Mean of Grouped Data

It is the process of calculating the average for data that is organized into groups or categories. The mean, which is used to find the central tendency of the data, is the average or computed central value of a collection of numbers. In statistics, the mean can be found by the sum of all observations divided by the total number of observations. That is: x = x₁, x₂, x₃,.………., x_n.

The mean, denoted x, is the mean of the n values x₁, x₂, x₃,.………., x_n. The mean is represented as x (x bar). 

The mean is calculated using the formula as the sum of the observations divided by the total number of observations. There are generally two formulas for calculating the mean for ungrouped data and the mean for grouped data. As we are learning to calculate the mean of the grouped data, let’s look into it further.

x = Σ(f × xi) / Σf

Where,
 

x = the mean value of the set of given data.
 

f = frequency of each class
 

∑fx = Sum of the product of midpoints and their frequencies
 

∑f = Total number of frequency

In order to find the mean of grouped data, the simplest way is by using a direct method. If the values of the observations are x₁, x₂,...., x_n with their corresponding frequencies are f₁, f₂,...., 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

Following these simple steps given below will help you out in finding the mean of grouped data. 

Step 1: Creating a table that organizes all the given groups or categories into their corresponding groups will make it easier to calculate the mean. A class interval is the range between the lower and upper boundaries of a group in grouped data. For example, class interval (denoted by x₁x₂....x_n) and class marks (denoted by f₁f₂.........f_n). 

Step 2: Calculate the given formulas of the table using the mean formula = \(\frac{\sum x_i f_i}{\sum f_i} \)

Step 3: Next, calculate the midpoint (that is x_i) using the formula x_i = (upper limit + lower limit)/2

Apart from the direct method to find the mean of grouped data, there are two other methods to find out the mean value of any grouped data. Let’s look into it. 

Assumed Mean Method: In this method, we assume that a is any assumed number of which the deviation of the observation is di = xi - a. By substituting it in the direct method, we get:


  \(\begin{align*} \bar{x} &= \frac{\sum (a + d_i) f_i}{\sum f_i} \\ \bar{x} &= \frac{\sum (a f_i) + \sum (d_i f_i)}{\sum f_i} \\ \bar{x} &= \frac{a \sum f_i + \sum (d_i f_i)}{\sum f_i} \\ \bar{x} &= a + \frac{\sum (d_i f_i)}{\sum f_i} \end{align*} \text{where } d_i = x_i - a \)

Step Deviation Method: This method is used when the values in a data set (class marks) are large and not close to the assumed mean (a number chosen to make calculations easier). These values will have a common factor, meaning they can all be divided by the same number. This helps simplify the calculations when finding the mean (average) of the data. 

Step Deviation of Mean = \(a + h \times \left( \frac{\Sigma (u \times f)}{\Sigma f} \right) \)
 

Where, 
 

a is the assumed mean
 

h is the class size
 

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

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.

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.