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.

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 x-bar, x̄.

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.

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.

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.

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.

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.

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.

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.

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.

• Average
• Mean, Median, Mode
• Categorical 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 mid-interval 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