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 just used to differentiate it from the other “means” such as harmonic and geometric mean. The arithmetic mean is a good average. It is sometimes known as 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.

Arithmetic Mean Formula

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 xˉ\bar{x}. 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,……, an are the values of variable “a”, then the mean of a is calculated as below:

Arithmetic Mean = a1+a2+a3++ann\frac{a_{1} + a_{2} + a_{3} +… + a_{n}}{n}

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 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, n in A.P.

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 in three different types of series as listed below

  • Individual series
  • Discrete series
  • Continuous series

Arithmetic Mean in Individual Series

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

xˉ\bar{x} =x1+x2+x3++xnn=xn\frac{x_{1} + x_{2} + x_{3}+ …+x_{n}}{n}=\frac{\sum x}{n}

Arithmetic Mean in Discrete Series

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

xˉ\bar{x} = f1x1+f2x2+f3x3..+fnxnN=fxN\frac{f_{1}x_{1} + f_{2}x_{2} + f_{3}x_{3}……..+ f_{n}x_{n}}{N}=\frac{\sum fx}{N} where N\text{N} = f\sum f

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”.

Mid value\text{Mid value} = (lower limit + upper limit)2\frac{(\text{lower limit + upper limit})}{2}

Weighted Arithmetic Mean

Weighted Mean is an average computed by giving different weights to some of the individual values. The weighted mean for given set of non negative data, X\text{X} = x1, x2,……, xn with non negative weights, W\text{W} = w1, w2,……, wn can be derived from the formula:

=> xwˉ\bar{x_w} = WXW\frac{\sum WX}{\sum W}

Where

xwˉ\bar{x_w} = Weighted arithmetic mean

x = Values of the items

w = Weight of the item

Arithmetic mean vs Geometric mean

Geometric mean of two numbers is the square root of product of the numbers. If a and b are two positive numbers, then geometric mean denoted by G.M = ab\sqrt{ab}

If there are n numbers, then G.M = a1.a2ann\sqrt[n]{a_{1}.a_{2}…a_{n}}

Relationship between A.M and G.M : A.M > G.M

Related articles:

Geometric mean

Harmonic mean

Solved Problems

Some of the problems are illustrated on AM below.

Question 1: Find the value of pq\mathbf{\frac{p}{q}}, if the arithmetic mean between p and q is twice as greater as the geometric mean.

Solution:

Since, Arithmetic Mean = p  +  q2\mathbf{\frac{p\;+\;q}{2}}

And, Geometric Mean = p  q\mathbf{\sqrt{p\;q}}

Also, 2 GM = AM

Therefore, 2 × p  q\mathbf{\sqrt{p\;q}} = p  +  q2\mathbf{\frac{p\;+\;q}{2}}

p  +  q    4  pq  =  0\mathbf{p\;+\;q\;-\;4\;\sqrt{pq}\;=\;0}

Now, on dividing the above equation by q we get

pq  +  1    4  pq  =  0\mathbf{\frac{p}{q}\;+\;1\;-\;4\;\frac{\sqrt{p}}{\sqrt{q}}\;=\;0}

Or, (pq)2    4  pq  +  1  +  4    4=  0\mathbf{\left ( \frac{\sqrt{p}}{\sqrt{q}} \right )^{2}\;-\;4\;\frac{\sqrt{p}}{\sqrt{q}}\;+\;1\;+\;4\;-\;4=\;0}

Or, (pq    2)2  =  3\mathbf{\left ( \frac{\sqrt{p}}{\sqrt{q}}\; -\;2 \right )^{2}\;=\;3}

Or, pq  =  2  ±    3\mathbf{\frac{\sqrt{p}}{\sqrt{q}}\;=\;2\;\pm \;\;\sqrt{3}}

Now, on squaring both LHS and RHS we get,

pq  =  (2  ±    3)2  =  4  +  3  ±  4  3  =  7  ±  43\mathbf{\frac{p}{q}\;=\;\left ( 2\;\pm \;\;\sqrt{3} \right )^{2}\;=\;4\;+\;3\;\pm \;4\;\sqrt{3}\;=\;7\;\pm \;4\sqrt{3}}

Therefore, pq  =  7  ±  43\mathbf{\frac{p}{q}\;=\;7\;\pm \;4\sqrt{3}}

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
fx=470\sum fx = 470 and f=53\sum f = 53

Mean is given by

xˉ\bar{x} = f1x1+f2x2+f3x3..+fnxnN\frac{f_{1}x_{1} + f_{2}x_{2} + f_{3}x_{3}……..+ f_{n}x_{n}}{N} = fxN\frac{\sum fx}{N}

= 470/53 = 8.87

Frequently Asked Questions

What are Characteristics of Arithmetic Mean

Arithmetic Mean is mainly used,
To get a general idea about the whole group.
To represent the whole group and hence summarizes the whole data.
For summarizing data.
For decision-making.

Is arithmetic mean the same as average?

The arithmetic mean is a good average. It is sometimes known as 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 give absurd answers. For example, the mean number of children in a family is 4.3.

How do you find the average arithmetic mean?

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 value can be calculated using formula: A = (The sum of the n numbers)/ (number of terms).