Mean

Mean is generally the average of a given set of numbers. It is the most important topic in statistics. It is the value that gives the central tendency from a given set of observations. In statistics, three measures define the central values of the observed data they are mean, median, and mode. The mean is usually calculated by dividing the sum of all the numbers and the number of observations in the set. Mean is used for many general purposes like to calculate the average mark of students in all subjects or to calculate the average salary of an employee in a company. There are three different types of means Arithmetic mean, Harmonic mean, and Geometric mean.

What is Mean?

Mean is the central tendency of the distributed data, which refers to the average value of the given set of data. The mean corresponds to the single value from the distributed data and represents an appropriate way to explain the data. This is the method most commonly used in statistics to understand and analyze the data. Mean, mode, and median are the three statistical measures of central tendency. For example, if we want to calculate the average salary of employees present in a company can be calculated by using the mean by dividing the sum of all employee salaries and the number of employees.

For discrete distribution kind of data, we need to calculate the mean by taking the sum of weighted value, which is calculated by taking the product of the random variable and the probability of the random variable.

Mean Definition

Mean is defined as the average of the given set of values and is used to measure of the central tendency. Central tendency is the measure that acknowledges the set of data or the distribution from a value. Therefore, it can be said that mean can provide the description of the entire data. In statistics, the mean can be obtained by dividing the sum of observations by the total number of observations. Let’s say the dataset given is X = x₁, x₂, x₃, …x_n. The mean of this dataset is denoted as . This mean is the average or arithmetic mean of all the n terms of data.

Mean Symbol: Mean is denoted as a bar on X or .

Mean Example

There are many uses and examples of mean in real-life. Following are some of the real-life examples of mean:

• The average (mean) marks obtained by the students in a class.
• Mean of the runs scored by a cricketer in the game.
• Average of the salaries of employees in an organization.

Mean Formula

Mean formula in statistics is defined as the sum of all the observations in the given dataset divided by the total number of observations. Furthermore, the mean formula can be divided and explained in two ways, they are – mean formula for grouped data and the mean formula for ungrouped data.

Mean Formula for Ungrouped data

Let’s say there is a set of given observations. The mean formula for ungrouped data can be expressed as:

Mean = (Sum of Observations) ÷ (Total number of observations)

Mean Formula for Grouped data

Similarly, for grouped data as well, mean formula is present, and the formula is expressed as:

Where,

•  = Mean of the given set of data
• f = Frequency of each class
• x = Mid-interval value of each class

Therefore, the average of the given dataset is called the mean of the data.

Example: Calculate the mean of the first 10 natural numbers.

Solution:

First 10 natural numbers = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Sum of first 10 natural numbers = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)

Mean () = Sum of 10 natural numbers/10

Mean () = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)/10

Mean () = 55/10

Mean () = 5.5

How to Find Mean?

To find the mean value, we use the mean formula, which calculates the average of the given data. It is calculated by dividing the sum of all observed data values by a number of observed values. The mean formula is given by

Mean = (Sum of observed values in data)/(Total number of observed values in data)

There are two steps involved in the calculation of the mean:

Step 1: Calculate the sum of observed values in the data.

Step 2: Divide the sum of observed values into the number of observed values in the data.

Based on the type of dataset given, we can find out the mean using different methods. Let’s take a look at the different cases to find their mean:

Case 1: If there are ‘n’ number of items in a list. The data is {x₁, x₂, x₃, … x_n}. Mean is calculated using the formula:

Or

Case 2: Let’s assume there are n number of items in a set {x₁, x₂, x₃, … x_n}, and the frequency of each item is given as {f1, f2, f3, …fn}. Then, the mean is calculated using the formula:

Or

Case 3: When items of the set are given in the form of a range, for example, 1-10, 10-20, etc. First class mark is required. Then, the mean is calculated using the given formula:

Mean of Ungrouped Data

The mean of ungrouped data is the sum of all the observations divided by the total number of observations. Ungrouped data is known as raw data, where the dataset simply contains all the data in no particular manner. Following are the steps that are to be followed in order to find the mean of ungrouped data:

• Note down the entire dataset for which the mean is to be calculated.
• Now, apply any of the two formulas based on the observation of data:

Or 

Example: Calculate the mean for the following set of data 2, 6, 7, 9, 15, 11, 13, 12.

Solution:

Given,

Observed values 2, 6, 7, 9, 15, 11, 13, 12

Total number of observed values = 8

Since, we know that 

Mean = (Sum of observed values in data) / (Total number of observed values in data)

Sum of observed values = 2 + 6 + 7 + 9 + 15 + 11 + 13 + 12 = 75

Total number of observed values = 8

Mean = 75/8

Mean = 9.375 

Therefore, mean for the given observed values = 9.375

Mean of Grouped Data

Grouped data is the set of data that is obtained by forming individual observations of variables into groups. Grouped data is divided into groups. A frequency distribution table is required for the grouped data, which helps in showcasing the frequencies of the given data. Mean of grouped data can be obtained using majorly three methods. The methods are:

• Direct Method
• Assumed Mean Method
• Step Deviation Method

Calculating Mean Using Direct Method

The simplest method used to find the mean of grouped data is the direct method. Following are the steps that can be used to find the mean of grouped data using direct method:

• Four columns are created in the table. The columns are Class interval, class marks (x_i), frequencies (f_i), the product of frequencies, and class marks (f_i x_i).
• Now, calculate the mean of grouped data using the formula: 

Calculating Mean Using Assumed Mean Method

There are cases when calculating mean using the direct method becomes tedious; then we calculate the mean using the assumed mean method. The following steps are to be followed to find the mean using the assumed mean method:

• Five columns are created in the table. They are class interval, class marks (x_i), corresponding deviations (d_i = x_i – A) where A is the central value from class marks as assumed mean, frequencies (f_i), mean of di using formula: di =
• Now, calculate the mean by adding the assumed mean (A) and mean of di.

Calculating Mean Using Step Deviation Method

Step deviation method is also famously known as the scale method or shift of origin method. When finding the mean of grouped data becomes tedious, using step deviation method can be used. Following are the steps that should be followed while using the step deviation method:

• Five columns are created in the table. They are class interval, class marks (x_i, here the central value is A), deviations (d_i), u_i = d_i/h (h is class width).
• Find the mean of u_i using 
• Finally, calculate the mean by adding A with the product of h and u_i.

Types of Mean

In statistics, there are four types of mean, and they are weighted mean, arithmetic mean (AM), geometric mean (GM), and Harmonic mean (HM). When not specified, mean is generally referred to as the arithmetic mean. Let’s take a look into all the types of mean: 

Arithmetic Mean

The arithmetic mean is calculated for a given set of data by calculating the ratio of the sum of all observed values to the total number of observed values. When the specification of mean is not given, it is presumed that the mean is an arithmetic mean. The general formula for arithmetic mean is given as:

Arithmetic Mean = (Sum of observed values)/(Number of observed values in data)

Or

Where,

•  = Arithmetic mean
• F_i = Frequency of each data
• N = Number of frequencies.

Geometric Mean

The geometric mean is calculated for a set of n values by calculating the nth root of the product of all n observed values. It is defined as the nth root of the product of n numbers in the dataset. The formula for the geometric mean is given as:

Geometric Mean = nth root of (x₁ × x₂ × x₃ × x₄ …. n values)

Or

GM =

Harmonic Mean

The harmonic mean is calculated by dividing the number of values in the observed set by the sum of reciprocals of each observed data value. Therefore, the harmonic mean can also be called the reciprocal of the arithmetic mean. The formula for harmonic mean is given as:

Harmonic Mean = (Number of observed values) / (1/n₁ + 1/n₂ + 1/n₃ …)

Or

HM = 1/

Weighted Mean

Weighted mean is calculated in certain cases of the dataset when the given set of data has some values more important than the other. In the dataset, a weight ‘w_i‘ is connected to each data ‘x_i‘, and the general formula for weighted mean is given as:

Weighted Mean = 

Where,

•  = mean of the given set of data.
• w_i = Weight of each observation.

Solved Examples on Mean

Example 1: Calculate the mean of the first 5 even natural numbers.

Solution: 

Given,

Observed first 5 even natural numbers 2, 4, 6, 8, 10.

Total number of observed values = 5

Mean = (Sum of observed values in data)/(Total number of observed values in data)

Sum of observed values = 2 + 4 + 6 + 8 + 10 = 30

Total number of observed values = 5

Mean = 30/5

Mean = 6 

Therefore, mean for first 5 even numbers = 6

Example 2: Calculate the mean of the first 10 natural odd numbers.

Solution: 

Given,

Observed first 5 odd natural numbers 1, 3, 5, 7, 9.

Total number of observed values = 5

Mean = (Sum of observed values in data)/(Total number of observed values in data)

Sum of observed values = 1 + 3 + 5 + 7 + 9 = 25

Total number of observed values = 5

Mean = 25 / 5

Mean = 5

Therefore, mean for first 5 odd numbers = 5

Example 3: Calculate missing values from the observed set 2, 6, 7, x, whose mean is 6.

Solution:

Given,

Observed values 2, 6, 7, x

Number of observed values = 4

Mean = 6

Since, we know that 

Mean = (Sum of observed values in data)/(Total number of observed values in data)

Sum of observed values = 2 + 6 + 7 + x = 15 + x

Total number of observed values = 4

6 = (15 + x)/4

6 × 4 = 15 + x

x = 9

Therefore, missing value from the set is 9

Example 4: There are 20 students in class 10. The marks obtained by the students in mathematics (out of 100) are given below. Calculate the mean of the marks.

Marks Obtained	Number of students
100	1
92	3
80	5
75	10
70	1

Solution:

Total number of students in class 10 = 20

x₁ = 100, x₂ = 92, x₃ = 80, x₄ = 75, x₅ = 70

f₁ = 1, f₂ = 3, f₃ = 5, f₄ = 10, f₅ = 1

Using the mean formula:

 = {(100 × 1) + (92 × 3) + (80 × 5) + (75 × 10) + (70 × 1)}/20

 = (100 + 276 + 400 + 750 + 70)/20 = 1596/20 = 79.8 marks

FAQs on Mean

Question 1: What is mean in statistics?

Answer:

Mean in statistics is one of the important measures of central tendency, it is defined as the average of the set of values given.

Question 2: What is mean formula?

Answer:

There area different mean formulas based on the type of data present. 

For ungrouped data: 

Mean = (Sum of Observations) ÷ (Total number of observations)

For grouped data:

Where,

•  = Mean of the given set of data
• f = Frequency of each class
• x = Mid-interval value of each class

Question 3: What are the different types of mean?

Answer:

There are four different types of mean. They are:

• Arithmetic mean
• Geometric mean
• Weighted mean
• Harmonic mean

Question 4: What are the applications of mean in real life?

Answer:

There are many applications of mean in real-life. For example, in schools, the mean of marks in subjects are taken by teachers, the HR managers often find out the mean of salaries of employees, etc.

Question 5: What is the mean formula for ‘n’ number of observations?

Answer:

The formula for ‘n’ number of observations is:

Mean = Sum of n observations/n