Mean of Grouped Data: Concept, Formula, Methods & Solved Examples

Mean is the simplest arithmetic average and a part of statistical mathematics. Mean of Grouped Data in simple terms means the mean of data that is grouped together in sets, groups or categories. Usually, the mean or average of a set of different values of a variable is equal to the sum of the values of all the observations divided by the total number of observations.

In this maths article, we will learn about the concept of the Mean of grouped data in brief, its different types and methods of calculation and solved examples.

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Mean of Grouped Data

The mean or the average we know is the sum of values of all observations divided by the total number of observations made.

If X1, X2, X3, X4, X5…..Xn are the number of observations for frequencies F1, F2, F3, F4, F5….Fn, then, the mean of the data is given by,

We can also represent it as,

which, more briefly is written as

We can use this formula to find the mean of grouped data.

The mean of Grouped data is known as the sum of observations divided by the total number of observations.

There are two methods to calculate the mean, one is for grouped data and the other is for ungrouped data.

In this article we are studying the mean of grouped data,

The formula for mean of grouped data is

where ,

is the mean value of the set of given data.

f = sum of the observations made

N = total number of observations

Hence, it is the average of all data points.

Different Methods for Calculating Mean of Grouped Data

Let us understand the different methods for finding the mean of grouped data.

There are three different methods:

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

Direct Method

Direct Method is one of the simpler methods in finding the mean of grouped data.

In this method, the sum of all observations made is divided by the total number of observations.

If,

X1, X2, X3….XN are the observations made and “N” are the total number of observations, then,

By direct method,

Let us understand the concept of finding the mean of grouped data by direct method, by an example.

Example 1. The marks obtained by 26 students of class 10th of a certain school in English paper consisting of 100 marks are presented in the table below. Find the mean marks obtained by the students.

Marks obtained	10	20	36	40	50	56	60	70	72	80	88
Number of students	1	1	3	4	3	2	4	4	1	1	2

Solution 1

According to the given data,

Marks obtained	Number of students
10	1	10
20	1	20
36	3	108
40	4	160
50	3	150
56	2	112
60	4	240
70	4	280
72	1	72
80	1	80
88	2	176
Total

By direct method formula,

Therefore the mean marks obtained is 54.15.

Assumed Mean Method

To calculate the estimated mean for a grouped data is known as the assumed mean method.

Sometimes when the numerical values of and are large, it becomes difficult to find each of its products and calculate the mean by direct method.

We can’t do anything about the observations made, i.e., but we can change each number of the observations, i.e., to a smaller number.

Let “a” be the assumed mean amongst a set of observations and the difference “d”.

Hence we get the assumed mean method formula as,

Let us understand the above concept with an example

Referring to the same example,

We will use the Assumed Mean Method to find the mean.

To find class mark, use the formula

Example: Find the mean of the following data using the Assumed Mean Method.

Class Interval	Frequency
0 - 20	8
20 - 40	12
40 - 60	15
60 - 80	20
80 - 100	10

Solution:
First, we calculate the midpoints of each class interval. To find the midpoint, we take the average of the lower and upper class limits.

So,

• Midpoint for 0 - 20 = (0 + 20)/2 = 10
• Midpoint for 20 - 40 = (20 + 40)/2 = 30
• Midpoint for 40 - 60 = (40 + 60)/2 = 50
• Midpoint for 60 - 80 = (60 + 80)/2 = 70
• Midpoint for 80 - 100 = (80 + 100)/2 = 90

Let’s take 50 as the assumed mean (A = 50).
Now we calculate the deviation for each class as 

di=xi−A, where xi is the midpoint.

Then we calculate fi×di for each class.

Class Interval	Frequency (f₁)	Midpoint (xᵢ)	Deviation dᵢ = xᵢ - A	fᵢ × dᵢ
0 - 20	8	10	-40	-320
20 - 40	12	30	-20	-240
40 - 60	15	50	0	0
60 - 80	20	70	20	400
80 - 100	10	90	40	400
Total	65	-	-	240

Now, using the Assumed Mean Formula:

Mean = A + (∑ fᵢdᵢ) / (∑ fᵢ)

Mean = 50 + 240 / 65

Mean = 50 + 3.69 = 53.69

Therefore, the estimated mean of the data is 53.69.

Step Deviation Method

Step deviation method is used when the deviations of the made observations are large and they all have a common factor.

We can derive the step deviation method from the direct and assumed mean methods.

The formula for step deviation method is

Where a is the assumed mean

h is the size of the observation made,

For example 10-20, 40-50, 90-100 are the class observations, then,

20-10 = 50-40 = 100-90 = 10 = h.

Let us understand the concept of Step Deviation Method from the following example,

Example. Find the mean of the following data

Class Interval	50 – 70	70 – 90	90 – 110	110 – 130	130 – 150	150 – 170
Frequency	15	10	20	22	16	17

Class Interval
50-70	15	60	-2	-30
70-90	10	80	-1	-10
90-110	20	100	0	0
110-130	22	120	1	22
130-150	16	140	2	32
150-170	17	160	3	51
Total	100	–	–	= 65

h = 20

Using the formula for step deviation method

= 113.

Therefore the mean of the data is 113.

Properties of Mean of Grouped Data

1. Uniqueness:
   The mean is a unique value — there is only one mean for a given data set.
2. Affected by Extreme Values:
   The mean is sensitive to very high or very low values (outliers) in the data.
3. Based on All Observations:
   The mean is calculated using all data values and their frequencies, so it gives a true average.
    
4. Mathematical Simplicity:
   It is easy to calculate and apply algebraically, which makes it useful in further statistical analysis.
5. Balancing Point:
   The mean can be thought of as a point of balance — it is the center of mass of the data distribution.
6. Can Be Misleading for Skewed Data:
   In a highly skewed distribution, the mean might not represent the "typical" value.

Applications of Mean of Grouped Data

1. Education:
   Used to find the average marks of students in exams or subjects when data is provided in intervals.
2. Economics:
   Helps in finding average income, expenditure, or production over time from grouped economic data.
3. Business & Marketing:
   Used to calculate average sales, customer footfall, or revenue in grouped formats for decision-making.
4. Healthcare:
   Average age, weight, or blood pressure readings of patients grouped in intervals can be analyzed using mean.
5. Quality Control:
   Industries use the mean of grouped data to monitor average product measurements and detect issues in manufacturing.
6. Surveys and Research:
   Commonly used in social or market research surveys where responses are grouped into class intervals.

Examples on Mean of Grouped Data

Q1. The distribution below shows the number of marks taken by a student on a weekend test. Find the mean number of marks by direct method.

Mark	20	60	100	150	250	350
Number of students	7	5	16	12	2	3

Solution 1.

According to the given data,

Marks	Number of students
20	7	140
60	5	300
100	16	1600
150	12	1800
250	2	500
350	3	1050
–

By direct method formula,

= 119.77

The mean number of marks is 119.77

Example 2. The distribution below shows the number of wickets taken by bowlers on one day cricket matches. Find the mean number of wickets by choosing a suitable method.

Solution 2.

Number of wickets	20-60	60-100	100-150	150-250	250-350	350-450
Number of bowlers	7	5	16	12	2	3

The class size varies and the value of are large.

Let us find the mean by step deviation method.

Here a = 200 and h = 20

Number of wickets taken	Number of bowlers
20-60	7	40	-160	-8	-56
60-100	5	80	-120	-6	-30
100-150	16	125	-75	-3.75	-60
150-250	12	200	0	0	0
250-350	2	300	100	5	10
350-450	3	400	200	10	30
Total	45	–	–	–	-106

therefore ,

200-47.11 = 152.89

On an average the number of wickets taken by 45 bowlers in one day cricket match is 152.89

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