Grouped Frequency Table
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 Frequency Distribution Grouped
1. What is Grouped Data?
Ans. Grouped data is a type of data that is classified after the data is collected. The raw data is categorized into different groups and a table is formed. The primary aim to create a table is to show the points occurring in each group. For example, when a test is done, the results are the data in this situation and there are different ways to group this data. For example, the number of students that scored above 50 can be recorded.
2. What is Frequency?
Ans. The number of times a particular observation occurs in a given data is known as frequency whereas the table in which such a distribution of frequencies is given is known as the frequency distribution table.
3. What is the Cumulative Frequency?
Ans. Cumulative frequency is defined as the total of all the frequencies. It is the addition of all the previous frequencies up to the present point. It is an important tool in statistics to tabulate the given data in an organized manner.
4. What is the Cumulative Frequency Table?
Ans. The cumulative frequency table is a simple visual representation of the cumulative frequencies for each distinct value or category. The cumulative frequency table is useful in determining the median of the series.
