Methods of Finding Arithmetic Mean for Different Types of Data
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  • Written By Preethu
  • Last Modified 18-05-2022
  • Written By Preethu
  • Last Modified 18-05-2022

Methods of Finding Arithmetic Mean for Different Types of Data

Methods of Finding Arithmetic Mean for Different Types of Data: A measure of central tendency is a numerical method to explain or summarise the data in brief. Mean, median, and mode are measures of central tendency. The mean is defined as the average of a set of numbers and is calculated by adding all the numbers in a given data set and dividing by the total number of observations.

Arithmetic mean is used in many real-life instances like finding the average rainfall in a place, the average monthly family income, average marks, etc. There are different ways to find the arithmetic mean like the direct method, assumed mean method and the step deviation method.

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What is Arithmetic Mean?

In everyday life, we can see many examples of summarising a large set of data, such as average marks obtained by the students in an exam, average rainfall in a locality, average production in a factory, and the average income of people living in a neighbourhood.

The arithmetic mean is also called the mean or arithmetic average. Average is calculated by adding all the numbers in a given data set and dividing by the total number of observations. For evenly distributed numbers, the arithmetic mean (AM) is the middle number. Furthermore, AM is calculated using different methods based on the amount of data and the distribution.

Formula for Arithmetic Mean

In general, if there are \(N\) observations as \({X_1},\;{X_2},\;{X_3},\;…,\;{X_N}\), then the arithmetic mean is given by

\(\bar X = \frac{{{X_1} + {X_2} + \;{X_3} + \;… + \;{X_N}}}{N}\)

This can also be expressed as

\(\bar X = \frac{{\mathop \sum \nolimits_{i = 1}^N {X_i}}}{N}\)

Here, \(X_i\) represents each observation and \(\sum\) denotes the summation of all the observations when \(i\) takes a value from \(1\) to \(N\).

This formula can also be simplified as,

\(\bar X = \frac{{\sum X}}{N}\)

Where \(\sum X\) denotes the sum of all observations, and \(N\) denotes the number of observations.

Properties of Arithmetic Mean

Two interesting properties of arithmetic mean are given below:

  • The sum of deviations of observations about arithmetic mean is always equal to \(0\). Symbolically, \(\sum \left( {X – \bar X} \right) = 0\).
  • The arithmetic mean is affected by the outliers or extreme values. Any value much smaller or much larger in the data set can affect the mean.

How is Arithmetic Mean Calculated?

The calculation of arithmetic mean can be done under two categories:

Arithmetic mean calculation

Arithmetic Mean for Ungrouped Data

The arithmetic mean for ungrouped data can be calculated through three methods:

calculation of three methods

Arithmetic Mean by Direct Method

In the direct method, the arithmetic mean can be calculated by the adding all observations in the data set and dividing it by the total number of observations. This is denoted as

\(\bar X = \frac{{\sum X}}{N}\)

Where \(\sum X\) denotes the sum of all observations, and \(N\) denotes the number of observations.

Arithmetic Mean by Assumed Mean Method

There can be instances where the number of observations in the data set is more and figures in the data set are large. In these cases, it is difficult to calculate the arithmetic mean by the direct method. Hence, we use another method called the assumed mean method.

In order to save time in calculating the mean of a data set with a large number of observations and large numerical figures, you can use the assumed mean method.

Step 1: On the basis of logic/experience, assume a specific figure in the data as the arithmetic mean.
Step 2: For each observation, compute the deviations from the assumed mean.
Step 3: The sum of these deviations can then be divided by the number of observations in the data.
Step 4: The actual arithmetic mean is calculated by adding the assumed mean and the deviation ratio to the number of observations.

Formula

Let, \(A =\) assumed mean
\(X =\) individual observations
\(N =\) total numbers of observations
\(d =\) deviation of assumed mean from individual observation
\(d = X – A\)
Then, the sum of all deviations, \(\sum d = \sum \left( {X – A} \right)\)
Then find \(\frac{{\sum d}}{N}\).
Then add \(A\) and \(\frac{{\sum d}}{N}\) to get \(\bar X\)
Hence, arithmetic mean, \(\bar X = A + \frac{{\sum d}}{N}\)
Whether existing in the data set or not, any value can be taken as the assumed mean. However, to simplify the calculation, a centrally located value in the data must be selected as the assumed mean.

Arithmetic Mean by Step Deviation Method

The calculations to find the arithmetic mean can be simplified by dividing all the deviations taken from the assumed mean by a common factor \(c\). The objective here is to avoid large numerical figures.

That is, if \(d = X – A\) is very large, then we will find \(d’\).

\(d’ = \frac{d}{c}\)

\(\therefore \,d’ = \frac{{\bar X – A}}{c}\)

Hence, the arithmetic mean, \(\bar X = A + \frac{{\sum d’}}{N} \times c\)

Calculation of Arithmetic Mean for Grouped Data

The arithmetic mean calculation for grouped data is different from the ungrouped data. The ungrouped data can be discrete or continuous. Different strategies are followed for discrete and continuous data to calculate the mean.

Arithmetic Mean of Discrete Series

As discussed for ungrouped data, the arithmetic mean of discrete data can be calculated in three ways.
The arithmetic mean for grouped data can be calculated through three methods:

  1. Direct Method
  2. Assumed Mean Method
  3. Step Deviation Method

Arithmetic Mean by Direct Method

In the case of discrete series, the observations will be different, but there can be frequency \(f\), for the observations.

Arithmetic mean, \({{\bar X}} = \frac{{\sum {{fX}}}}{{\sum {{f}}}}\)

Steps to calculate the arithmetic mean are described below.

Step 1: Multiply the frequency against each observation by the value of the observation to get \(fX\).
Step 2: Calculate the sum of all the values from Step 1 to get \(\sum fX\)
Step 3: Divide the sum by the total number of frequencies \(\sum f\) to get the mean.

Assumed Mean Method

As in the case of ungrouped data, the calculations can be simplified using the assumed mean method with a simple modification.

Arithmetic mean, \({{\bar X}} = A + \frac{{\sum {{fd}}}}{{\sum {{f}}}}\)

Step 1:  Assume a mean for the given data, \(A\).
Step 2: Calculate the deviation, \(d\), for each data from the assumed mean, \(A\).
Step 3: Multiply each deviation \((d)\) by the frequency to get \(fd\).
Step 4: Calculate the sum of all the values from Step 1 to get \(\sum fd\).
Step 5: Add all the values of \(f\) to get \(\sum f\).
Step 6: Divide \(\sum fd\) by \(\sum f\).
Step 7: Add the quotient to the assumed mean to get the actual mean of the data.

Step Deviation Method

In the step deviation method, the deviations are divided by the common factor \(c\) to simplify the calculation:

Calculate \(d’ = \frac{d}{c} = \frac{{X – A}}{c}\) to ease the calculations. Then get \(fd’\) and \(\sum fd’\).

Hence, the arithmetic mean \(\bar X = A + \frac{{\sum fd’}}{{\sum f}} \times c\)

Arithmetic Mean of Continuous Series

The observations are provided in class intervals when a data set is continuous. Calculating the arithmetic mean for a continuous series is the same as that of a discrete series. The only difference here is that the mid-points of the class intervals are considered here instead of each observation.

We know that the class intervals may be exclusive, inclusive, or of unequal size.

  • An example of an exclusive class interval is, say, \(0 – 10,\,10 – 20\) and so on.
  • An example of an inclusive class interval is, say, \(0 – 9,\,10 – 19\) and so on.
  • An example of an unequal class interval is, say, \(0 – 20,\,20 – 50\) and so on.

In all these cases, the arithmetic mean calculation is done similarly as discrete series, but mid-points of class intervals are considered.

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Solved Examples – Methods of Finding Arithmetic Mean for Different Types of Data

Check below the solved examples for Methods of Finding Arithmetic Mean for Different Types of Data:

Q.1. Compute the arithmetic mean of the first \(6\) odd, whole numbers.
Ans:
The first \(6\) odd, whole numbers are \(1,\;3,\;5,\;7,\;9,\;11\)
Arithmetic mean, \(\bar X = \frac{{\sum X}}{N}\)
\(\Rightarrow \bar X = \frac{{\left( {1 + 3 + 5 + 7 + 9 + 11} \right)}}{6}\)
\(\Rightarrow \bar X = \frac{{36}}{6}\)
\(\therefore \,\bar X = 6\)

Q.2. The data set below represents the number of books sold from a shop from \(7\) different days: \(7,\;9,\;12,\;15,\;5,\;4,\;11\). Find the mean of the sold number of books.
Ans:
The data is ungrouped and simple.
Arithmetic mean, \(\bar X = \frac{{\sum X}}{N}\)
\(\Rightarrow \bar X = \frac{{\left( {7 + 9 + 12 + 15 + 5 + 4 + 11} \right)}}{7}\)
\(\Rightarrow \bar X = \frac{{63}}{7}\)
\(\therefore \,\bar X = 9\)

Q.3. A proofreader read through \(73\) pages of the report. The number of mistakes he found on each of the pages is given in the table below. Find the mean of the number of mistakes found per page.

Number of mistakes\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)
Number of pages\(5\)\(9\)\(12\)\(17\)\(14\)\(10\)\(6\)

Ans:
The data is grouped and discrete. Let us calculate the mean using the assumed mean method.

\(x_i\)\(f_i\)\(d_i = x_i – A\)\(f_i d_i\)
\(1\)\(5\)\(-3\)\(-15\)
\(2\)\(9\)\(-2\)\(-18\)
\(3\)\(12\)\(-1\)\(-12\)
\(4\)\(17\)\(0\)\(0\)
\(5\)\(14\)\(1\)\(14\)
\(6\)\(10\)\(2\)\(20\)
\(7\)\(6\)\(3\)\(18\)
\(\sum f = 73\)\(\sum fd = 7\)

Now, the arithmetic mean \(\bar X = A + \frac{{\sum fd}}{{\sum f}}\)
\(= 4 + \frac{{73}}{7}\)
\(\therefore \,\bar X = 4.09\)

Q.4. Find the arithmetic mean of the data given in the table below using the step-deviation method.

Class Intervals\(0 – 10\)\(10 – 20\)\(20 – 30\)\(30 – 40\)\(40 – 50\)\(50 – 60\)\(60 – 70\)Total
Frequency\(4\)\(4\)\(7\)\(10\)\(12\)\(8\)\(5\)\(50\)

Ans: The given data is grouped and continuous. The arithmetic mean is asked to calculate by the step deviation method.
\(\bar X = A + \frac{{\sum fd’}}{{\sum f}} \times c\)
Let us consider \(A = 35\)
Class width \(= 10\)
Let us find the rest of the values by extending the table.

Class Interval\(x_i\)\(f_i\)\(d_i = \frac {x_i – A}{c}\)\(f_i d_i\)
\(0 – 10\)\(5\)\(4\)\(-3\)\(4 \times \left( { – 3} \right) = – 12\)
\(10 – 20\)\(15\)\(4\)\(-2\)\(4 \times \left( { – 2} \right) = – 8\)
\(20 – 30\)\(25\)\(7\)\(-1\)\(7 \times \left( { – 1} \right) = – 7\)
\(30 – 40\)\(35\)\(10\)\(0\)\(10 \times 0 = 0\)
\(40 – 50\)\(45\)\(12\)\(1\)\(12 \times 1 = 12\)
\(50 – 60\)\(55\)\(8\)\(2\)\(8 \times 2 = 16\)
\(60 – 70\)\(65\)\(5\)\(3\)\(5 \times 3 = 15\)
Total\(\sum f_i = 50\)\(\sum f_i d_i = 16\)

Hence, arithmetic mean \(\bar X = A + \frac{{\sum fd’}}{{\sum f}} \times c\)
\(= 35 + \frac{{16}}{{50}} \times 10\)
\(= 35 + 3.2\)
\(\therefore \,\bar X = 38.2\)

Q.5. In a class of \(30\) students, the marks obtained by students in English out of \(50\) are tabulated below. Calculate the arithmetic mean of the data.

Marks ScoredNumber of Students
\(10 – 20\)\(5\)
\(20 – 30\)\(5\)
\(30 – 40\)\(8\)
\(40 – 50\)\(12\)

Ans:
The given data is continuous. Hence, we can calculate the mean using the direct method.

Marks ScoredNumber of StudentsClass Mark\(f_i x_i\)
\(10 – 20\)\(5\)\(15\)\(75\)
\(20 – 30\)\(5\)\(25\)\(125\)
\(30 – 40\)\(8\)\(35\)\(280\)
\(40 – 50\)\(12\)\(45\)\(540\)
Total\(\sum f_i = 30\)\(\sum f_i x_i = 1020\)

The arithmetic mean using the direct method is
\(\bar X = \frac{{\sum {f_i}{X_i}}}{{\sum f}}\)
\(= \frac{{1020}}{{30}}\)
\(\therefore \,\bar X = 34\)

Summary

The arithmetic mean is also called the mean or arithmetic average. It is calculated by adding all the numbers in a given data set and dividing the sum by the total number of observations. The arithmetic mean is calculated using different methods based on the amount of data and the distribution.

The calculation is classified under two categories: Ungrouped data and grouped data. The calculation for each type is different. The arithmetic mean can be calculated using three methods: Direct method, assumed mean method and step deviation method.

To save time in calculating the mean of a data set with a large number of observations and large numerical figures, we can use the assumed mean method. It can be further simplified using the step deviation method.

Frequently Asked Questions (FAQs)

The frequently asked questions on methods of finding arithmetic mean for different types of data are given below:

Q.1. What are the different methods of calculating arithmetic mean?
Ans: There are three methods to calculate the arithmetic mean:
• Direct method
• Assumed mean method
• Step-deviation method

Q.2. Why step deviation method is used to calculate the arithmetic mean?
Ans: The calculations to find the arithmetic mean can be simplified by dividing all the deviations taken from the assumed mean by a common factor \(c\). The objective here is to avoid large numerical figures.

Q.3. Which data is suitable for the arithmetic mean?
Ans: Arithmetic mean can be calculated for grouped or ungrouped data. In grouped data, both discrete and continuous data can have a mean.

Q.4. How to calculate Arithmetic Mean?
Ans: Arithmetic mean is the ratio of the sum of the values of all observations and the number of observations. It is denoted as \(\bar X\).
In general, if there are \(N\) observations as \({X_1},\;{X_2},\;{X_3},\;…,\;{X_N}\), then
\(\bar X = \frac{{{X_1} + {X_2} + \;{X_3} + \;… + \;{X_N}}}{N}\)

Q.5. How to find the arithmetic mean between 2 numbers?
Ans: If \(x_1\) and \(x_2\) are two numbers, then the mean between them is \(\bar X = \frac{{{x_1} + {x_2}}}{2}\).

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