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Mean of Grouped Data
Mean of grouped data is the data set formed by aggregating individual observations of a variable into different groups. Grouped data is data that is grouped together in different categories. Mean is considered as the average of the data. For the mean of grouped data, it might be difficult to find the exact value however, we can always estimate it. Let us learn more about the mean of grouped data, the methods to find the mean of grouped data, and solve a few examples to understand this concept better.
1.  What is Mean of Grouped Data? 
2.  Mean of Grouped Data Formula 
3.  Methods of Calculating Mean of Grouped Data 
4.  FAQs on Mean of Grouped Data 
What is Mean of Grouped Data?
Mean of grouped data is the process of finding the average of a set of data that are grouped together in different categories. To determine the mean of a grouped data, a frequency table is required to set across the frequencies of the data which makes it simple to calculate. There are three main methods of calculating the mean of grouped data, they are  direct method, assumed mean method, and step deviation method. Each of these methods has its own formulas and ways to calculate the mean.
Definition of Mean
The mean is the average or a calculated central value of a set of numbers that is used to measure the central tendency of the data. Central tendency is the statistical measure that recognizes the entire set of data or distribution through a single value. In statistics, the mean can also be defined as the sum of all observations to the total number of observations. Given a data set, \( X = x_{1},x_{2}, . . . ,x_{n}\), the mean (or arithmetic mean, or average), denoted x̄, is the mean of the n values \(x_{1},x_{2}, . . . ,x_{n}\). The mean is represented as xbar, x̄.
Mean of Grouped Data Formula
The mean formula is defined as the sum of the observations divided by the total number of observations. There are two different formulas for calculating the mean for ungrouped data and the mean for grouped data. Let us look at the formula to calculate the mean of grouped data. The formula is: x̄ = Σf\(_i\)/N
Where,
 x̄ = the mean value of the set of given data.
 f = frequency of the individual data
 N = sum of frequencies
Hence, the average of all the data points is termed as mean.
Methods of Calculating Mean of Grouped Data
To calculate the mean of grouped data we have three different methods  direct method, assumed mean method, and step deviation method. The mean of grouped data deals with the frequencies of different observations or variables that are grouped together. Let us look at each of these methods separately.
Direct Method
The direct method is the simplest method to find the mean of the grouped data. If the values of the observations are x\(_1\), x\(_2\), x\(_3\),.....x\(_n\) with their corresponding frequencies are f\(_1\), f\(_2\), f\(_3\),.....f\(_n\) then the mean of the data is given by,
x̄ = x\(_1\)f\(_1\) + x\(_2\)f\(_2\) + x\(_3\)f\(_3\) +.....x\(_n\)f\(_n\) / f\(_1\) + f\(_2\) + f\(_3\) +.....f\(_n\)
x̄ = ∑x\(_i\)f\(_i\) / ∑f\(_i\), where i = 1, 2, 3, 4,......n
Here are the steps that can be followed to find the mean for grouped data using the direct method,
 Create a table containing four columns such as class interval, class marks (corresponding), denoted by x\(_i\), frequencies f\(_i\) (corresponding), and x\(_i\)f\(_i\).
 Calculate Mean by the Formula Mean = ∑x\(_i\)f\(_i\) / ∑f\(_i\). Where f\(_i\) is the frequency and x\(_i\) is the midpoint of the class interval.
 Calculate the midpoint, x\(_i\), we use this formula x\(_i\) = (upper class limit + lower class limit)/2.
Let us look at an example.
Example: Find the mean of the following data.
Class Interval  0  10  10  20  20  30  30  40  40  50 
Frequency (f\(_i\))  9  13  8  15  10 
Solution: The first step is to create the table with the midpoint or marks and the product of the frequency and midpoint. To calculate the midpoint we find the average between the class interval by using the formula mentioned above.
Midpoint x\(_i\) = 0  10 = 5 ([10 + 0]/2), 10  20 = 15 ([20 + 10]/2) and so on.
x\(_i\)f\(_i\) = For the class interval 0  10 = 5 × 9 = 45, For the class interval 10  20 = 13 × 15 = 195 and so on.
Class Interval  Frequency (f\(_i\))  Class Mark (x\(_i\))  x\(_i\)f\(_i\) 
0  10  9  5  45 
10  20  13  15  195 
20  30  8  25  200 
30  40  15  35  525 
40  50  10  45  450 
Total  55  1415 
Once we have determined the totals, let us use the formula to calculate the estimated mean.
Estimated Mean = ∑x\(_i\)f\(_i\) / ∑f\(_i\) = 1415/55 = 25.73.
Assumed Mean Method
The next method to calculate the estimated mean for a group of data is the assumed mean method. In the direct method, we know the observations and the corresponding frequencies. In the assumed mean method, let us assume that a is any assumed number which the deviation of the observation is d\(_i\) = x\(_i\)  a. Substituting this in the direct method formula we get,
x̄ = ∑[a + d\(_i\)]f\(_i\) / ∑f\(_i\)
x̄ = ∑af\(_i\) + ∑d\(_i\)]f\(_i\) / ∑f\(_i\)
x̄ = a∑f\(_i\) + ∑d\(_i\)]f\(_i\) / ∑f\(_i\)
x̄ = a + ∑d\(_i\)f\(_i\) / ∑f\(_i\)
Therefore, the assumed mean formula = a + ∑d\(_i\)]f\(_i\) / ∑f\(_i\), where d\(_i\) = x\(_i\)  a.
To calculate the mean of grouped data using the assumed mean method, here are the steps:
 Calculate the midpoint or x\(_i\) for the class interval as we did in the direct method.
 Take the central value from the class marks as the assumed mean and denote it as A.
 Calculate the deviation d\(_i\) = x\(_i\)  A for each i.
 Calculate the product of d\(_i\)f\(_i\) for each i.
 Find the total of f\(_i\)
 Calculate the mean by using the assumed mean method = a + ∑d\(_i\)f\(_i\) / ∑f\(_i\).
Let us look at an example to understand this better.
Example: Find the mean of the following data.
Class Interval  0  10  10  20  20  30  30  40  40  50 
Frequency  12  15  10  25  20 
Solution: The first step is to create the table with the midpoint or marks and the product of the frequency and midpoint. To calculate the midpoint we find the average between the class intervals.
Midpoint x\(_i\) = 0  10 = 5 ([10 + 0]/2), 10  20 = 15 ([20 + 10]/2) and so on. From the midpoint let us select the assumed mean, so A = 25.
Find the deviation, d\(_i\) = x\(_i\)  A for each i. So, d\(_i\) = x\(_i\)  25. Let find for each i, 2  25 =  20 , 15  25 = 10 and so on.
f\(_i\) × d\(_i\) =  20 × 12 =  240 ,  10 × 15 =  150 and so on.
Class Interval  (f(_i\))  (x\(_i\))  Deviation d\(_i\) = x\(_i\)  A  f\(_i\)d\(_i\) 
0  10  12  5   20   240 
10  20  15  15   10   150 
20  30  10  25  0  0 
30  40  25  35  10  250 
40  50  20  45  20  400 
Total  82  260 
Once we have determined the totals, let us use the assume mean formula to calculate the estimated mean.
Assumed Mean Formula = a + ∑d\(_i\)f\(_i\) / ∑f\(_i\)
= 25 + 260/82
= 25 + 3.170
= 28.17.
Therefore, the assumed mean for the data is 28.17.
Step Deviation Method
The step deviation method is used when the deviations of the class marks from the assumed mean are large and they all have a common factor. We already know the direct method mean formula, let us derive the step deviation formula by using the direct method formula and the deviation process in the assumed mean method. Let us consider the class size to be h and the assumed mean to be A. Using the same formula used in assumed mean, d\(_i\) = x\(_i\)  A and calculating the value of u\(_i\) with the equation u\(_i\) = d\(_i\)/h, where h is the class width and d\(_i\) = x\(_i\)  A. Hence, the formula for step deviation method for grouped data is:
Step Deviation of Mean = a + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)],
where,
 a is the assumed mean
 h is the class size
 u\(_i\) = d\(_i\)/h
Let us look at an example to understand this better.
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 
Solution: The first step is to create the table with the midpoint or marks and the product of the frequency and midpoint. To calculate the midpoint we find the average between the class intervals.
Midpoint x\(_i\) = 50  70 = 60 ([70 + 50]/2), 70  90 = 80 ([90 + 70]/2) and so on. From the midpoint let us select the assumed mean, so A = 100 and the value of h = 20 which is the class size.
Find the value, u\(_i\) = d\(_i\)/h, where d\(_i\) = x\(_i\)  A. Hence, we can write it as u\(_i\) = x\(_i\)  A / h. Let us find the value for each class interval. So, 50  70 = (60  100) / 20 = 2 , 70  90 = (80  100) / 20 = 1 and so on.
f\(_i\) × d\(_i\) =  20 × 12 =  240 ,  10 × 15 =  150 and so on.
Class Interval  (f(_i\))  (x\(_i\))  u\(_i\) = x\(_i\)  A / h  f\(_i\)u\(_i\) 
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 
Once we have determined the totals, let us use the assume mean formula to calculate the estimated mean.
Step Deviation of Mean = a + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)]
= 100 + 20 [65/100]
= 100 + 13
= 113.
Therefore, the mean of the data is 113.
Related Topics
Listed below are a few interesting topics that are related to mean of grouped data, take a look.
Examples on Mean of Grouped Data

Example 1: There are 40 students in Grade 8. The marks obtained by the students in mathematics are tabulated below. Calculate the mean marks.
Marks Obtained Number of students 100 6 95 8 88 10 76 9 69 7 Solution:
The total number of students in Grade 8 = 40
x\(_1\) = 100, x\(_2\) = 95, x\(_3\) = 88, x\(_4\) = 76, x\(_5\) = 69, f\(_1\) = 6, f\(_2\) = 8, f\(_3\) = 10, f\(_4\) = 9, f\(_5\) = 7
\(x_1f_1\) = 100 × 6 = 600
\(x_2f_2\) = 95 × 8 = 760
\(x_3f_3\) = 88 × 10 = 880
\(x_1f_1\) = 76 × 9 = 684
\( x_1f_1\) = 69 × 7 = 483\(f_1x_1 + f_2x_2 + f_3x_3 + f_4x_4 + f_5x_5\) = 600 + 760 + 880 + 684 + 483 = 3,407
\(f_1 + f_2 + f_3 + f_4 + f_5\) = 6 + 8 + 10 + 9 + 7 = 40.
We will use the formula given below.
x̄ = Σf\(_i\)x\(_i\)/Σf\(_i\)
Mean marks = 3407/40 = 85.175
Therefore, the mean marks = 85.175.

Example 2: Find the mean percentage of the work completed for a project in a country where the assumed mean is 50, the class size is 20, frequency is 100, and the product of the frequency and deviation is  42. Solve this by using the stepdeviation method.
Solution: Given,
a = 50, h = 20, f\(_i\) = 100, f\(_i\)u\(_i\) =  42
Using the step deviation method formula,
Step Deviation of Mean = a + h [∑u\(_i\)f\(_i\) / ∑f\(_i\)]
= 50 + 20 [42/100]
= 50  42/5
= 50  8.4
= 41.6
Therefore, the mean percentage is 41.6.

Example 3: The marks obtained by 8 students in a class test are 12, 14, 16, 18, 20, 10, 11, and19. Use the mean formula and find out what is the mean of the marks obtained by the students?
Solution:
To find: Mean of marks obtained by 8 students
Marks obtained by 8 students in class test = 12, 14, 16, 18, 20, 10, 11, and19 (given)
Total marks obtained by 8 students in class test = (12 + 14 + 16 + 18 + 20 + 10 + 11 + 19) = 120
Using the mean formula,
Mean = (Sum of Observation) ÷ (Total numbers of Observations) = 120/8 = 15Therefore, the mean of marks obtained by 8 students is 15.
FAQs on Mean of Grouped Data
What is Mean of Grouped Data?
Mean of grouped data is expressed as a data set formed by aggregating individual observations of a variable into different groups. To determine the mean of a grouped data, a frequency table is required to set across the frequencies of the data which makes it simple to calculate. There are three main methods of calculating the mean of grouped data, they are  direct method, assumed mean method, and step deviation method.
What is Mean Formula for Grouped Data?
The mean formula to find the mean of a grouped set of data can be given as, x̄ = Σfx/Σf, where, x̄ is the mean value of the set of given data, f is the frequency of each class and x is the midinterval value of each class
What is the Mean Formula for Ungrouped Data?
The mean formula to find the mean for an ungrouped set of data can be given as, Mean = (Sum of Observations) ÷ (Total Numbers of Observations)
How Do You Find the Mean of Grouped Data From a Frequency Table?
While calculating mean of a grouped data, we always created a frequency table with the midpoint, derivation, and the product of the frequency and midpoint or frequency and derivation. This criteria depends on the type of method used. To calculate the mean from the frequency table we add all the numbers and then divide it by the numbers there are.
What are Different Types of Mean?
The different types of means in mathematics are,
 Arithmetic Mean
 Weighed Mean
 Geometric Mean
 Harmonic Mean
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