Arithmetic Mean

Arithmetic Mean, commonly known as the average, is a fundamental measure of central tendency in statistics. It 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.

Arithmetic mean is used in various scenarios such as in finding the average marks obtained by the student , the average rainfall in any area, etc. The Arithmetic Mean provides a single value that represents the central point of the dataset, making it useful for comparing and summarizing data. The arithmetic mean takes into account every value in the dataset, offering a comprehensive overview of the data's overall behavior.

In this article we will discuss about Arithmetic Mean, its formula, properties and various methods to calculate the mean for grouped and Ungrouped Data

What is Arithmetic Mean?

Arithmetic Mean, often referred to simply as the mean or average, is a measure of central tendency used to summarize a set of numbers.

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.

Example. Find the Arithmetic mean of 3, 6, 7, and 4.

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.

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, ........n_n. Then the arithmetic mean is calculated as,

A.M. = (n_{1} + n_{2} + n_{3} + n_{4} + ... + n_{n})/n

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

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
a_i are set values.

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.

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.

(x₁−x̄)+(x₂−x̄)+(x₃−x̄)+...+(xₙ−x̄) = 0

• For Discrete Data Set, we can say that ∑(x_i − x̄) = 0
• For Grouped Frequency Distribution, we can say that ∑f(x_i − ∑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.

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.

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 for Mean
• Short-Cut Method
• Step-Deviation Method for Mean

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

Example: Find the mean of the following data.

Solution:

For mean,

xi	5	10	15	20	25
fi	5	2	2	3	4
fixi	25	20	30	60	100

∑f_i = 5+2+2+3+4 = 16

∑f_ix_i = 25+20+30+60+100 = 235

x̄ = (x₁f₁+x₂f₂+......+xₙfₙ) / ∑f_i

x̄ = 235/16 = 14.6875

Thus, the mean of the given data set is 14.6875

Short-cut Method for Finding the Arithmetic Mean

We can easily find the arithmetic mean using the shortcut method also called the assumed mean method by using the steps discussed below,

Step 1: Find the midpoint of each class interval say x_i

Step 2: Assumed a random number as the assumed mean say A.

Step 3: Find the deviation of each class interval midpoint as, (d_i) = x_i – A

Step 4: Use the formula for finding the arithmetic mean

x̄ = A + (∑f_id_i/∑f_i)

Example: Find the mean of the given data using the short-cut method.

Solution:

For Arithmetic Mean,

Let the assumed mean be 20 

Class Interval (CI)	xi	Frequency(fi)	di = (xi - A)	fidi
5-15	10	4	10 - 20 = -10	-40
15-25	20	12	20 - 20 = 0	0
25-35	30	8	30 - 20 = 10	80
35-45	40	6	40 - 20 = 20	120

∑f_i = 4+12+8+6 = 20

∑f_id_i = -40+0+80+120 = 160

Using the Formula,

x̄ = A + (∑f_id_i/∑f_i)

x̄ = 20 + 160/20

   = 20 + 8

   = 28

Thus, the Arithmetic mean is, 28

Step-Deviation Method for Finding the Arithmetic Mean

We can easily find the arithmetic mean using the step-deviation method also called the scale method by using the steps discussed below,

Step 1: Find the midpoint of each class interval say x_i

Step 2: Assumed a random number as the assumed mean say A.

Step 3: Find the u_i = (x_i-A)/h, where, h is the class interval.

Step 4: Use the formula for finding the arithmetic mean

x̄ = A + h(∑f_iu_i/∑f_i)

Example: Find the mean of the given data using the short-cut method.

Solution:

For Arithmetic Mean,

Let the assumed mean be 20 

The class interval is 10.

Class Interval (CI)	xi	Frequency(fi)	ui = (xi-A)/h	fiui
5-15	10	4	-1	-4
15-25	20	12	0	0
25-35	30	8	1	8
35-45	40	6	2	12

∑f_i = 4+12+8+6 = 20

∑f_iu_i = -4+0+8+12 = 16

Using the Formula,

x̄ = A + h(∑f_id_i/∑f_i)

x̄ = 20 + 10(16/20)

   = 20 + 8

   = 28

Thus, the Arithmetic mean is, 28

Advantages of Arithmetic Mean

Arithmetic mean is a widely used concept in mathematics. It is not only used in statistics and mathematics but also in various other fields such as economics, marketing, investments, and others. Some of the major advantages of the arithmetic mean are,

• The formula for arithmetic mean is a rigid formula and it does not change with the deviation in the values of the data set.
• Arithmetic mean considers all the values of the data set.
• It takes into consideration each value of the data set.
• The arithmetic mean formula is very easy to use.
• Other mathematical measures such as median, mode, etc are easily calculated using the arithmetic mean.
• It is used to find the various geometrical concepts such as midpoints, centroids, etc.

Disadvantages of Arithmetic Mean

There are also various disadvantages of using the arithmetic mean that include,

• Arithmetic mean gets easily affected by extreme values and thus changing the extreme values easily changes the arithmetic mean.
• The arithmetic mean can not be easily calculated if the data set is given as an open interval i.e., if the data set,

In the above-given data set finding the arithmetic mean is a difficult task as finding the midpoint of class interval less than 25 and more than 75 is very tough until we assume its starting and ending point.

• Finding arithmetic means using the graphical method is practically impossible.
• If the value of a single data set gets missing the mean of the data set changes drastically.
• Sensitivity to Extreme Values, The arithmetic mean can be affected by extremely high or low values, which can distort the average( a single value can change the mean drastically).

Read More,

• Mean Deviation Formula
• Mean, Median and Mode
• Statistics Formulas

Solved Examples on Arithmetic Mean

Example 1: Find the arithmetic mean of the first five prime numbers.

Solution:

Arithmetic mean of first five prime numbers,

First Five Prime Numbers = 2, 3, 5, 7 and 11

Number of observations (n) = 5

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

x̄  = (2 + 3 + 5 + 7 + 11)/5 = 28/5

x̄ = 5.6

Hence, the arithmetic mean of the first five prime numbers is 5.6.

Example 2: If the arithmetic mean of five observations 5, 6, 7, x, and 9 is 6. Find the value of x.

Solution:

Given observations are 5, 6, 7, x, and 9

Number of Observations = 5

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

6 = (5 + 6 + 7 + x + 9)/5

30 = 27 + x

x = 30 - 27

x = 3

Hence, the value of x is 3

Example 3: If the arithmetic mean of five observations 10, 20, 30, x, and 50 is 30. Find the value of x.

Solution:

Given, observations are 10, 20, 30, x and 50

Number of observations = 5

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

30 = (10 + 20 + 30 + x + 50)/5

150 = 110 + x

150 - 110 = x

x = 40

Hence, the value of x is 40

Read More,

• What is Mean in Statistics?
• Mean, Median and Mode
• Mean, Variance and Standard Deviation
• Geometric Mean Formula

Conclusion

Arithmetic Mean is a fundamental concept in mathematics, statistics, and various other fields. The Arithmetic Mean, also known as the average, is a measure of central tendency that provides a simple yet powerful way to summarize a set of numbers. By calculating the sum of all observations and dividing it by the number of observations, one can easily determine the average or mean value.

Arithmetic Mean remains a key tool in data analysis and problem-solving. As it provides a single value to represent the central point of the dataset, making it useful for comparing and summarizing data. This formula is widely applicable, whether dealing with ungrouped data or grouped data. Its simplicity and utility make it indispensable in fields such as economics, finance, and data analysis.