What Is Grouped Frequency Distribution and Why Is It Important?
The fundamental aim of statistics is to organize or summarise data. The data collected from any research project is unorganized and in raw form. No exact meaning can be conveyed from raw data unless the data is arranged or grouped in a certain manner to give more insight into the data. The most convenient form of organized data is to construct a frequency distribution. A frequency distribution is a graph or data set organized to show the frequency of occurrences of each possible outcome of repeatable events observed many times. It allows researchers to observe the entire data conveniently.
When data includes hundreds of values, it is preferable to group them into smaller parts to make the data more understandable. The arrangement of large data sets can be done by grouping the observations into intervals and tabulating the frequencies for each interval. The result is known as a grouped frequency distribution or grouped frequency tables. In the grouped frequency distribution, the intervals are known as classes.
Mean of The Frequency Distribution
In statistics, the mean is the arithmetic average of a given data set. The mean can be calculated easily by adding all the numbers and dividing the result by the total numbers.
Example:
Find the Mean of the Following Numbers.
5, 12 , and 6
Add up all the numbers : 5 + 12 + 7 = 24
Dividing the result by total numbers ( there are 3 numbers) = 24 3
The Mean is 8
But sometimes, we don't have a simple list of numbers. It might be given in a frequency table as shown below ( the frequency says how often they occur).
(It says scored one occurs twice, score 2 occurred 5 times, score 3 occurred 4 times, etc).
The Mean of the Above Frequency Distribution Table Can be Calculated as:
\[Mean=\frac{(2\times 1)+(5\times2)+(3\times4)+(4\times2)+(5\times1)}{2+5+4+2+1}\]
\[Mean=\frac{2+10+12+8+5}{14}=\frac{37}{14}=2.64\]
Therefore, the mean is 2.64
Frequency Distribution Table For Grouped Data
Following are steps to construct a frequency distribution table for grouped data:
Find the highest score (H) and the lowest score (L). Find the range which is the difference between these two scores.
Estimate the class width(w) by dividing the range by a number of groupings or classes. If the result is not an integer, round off the values to the nearest integer to get the class width. It is convenient if the class width is an odd number. (The number 7 or 10 are selected so that the number of intervals will neither be small or large).
Construct the class intervals. Start with the lowest score or convenient value slightly less than the lowest score. Then add the class width to the starting point to get the next interval. Continue this, until the highest score is obtained in the class interval.
Tally the corresponding number of scores in each class interval. Then sum up the tally under the frequency column.
Example:
Construct a grouped frequency table of students scores on a Maths Test given below:
Solution:
Highest Score (H) - 48, Lowest score (L) - 15. Range (d) = 48 - 15 = 33.
Width = 33/7 = 4.7 which can be rounded off to 5, Therefore width (w = 5). An odd number w is convenient because the midpoint of the interval is an integer.
Starting with 15 and considering w = 5, the classes are 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49.
Let’s now construct a grouped frequency table for the above data.
Grouped Frequency Table of Student’s Score on Maths Test
The frequency distribution table for the grouped data gives more precise information about the gathered data. For example, in the above table, the greatest frequency is found in the fifth- interval (35-39), and more than half of the students scored between 30-39.
Mean of Grouped Data
An approximate mean \[\overline{x}\] of the population from which data are collected, can be calculated from grouped data by using the formula of the mean of grouped data given below:
\[\overline{x}=\frac{\sum fx}{\sum f}\]
In the above formula of the mean of grouped data,\[\overline{x}\] refers to sample mean, f is the class frequency, and x is the midpoint of the class interval.
Ungrouped Frequency Distribution
Ungrouped data is data that has not been placed in any group or category after collection. It is often known as raw data. For example, 240 people are living in your locality. This is raw data as it is not grouped in any category.
Let us now understand how to construct an ungrouped frequency distribution table.
Given below are marks scored by 20 students in English out of 25.
25, 17, 19, 23, 12, 19, 15, 15, 17, 17, 19, 23, 23, 19, 21, 23, 21, 25, 21, 19.
Ungrouped Frequency Distribution Table of Students Marks in English
Mean of Ungrouped Frequency Distribution
The mean of ungrouped frequency distribution can be calculated by adding up all of the observations and dividing the result by the total number of observations (n). Hence, the mean or arithmetic mean of n observations i.e. x₁, x₂ , x₃, x₄ , x₅ … xₙ is given by
\[Mean=\frac{x_{1}+x_{2}+x_{3}+x_{4}+...+x_{n}}{n}\]
In other words, \[Mean=\frac{Sum of Observations}{Total Number of Observations}\]
Symbolically, \[A=\frac{\sum x_{i}}{n}\] where, i = 1,2,3,4,......,n.
Solved Example
1. Find the Mean of the Following Grouped Data Set.
Solution:
Step 1. Find the class mark ( also known as midpoints) of each interval by calculating the average of the upper and lower limits. For example, the class mark of interval:
10 < 20 is (20 + 10)/2 = 15
20 < 30 is (20 + 30)/2 = 25
30 < 40 is ( 30 + 40)/2 = 35
30 < 40 is ( 30 + 40)/2 = 35
40 < 50 is ( 40 + 50)/2 = 45
50 < 60 is ( 50 + 60)/2 = 55
Step 2: Find the product of the class mark or midpoint and frequency for each interval as shown below:
Step 3: Applying the formula of the mean of grouped data to calculate the mean.
\[\overline{x}=\frac{\sum fx}{n}\]
Substituting the values in the above formula, we get:
\[\overline{x}=\frac{845}{25}\]
\[\overline{x}\] = 33.8
Hence, the mean of the given grouped data set is 33.8
2. A Class 7 student scored 80%, 72%, 50%, 64% and 74% marks in five subjects in an examination. Find the mean percentage of marks obtained by him.
Solution:
The observations in the percentage are: x₁ = 80, x₂ = 72, x₃ = 50, x₄ = 64, x₅ =74
Therefore, the mean can be calculated as \[\frac{x_{1}+x_{2}+x_{3}+x_{4}+x_{5}}{5}\]
\[=\frac{80+72+50+64+74}{5}\]
\[=\frac{340}{5}\]
= 68
FAQs on Grouped Frequency Distribution: Concepts & Examples
1. What is a grouped frequency distribution, and what is its main purpose in statistics?
A grouped frequency distribution is a method used in statistics to organise and summarise a large set of data by grouping it into well-defined categories called class intervals. Its main purpose is to make a large dataset more manageable and easier to interpret. Instead of listing every single data point, it shows how many data points fall within each specific range, revealing patterns, concentration, and the overall shape of the data.
2. What are the essential components of a grouped frequency distribution table?
A standard grouped frequency distribution table, as per the CBSE/NCERT curriculum for the academic year 2025-26, includes the following key components:
Class Intervals: These are the non-overlapping ranges into which the data is divided (e.g., 10-20, 20-30). Each interval has a lower class limit and an upper class limit.
Frequency (f): This is the count of how many data points fall into a particular class interval.
Class Mark (or Mid-point): This is the central value of a class interval, calculated by averaging the upper and lower class limits. It is used for calculating the mean of grouped data.
Cumulative Frequency (cf): This is the running total of the frequencies, showing the total number of observations up to a certain class interval.
3. How do you decide the appropriate class intervals when creating a grouped frequency distribution?
Choosing appropriate class intervals is crucial for a meaningful distribution. First, find the range of the data (highest value - lowest value). Next, decide on the number of classes you want; typically between 5 and 15 classes work well. The width of each class interval can then be estimated by dividing the range by the number of classes. It's important to ensure that the intervals are mutually exclusive (non-overlapping) and exhaustive (cover all data points). For example, using intervals like 0-10, 10-20, 20-30 (where 10 is included in the second interval, not the first) is a common practice.
4. What is the difference between a grouped and an ungrouped frequency distribution?
The primary difference lies in how data is presented. An ungrouped frequency distribution lists each individual data value and its corresponding frequency. It is suitable for small datasets with a limited number of distinct values. In contrast, a grouped frequency distribution organises data into class intervals and shows the frequency for each group, not individual values. It is essential for large datasets where listing each value would be impractical and messy.
5. How does a histogram visually represent a grouped frequency distribution?
A histogram is the direct graphical representation of a grouped frequency distribution for continuous data. The class intervals are marked on the horizontal axis (x-axis), and the corresponding frequencies are represented by the height of rectangular bars on the vertical axis (y-axis). A key feature of a histogram is that there are no gaps between the bars, signifying the continuous nature of the class intervals.
6. What is the importance of 'cumulative frequency' in analysing grouped data?
Cumulative frequency is important because it provides a running total of the data, which helps in understanding the data distribution at a glance. Its primary application is in finding the median of the grouped data. By creating a cumulative frequency curve, known as an ogive, one can graphically estimate the median and other partition values like quartiles and percentiles, which are key concepts in higher statistics.
7. What are some common mistakes students make when constructing a grouped frequency distribution table?
Some common mistakes to avoid include:
Overlapping Class Intervals: Creating intervals like 10-20 and 20-30 without specifying where the value 20 belongs. This can lead to double-counting or incorrect classification.
Inconsistent Class Widths: Using varying class sizes (e.g., 0-10, 10-15, 15-25) without a specific reason, which can distort the visual representation in a histogram.
Incorrect Tallying: Miscounting the data points that fall into each interval, leading to an incorrect frequency count.
Ignoring the Range: Creating intervals that do not cover the smallest or largest values in the dataset.
8. Can you provide a real-world example where a grouped frequency distribution is useful?
A classic real-world example is analysing the monthly income of 1,000 households in a city. Instead of dealing with 1,000 individual income figures, a researcher can group them into intervals like ₹10,000-₹20,000, ₹20,001-₹30,000, and so on. This grouped frequency distribution would quickly show how many households fall into each income bracket, making it easy for economists and policymakers to understand the city's economic structure and identify the low, middle, and high-income groups.
