic mean of the age of 10 players is 61.4.

Arithmetic Mean between Two Numbers

Consider any two numbers, say m and n, and P be the arithmetic mean between two numbers.

The sequence will be m, P, and n in AP.

P – m = n – P

P = (n + m)/2 = (Sum of the numbers)/(number of terms).

How to Find the Arithmetic Mean of a Series?

We can calculate the arithmetic mean in three different types of series, as listed below:

Individual series
Discrete series
Continuous series

Arithmetic Mean in Individual Series

If x₁, x₂,……, x_n are the n items, then the AM is defined as,

\(\begin{array}{l}\bar{x}=\frac{x_{1} + x_{2} + x_{3}+ …+x_{n}}{n}=\frac{\sum x}{n}\end{array} \)

Arithmetic Mean in Discrete Series

If x₁, x₂,……, x_n are the n items and f₁, f₂,……,f_n are the corresponding frequencies, then the mean is given by,

\(\begin{array}{l}\bar{x} = \frac{f_{1}x_{1} + f_{2}x_{2} + f_{3}x_{3}……..+ f_{n}x_{n}}{N}=\frac{\sum fx}{N}\end{array} \)

where

\(\begin{array}{l}N = \sum f\end{array} \)

Arithmetic Mean in Continuous Series

In the case of continuous series, we use the same formula as in discrete series. In this case, mid values will be taken as “x”.

\(\begin{array}{l}\text{Mid value}=\frac{(\text{lower limit + upper limit})}{2}\end{array} \)

Weighted Arithmetic Mean

The weighted mean is an average computed by giving different weights to some of the individual values. The weighted mean for a given set of non-negative data, X = x₁, x₂,……, x_n with non-negative weights, W = w₁, w₂,……, w_n can be derived from the formula:

\(\begin{array}{l}\Rightarrow \bar{x_w}=\frac{\sum WX}{\sum W}\end{array} \)

Where,

\(\begin{array}{l}\bar{x_w}=\text{Weighted arithmetic mean}\end{array} \)

x = Values of the items

w = Weight of the item

Arithmetic Mean vs Geometric Mean

The geometric mean of two numbers is the square root of the product of the numbers. If a and b are two positive numbers, then the geometric mean is denoted by,

\(\begin{array}{l}GM = \sqrt{ab}\end{array} \)

If there are n numbers, then

\(\begin{array}{l}GM = \sqrt[n]{a_{1}.a_{2}…a_{n}}\end{array} \)

Relationship between AM and GM: AM > GM

Related articles:

Geometric Mean

Harmonic Mean

Solved Problems

Some problems on AM are illustrated below.

Question 1: Find the value of p/q, if the arithmetic mean between p and q is twice as greater as the geometric mean.

Solution:

Since, Arithmetic Mean = (p + q)/2

And, Geometric Mean = √(pq)

Also, 2 GM = AM

Therefore,

\(\begin{array}{l}2\times \sqrt{pq} = \frac{p+q}{2}\end{array} \)

\(\begin{array}{l}p + q – 4\sqrt{pq} = 0\end{array} \)

Now, dividing the above equation by q, we get

\(\begin{array}{l}\mathbf{\frac{p}{q}\;+\;1\;-\;4\;\frac{\sqrt{p}}{\sqrt{q}}\;=\;0}\end{array} \)

Or,

\(\begin{array}{l}\mathbf{\left ( \frac{\sqrt{p}}{\sqrt{q}} \right )^{2}\;-\;4\;\frac{\sqrt{p}}{\sqrt{q}}\;+\;1\;+\;4\;-\;4=\;0}\end{array} \)

Or,

\(\begin{array}{l}\mathbf{\left ( \frac{\sqrt{p}}{\sqrt{q}}\; -\;2 \right )^{2}\;=\;3}\end{array} \)

Or,

\(\begin{array}{l}\mathbf{\frac{\sqrt{p}}{\sqrt{q}}\;=\;2\;\pm \;\;\sqrt{3}}\end{array} \)

Now, on squaring both LHS and RHS, we get,

\(\begin{array}{l}\mathbf{\frac{p}{q}\;=\;\left ( 2\;\pm \;\;\sqrt{3} \right )^{2}\;=\;4\;+\;3\;\pm \;4\;\sqrt{3}\;=\;7\;\pm \;4\sqrt{3}}\end{array} \)

Therefore,

\(\begin{array}{l}\mathbf{\frac{p}{q}\;=\;7\;\pm \;4\sqrt{3}}\end{array} \)

Question 2: Find the mean of the following data:

x	2	4	6	8	10	12	14	16
y	5	4	8	12	7	6	9	2

Solution:

x	2	4	6	8	10	12	14	16	Total
y	5	4	8	12	7	6	9	2	53
xy	10	16	48	96	70	72	126	32	470

\(\begin{array}{l}\sum fx = 470\end{array} \)

and

\(\begin{array}{l}\sum f = 53\end{array} \)

Mean is given by

\(\begin{array}{l}\bar{x}=\frac{f_{1}x_{1} + f_{2}x_{2} + f_{3}x_{3}……..+ f_{n}x_{n}}{N}\end{array} \)

\(\begin{array}{l}=\frac{\sum fx}{N}\end{array} \)

= 470/53 = 8.87

AM, GM and HM

Frequently Asked Questions

What are the characteristics of arithmetic mean?

The arithmetic mean is mainly used,
To get a general idea about the whole group.
To represent the whole group and hence summarises the whole data.
For summarising data.
For decision-making.

Is arithmetic mean the same as average?

The arithmetic mean is a good average, so it is also known as the average. But, it cannot be used when,
The distribution has open end classes.
The distribution is highly skewed.
Averages are taken for ratios and percentages.

What are the merits and demerits of arithmetic mean?

Merits: Simple to understand, easily calculated, determined in most cases,
capable of further algebraic treatment, the result is stable.

Demerits:
Arithmetic mean need not coincide with any of the observed values, affected by extreme values, not good in the case of ratios and percentages and sometimes gives absurd answers. For example, the mean number of children in a family is 4.3.

How do you find the average arithmetic mean?

The arithmetic mean is simply the mean or average for a set of data or a collection of numbers. If A represents the average (arithmetic mean) of a set of n numbers, then the value can be calculated using the formula: A = (The sum of the n numbers)/(number of terms).