Arithmetic Mean

The mean or average of a set of data or a collection of data is known as the arithmetic mean. The arithmetic mean is evaluated by adding the given collection of numbers and dividing the sum by the count of numbers in the collection. The arithmetic mean is used in surveys and experimental studies. It comes under the statistics part of mathematics. In mathematics, there are 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.

The arithmetic mean is mostly the same as the average, but due to some cases such as the distribution has open-end classes, the distribution is highly skewed, we cannot agree to that.

Arithmetic Formula

The simplest and the best measure of central tendency is arithmetic mean. It is the ratio of the sum of the items to the number of items. The selection of average depends on the distribution of 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.

Arithmetic mean = (a₁ + a₂ + ……..+ a_𝑛)/𝑛

Where 𝑛 is the total number of variables in the set.

\[\overline{X} = \frac{X_{1} + X_{2} + X_{3} . . . X_{N}}{N}\]

Where

X̅ = The Mean

X₁ = The First Value

X₂ = The Second Value

X₃ = The Third Value

X_N = The Last Value

N = The Number of Value

Arithmetic Mean for Ungrouped Data

The simplest and the most used central tendency measure is arithmetic mean. It is the ratio of the sum of variables to the number of variables present in a given set. Let’s understand this with the help of an example.

1. Calculate the arithmetic mean from the following data.

20,30,40,70,80

Solution

Sum of terms = 20 + 30 + 40 + 70 + 80 =  240

Number of terms = 5

Arithmetic mean = 240/5 = 48

Therefore the arithmetic mean of the above five numbers is 48.

Arithmetic Mean Between Two Numbers

How to find the arithmetic mean between 2 numbers?

Consider any two numbers say x and y.

And m be the arithmetic mean of these numbers x and y.

The sequence will be x, m, y in arithmetic progression.

m - x = y - m

M = (x + y)/2 = (sum of the numbers)/(number of terms)

This is the answer to the question.

How to Find the Arithmetic Mean of a Series

The arithmetic mean can be found out for three different series namely:

• Individual series

• Discrete series

• Continuous series

Arithmetic Mean in Individual Series

An individual data series is a set of numbers that are arranged in ascending order. The arithmetic mean of individual series is calculated as below:

If x₁, x₂, x₃, ……, x_𝑛 are the 𝑛 items, then

Arithmetic mean = (x₁+x₂+x₃+....+x_𝑛)/𝑛 = Σx/𝑛

Arithmetic Mean in Discrete Series

Discrete series is a series where the variables have corresponding frequencies, but the variables are without class intervals. The arithmetic mean of discrete series is calculated as below:

If x₁, x₂, x₃, ……, x_𝑛 are the 𝑛 items and f₁, f₂, f₃,....,f_𝑛 are the corresponding frequencies, then

Arithmetic Mean =( f₁x₁+f₂x₂+f₃x₃+...+f_𝑛x_𝑛)/N = Σf_x/N

Where,

N = Σf

Arithmetic Mean in Continuous Series

Continuous series is a series where the variables are in the class interval, and each has a corresponding frequency. The formula for calculating the arithmetic mean of continuous series is the same as the discrete series; you only need to find out ‘x’. In this case, mid-value is considered as x.

Mid Value = (Lower Limit + Upper Limit)/2

Weighted Arithmetic Mean

In weighted arithmetic mean, different weights are given to the individual values based upon which the mean is calculated. This mean is calculated as below:

If  x₁, x₂, x₃, ……, x_𝑛 are the 𝑛 items and w₁, w₂, w₃,.....,w_𝑛 are the corresponding weights, then

Weighted arithmetic mean = (w₁x₁+w₂x₂+w₃x₃+...+w_𝑛x_𝑛)/Σw = Σw_x/Σw

Where, 

w = number of weights

Arithmetic Mean and Geometric Mean

The geometric mean of two numbers is calculated by finding out the square root of the product of two numbers.

If the two numbers are x and y

Geometric Mean = √xy

Relationship between A.M and G.M

A.M > G.M

Solved Problems

Example 1

Calculate the arithmetic mean from the following data

Solution:

Arithmetic mean = Σf_x/Σf

= 150 / 50 

= 3