Mean of Grouped Data

The mean of grouped data is the average value of the data organized into classes or groups. Such data, arranged in groups or categories, is called grouped data, in which each class interval shows a range of values, and the frequency indicates how many observations fall within that range. The mean, also known as the central value, helps us understand the overall trend of the data. For grouped data, we use the midpoint of each class to calculate the mean. 

 

In statistics, the mean is the sum of all observations divided by the total number of observations. The mean is denoted as \(\bar x\) (x bar). That is:
 

\( x = x_1 + x_2 + x_3 + …. + x_n.\)

The mean of grouped data is calculated using the midpoint (class mark) of each class interval along with its frequency. The central value of its interval is known as the midpoint. By multiplying these midpoints by their respective frequencies and dividing the total by the overall frequency. The formula for the mean of grouped data is: 
 

\({\bar x} = {{∑{f_i} {x_i} \over N }}\)
 

Where,

 
\(\bar x\) is the mean value of the set of given data


\(f_i\) is the frequency of the individual data


\(x_i\) is the midpoint of each class interval 


N is the sum of the frequencies
 

The mean of grouped data is a measure of central tendency that represents the average value of a dataset organized into classes. The different methods to calculate the mean of grouped data are: 
 

• Direct Method
   
• Assumed Mean Method
   
• Step Deviation Method
  
   

Direct Method

The direct method is the simplest method used to calculate the mean of grouped data. If the values of the observations are \(x_1, x_2,..., x_n\), with their corresponding frequencies being \(f_1, f_2,..., f_n\). Then the mean of the data is given by,
 

 \(x = \frac{x_1f_1 + x_2f_2 + \cdots + x_n f_n}{f_1 + f_2 + \cdots + f_n} \)

 

    \(x = \frac{\Sigma (f \times x)}{\Sigma f} \), where i = 1, 2, 3, 4,...n

 

To find the mean of grouped data, follow these steps: 
 

• Create a table and organize the data into class intervals with their frequencies. Each class interval represents the range of a group, and the midpoints xi represent the central value of the interval: 
  
  \(x_i = \frac{\text{lower limit} + \text{upper limit}}{2} \)
   
• Multiply each point xi by its frequency \(f_i\).
   
• Then substitute the values in the formula:  \({\bar x} = {{\frac{\sum x_i f_i}{\sum f_i} }}\)

 

For example, a class of students scored the following marks in a test. Find the mean marks. 

 

Mark	Number of Students
0-10	2
10-20	3
20-30	5
30-40	4
40-50	6

 

To find the midpoints (xi) of each class using the formula: \(x_i = \frac{\text{lower limit} + \text{upper limit}}{2} \)

 

Mark	Number of Students	Midpoint (\(x_i\))	\({{f_i} {x_i}}\)
0-10	2	\({{0 + 10}\over 2 } = 5\)	\(2 \times 5 = 10 \)
10-20	3	\({{10 + 20}\over 2 } = 15\)	\(3 \times 15 = 45\)
20-30	5	\({{20 + 30}\over 2 } = 25\)	\(5 \times 25 = 125 \)
30-40	4	\({{30 + 40}\over 2 } = 140\)	\(4 \times 35 = 140\)
40-50	6	\({{40 + 50}\over 2 } = 270\)	\( 6 \times 45 = 270\)

 

Sum the frequencies and sum the products: 
 

\(Σf_i = 2 + 3 + 5 + 4 + 6 = 20\)

 

\(Σf_ix_i = 10 + 45 + 125 + 140 + 270 = 590\)

 

Finding the mean of grouped data using the formula: \(\bar{x} = \frac{\sum f_i x_i}{N} \)


\({\bar x} ={{590 \over 20 }} \)


= 29.5

 

Assumed Mean Method

The assumed mean method is used to estimate the mean of a large dataset. In this method, a value from the dataset is chosen as an assumed mean (a), and the deviations of all midpoints from this assumed mean are calculated. 

Let the deviation of each midpoint from the assumed mean be: \(d_i = x_i - a\)

Then the mean is calculated as: 

  \(\bar{x} = \frac{\sum (a + d_i) f_i}{\sum f_i} \)

 

\(\bar{x} = a + \frac{\sum ( f_i d_i)}{\sum f_i}\)

Find the mean of the following grouped data using the assumed mean method. 

 

Class Interval	Frequency (f)
10-20	5
20-30	8
30-40	12
40-50	7
50-60	3

 

To find the midpoint, xi, we use the formula: 

\(x_i = \frac{\text{lower limit} + \text{upper limit}}{2} \)

 

Here, the midpoints are 15, 25, 35, 45, and 55

Let a = 35

Then deviations, di is calculated using the formula \(d_i = x_i - a\)

Class Interval	Frequency (f)	\(x_i\)	\({d_i} = {x_i} - a\)	\({f_i} \cdot {d_i}\)
10-20	5	15	-20	-100
20-30	8	25	-10	-80
30-40	12	35	0	0
40-50	7	45	10	70
50-60	3	55	20	60

 

Applying the assumed mean formula: \(\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \)
 

\({d_if_i} = (-100) + (-80) + 0 + 70 + 60 = -50\)
 

\(Σf_i = 5 + 8 + 12 + 7 + 3 = 35 \)

 

\( = {{35 + -50} \over { 35 }}\)

 

\(= 35 - 1.43\)

 

= 33.57

 

Step Deviation Method     

The step deviation method is used when the values in a data set are large and far from the assumed mean. Here, deviations are divided by the class size h to simplify calculations:     

\(\bar{x} = a + h \cdot \frac{\sum f_i u_i}{\sum f_i} \)
 

Where, 
 

a is the assumed mean
 

h is the class size
 

u_i = \(\frac{d_i}{h} \)

 

For example, find the mean of the following data using the step deviation method. 

 

Class Interval	Frequency
100-110	5
110-120	8
120-130	12
130-140	7
140-150	3

 

Finding the midpoints xi using:

\(x_i = \frac{\text{lower limit} + \text{upper limit}}{2} \)

 

Here, the midpoints are 105, 115, 125, 135, 145

Let the assumed mean: a = 125

Calculating deviation and step deviation:

 

\(d_i = x_i - a \)
 

\(u_i = \frac{d_i}{h} \), where h = 10
 

Class Interval	Frequency	\(x_i\)	\(d_i = x_i - a\)	\(u_i = {d_i \over h}\)	\({f_i u_i}\)
100-110	5	105	-20	-2	-10
110-120	8	115	-10	-1	-8
120-130	12	125	0	0	0
130-140	7	135	10	1	7
140-150	3	145	20	2	6

 

\(f_iu_i = -10 + -8 + 0 + 7 + 6 = -5 \)
 

\(f_i = 5 + 8 + 12 + 7 + 3 \)

Substituting the values in the equation: \(\bar{x} = a + h \cdot \frac{\sum f_i u_i}{\sum f_i} \)
 

\(\bar{x} = 125 + 10 \left( -\frac{5}{35} \right) \)

 

\(= {125 - 1.43}\)

 

\(\bar {x} = 123.57\)

Mean of Grouped data is a complex mathematical topic and therefore some tips and tricks can be helpful. In this section we discuss such tips and tricks.

 

• Check for equal class intervals: Make sure all class intervals are equal before applying short-cut or step-deviation methods. If not, use the direct method.
   
• Cross-verify the mean: After calculating, quickly check your results using both the direct and short-cut methods for accuracy.
   
• Practice with real data: Use practical examples (like marks, income, or temperature data) to strengthen your conceptual understanding.
   
• Use a calculator or spreadsheet for speed: When dealing with large grouped data, use tools like Excel or a calculator to speed up computation while focusing on the logic behind the steps.
   
• Choose an appropriate assumed mean (a): Select a class mark from the middle of the data as the assumed mean — it keeps deviations small and calculations easier.
   
• Encourage understanding: Parents can help students understand the formula and its logic rather than memorize it. 
   
• Provide step-by-step practice: Teachers can use structured tables for midpoints, frequencies, and deviations to help students visualize and organize calculations.

The mean of grouped data is used in many real-life situations where data is collected in ranges. Here are some important applications: 

 

• Education: Teachers use the mean of grouped data to evaluate students’ overall performance based on their scores in exams. 

 

• Business and economics: Companies use grouped data to analyze the average salary of employees across different departments.
   
• Healthcare: Hospitals analyze patient data, such as average blood pressure, weight, or cholesterol levels, using grouped data.
   
• Agriculture: Farmers use the mean of grouped data to determine the average crop yield from different plots of land or over several seasons.
   
• Weather analysis: Meteorologists calculate the mean temperature, rainfall, or humidity over grouped time intervals (like weeks or months) to study climate patterns.