How to Construct a Grouped Frequency Distribution Table with Formula and Examples
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 in Statistics
1. What is a grouped frequency distribution?
A grouped frequency distribution is a table that organizes data into class intervals and shows the frequency of observations in each interval. It is used when data values are large or spread out.
- Data is divided into class intervals (e.g., 0–10, 10–20).
- Each interval has a corresponding frequency.
- It helps summarize and analyze large datasets easily.
2. How do you construct a grouped frequency distribution table?
To construct a grouped frequency distribution table, divide the data into equal class intervals and count how many values fall into each class.
- Step 1: Find the range = Maximum − Minimum.
- Step 2: Decide the number of classes (usually 5–10).
- Step 3: Calculate class width = Range ÷ Number of classes.
- Step 4: Form class intervals.
- Step 5: Tally and record the frequency.
3. What is the formula for class width in grouped data?
The formula for class width is Class Width = (Maximum Value − Minimum Value) ÷ Number of Classes.
- First calculate the range.
- Divide the range by the chosen number of class intervals.
- Round to a convenient number if necessary.
4. What is a class interval in grouped frequency distribution?
A class interval is a range of values used to group data in a frequency table. It has a lower limit and an upper limit.
- Example: 10–20 is a class interval.
- 10 is the lower class limit.
- 20 is the upper class limit.
5. What is the difference between grouped and ungrouped frequency distribution?
The main difference is that grouped frequency distribution uses class intervals, while ungrouped frequency distribution lists individual values with their frequencies.
- Grouped: Data is summarized into intervals (e.g., 0–10, 10–20).
- Ungrouped: Each exact value is shown separately.
- Grouped is better for large datasets.
6. How do you find the mean of grouped data?
The mean of grouped data is calculated using the formula Mean = (Σ fᵢxᵢ) ÷ Σ fᵢ, where xᵢ is the class midpoint and fᵢ is the frequency.
- Step 1: Find each class midpoint = (Lower limit + Upper limit) ÷ 2.
- Step 2: Multiply midpoint by frequency.
- Step 3: Add all fᵢxᵢ values.
- Step 4: Divide by total frequency.
7. How do you find the median of grouped frequency distribution?
The median of grouped data is found using the formula Median = L + [(N/2 − cf) ÷ f] × h.
- L = Lower boundary of median class
- N = Total frequency
- cf = Cumulative frequency before median class
- f = Frequency of median class
- h = Class width
8. How do you calculate cumulative frequency in grouped data?
The cumulative frequency is calculated by successively adding frequencies of each class interval.
- Start with the first class frequency.
- Add the next class frequency to the previous total.
- Continue until the last class.
9. What is the purpose of grouped frequency distribution?
The purpose of a grouped frequency distribution is to simplify large datasets and make patterns easier to analyze.
- Summarizes continuous data.
- Helps calculate mean, median, and mode.
- Supports graphical representation like histograms.
10. Can you give an example of grouped frequency distribution?
A simple example of a grouped frequency distribution is shown below for marks of students.
- 0–10 : 2
- 10–20 : 5
- 20–30 : 8
- 30–40 : 4
