Arithmetic Mean is defined as the ratio of all the values or observations to the total number of values or observations. Arithmetic Mean is one of the fundamental formulas used in mathematics and it is highly used in various solving various types of problems. We can understand it with some examples, if in a family the husband earns 35,000 rupees and his wife earns 40,000 rupees then what is their average salary? This average is also called the arithmetic mean of 35,000 rupees and 40,000 rupees, which is calculated by adding these two salaries and then dividing it by 2.

Average Salary (Arithmetic Mean of Salary) = (35000 + 40000)/2 = 37500. 

Thus, the arithmetic mean is used in various scenarios such as in finding the average marks obtained by the student in marks, the average rainfall in any area, etc.

Arithmetic Mean Definition

Arithmetic Mean OR (AM) is calculated by taking the sum of all the given values and then dividing it by the number of values. For evenly distributed terms arranged in ascending or descending order arithmetic mean is the middle term of the sequence. The arithmetic mean is sometimes also called mean, average, or arithmetic average.

We can understand the concept of Arithmetic mean by understanding the following example. Find the mean of 3, 6, 7, and 4.

Here, the mean is calculated first by taking the sum of all the values 3+6+7+4 = 20 and then dividing it by, 4 as we have a total of 4 terms. Arithmetic mean =  20/4 = 5. Thus, the arithmetic mean of the given value is 5.

The formula to calculate the arithmetic mean is discussed in the image below:

Arithmetic Mean Formula

 

Arithmetic Mean Formula

Arithmetic Mean Formula is used to determine the mean or average of a given data set. The symbol used to denote the arithmetic mean is ‘x̄’ and read as x bar. The arithmetic mean of the observations is calculated by taking the sum of all the observations and then dividing it by the total number of observations.

The formula for calculating the arithmetic mean is,

Arithmetic Mean (x̄) = Sum of all observations / Number of observations

Let there be n observations in a data set namely n1, n2, n3, n4, n5, ……..nn. Then the arithmetic mean is calculated as,

A.M. = (n1 + n2 + n3 + n4 + … + nn)/n

If the frequency of various numbers in a data set is f1, f2, f3, f4, f5, …, fn for the numbers n1, n2, n3, n4, n5, … nn.

A.M. = \frac{f_1n_1 + f_2n_2 + f_3n_3 + f_4n_4 + ... + f_nn_n}{f_1 + f_2 + f_3 + f_4 + ... + f_n}

The arithmetic mean formula is given by,

A.M = {\frac{1}{n}\sum_{i=1}^{n}a_{i}}

where,
n is number of items
A.M is arithmetic mean
ai are set values.

Derivation of Arithmetic Mean Formula 

Let n be the number of observations in the operation and n1, n2, n3, n4, …, nn be the given numbers. Now as per the definition, the arithmetic means formula can be defined as the ratio of the sum of all numbers of the group by the number of items.

A.M. = (n1 + n2 + n3 + n4 + … + nn)/n

By solving the equation, the formula of arithmetic mean is obtained which is,

A.M = {\frac{1}{n}\sum_{i=1}^{n}a_{i}}

Thus, the Arithmetic mean formula is derived.

Properties of Arithmetic Mean

Arithmetic Mean has various Properties and some of the important properties of the arithmetic mean are discussed below. If we take “n” observations, i.e. x₁, x₂, x₃, ….,xₙ and let x̄ be its arithmetic mean then,

  • If all the values in the data set are equal then the arithmetic mean of the data set is the individual value of the data set, i.e. if the values of observation are say m, m, m, …, m upto n terms then the arithmetic mean is m. We can understand this with the help of the following example,

Find the arithmetic mean of the data set, 6, 6, 6, 6, and 6

Solution:

Arithmetic Mean = (6 + 6 + 6 + 6 + 6)/6
                          = 30/5
                          = 6

  • The sum of the deviation of all the values in a set of observations from the arithmetic mean is zero. This can be understood as, if we have n terms in any observations say x1, x2, x3, …, xn, and if their arithmetic mean is, x̄ then we can say that,

(x₁−x̄)+(x₂−x̄)+(x₃−x̄)+…+(xₙ−x̄) = 0

  • For Discrete Data Set, we can say that ∑(xi − x̄) = 0
  • For Grouped Frequency Distribution, we can say that ∑f(xi − ∑x̄) = 0
  • If we increase or decrease all the values of the data set by a fixed value then the arithmetic is increased or decreased by the same value. This can be understood as, if we have n terms in any observations say x1, x2, x3, …, xn, and if their arithmetic mean is, x̄ then if all the term is increased or decreased by “m” then the arithmetic mean x̄ is increased or decreased by m. We can understand this with the help of the following example,

If the arithmetic mean of the data set, 4, 5, 6, 7, and 8 is 6 and if each value is increased by 3 find the new mean.

Solution:

New data set = 4+3, 5+3, 6+3, 7+3, 8+3

                      = 7, 8, 9, 10, 11

Arithmetic Mean = (7 + 8 + 9 + 10 + 11)/5
                          = 45/5
                          = 9…(i)

Also, 

Old AM = 6

Change in each value, 3

New AM = 6 + 3 =  9…(ii)

From (i) and (ii) above property is proved.

  • If we multiply or divide all the values of the data set by a fixed value then the arithmetic is multiplied or divided by the same value. This can be understood as, if we have n terms in any observations say x1, x2, x3, …, xn, and if their arithmetic mean is, x̄ then if all the term is multiplied or divided by “m” then the arithmetic mean x̄ is multiplied or divided by m. We can understand this with the help of the following example,

If the arithmetic mean of the data set, 4, 5, 6, 7, and 8 is 6 and if each value is multiplied by 3 find the new mean.

Solution:

New data set = 4×3, 5×3, 6×3, 7×3, 8×3

                      = 12, 15, 18, 21, 24

Arithmetic Mean = (12+15+18+21+24)/5
                          = 90/5 
                          = 18…(i)

Also, 

Old AM = 6

Each value is multiplied by 3

New AM = 6 × 3 =  18…(ii)

From (i) and (ii) above property is proved.

Arithmetic Mean can easily be calculated for,

  • Ungrouped Data
  • Grouped Data

Calculating Arithmetic Mean for Ungrouped Data

For ungrouped data, the arithmetic mean is easily calculated using the formula,

Mean (x̄) = Sum of All Observations / Number of Observations

We can understand this with the help of the example discussed below,

Example: Find the mean of the first 5 even numbers.

Solution: 

First 5 even numbers are: 0, 2, 4, 6, 8

x̄ = (0+2+4+6+8) / 5 
   = 20/5
   = 4

Thus, the arithmetic mean of first five even numbers is 4.

Calculating Arithmetic Mean for Grouped Data

The grouped data is the data given as the continuous interval, i.e. in grouped data the class interval is given along with the frequency of each class. There are three different methods which are used to find the arithmetic mean for grouped data, they are

  • Direct Method
  • Short-Cut Method
  • Step-Deviation Method

We can use any of the three methods for finding the arithmetic mean for grouped data depending on the value of frequency and the mid-terms of the interval. Now let’s discuss the three methods for finding the arithmetic mean for grouped data in detail.

Direct Method for Finding the Arithmetic Mean

We can easily find the arithmetic mean using the direct method as,

Let we have to find the mean of n observation say x₁, x₂, x₃ ……xₙ, and their frequency is f₁, f₂, f₃ ……fₙ respectively. Then the formula for arithmetic mean is,

x̄ = (x₁f₁+x₂f₂+……+xₙfₙ) / ∑fi

where 
x̄ is the arithmetic mean
f₁+ f₂ + ….fₙ = ∑fi indicates the sum of all frequencies

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