presentation of grouped data

Grouped Data / Ungrouped Data: Definition, Examples

What is grouped data.

• Ungrouped data
• Mean of Grouped Data

Grouped data is data that has been bundled together in categories. Histograms and frequency tables can be used to show this type of data:

The data is grouped together by classes or bins .

Grouped vs. Ungrouped Data

Ungrouped data is the data you first gather from an experiment or study. The data is raw — that is, it’s not sorted into categories, classified, or otherwise grouped. An ungrouped set of data is basically a list of numbers.

Calculating the Sample Mean for Grouped Data

• x̄ is the sample mean,
• x is the class (or category) midpoint,
• f is the class frequency.

Example question: Find the sample mean for the following frequency table.

Step 1: Find the midpoint for each class interval. the midpoint is just the middle of each interval. For example, the middle of 10 and 15 is 12.5:

Step 2: Multiply the midpoint (x) by the frequency (f):

Add up all of the totals for this step. In other words, add up all the values in the last column (you should get 405).

Step 3: Divide the last column (f*x) by the second column (f): The mean (x̄) = 405 / 20 = 20.25.

Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York. Klein, G. (2013). The Cartoon Introduction to Statistics. Hill & Wamg.

• Math Article

Grouping Data

Grouping of data plays a significant role when we have to deal with large data. This information can also be displayed using a pictograph or a bar graph. Data formed by arranging individual observations of a variable into groups, so that a frequency distribution table of these groups provides a convenient way of summarizing or analyzing the data is termed as grouped data.

Frequency distribution table for grouped data

When the collected data is large, then we can follow the below approach to analyse it easily using tally marks .

Consider the marks of 50 students of class VII obtained in an examination. The maximum marks of the exam are 50.

23, 8, 13, 18, 32, 44, 19, 8, 25, 27, 10, 30, 22, 40, 39, 17, 25, 9, 15, 20, 30, 24, 29, 19, 16, 33, 38, 46, 43, 22, 37, 27, 17, 11, 34, 41, 35, 45, 31, 26, 42, 18, 28, 30, 22, 20, 33, 39, 40, 32

If we create a frequency distribution table for each and every observation, then it will form a large table. So for easy understanding, we can make a table with a group of observations say 0 to 10, 10 to 20 etc.

The distribution obtained in the above table is known as the grouped frequency distribution. This helps us to bring various significant inferences like:

(i) Many students have secured between 20-40, i.e. 20-30 and 30-40.

(ii) 8 students have secured higher than 40 marks, i.e. they got more than 80% in the examination.

In the above-obtained table, the groups 0-10, 10-20, 20-30,… are known as class intervals (or classes). It is observed that 10 appears in both intervals, such as 0-10 and 10-20. Similarly, 20 appears in both the intervals, such as as10-20 and 20-30. But it is not feasible that observation either 10 or 20 can belong to two classes concurrently. To avoid this inconsistency, we choose the rule that the general conclusion will belong to the higher class. It means that 10 belongs to the class interval 10-20 but not to 0-10. Similarly, 20 belongs to 20-30 but not to 10-20, etc.

Consider a class say 10-20, where 10 is the lower class interval and 20 is the upper-class interval. The difference between upper and lower class limits is called class height or class size or class width of the class interval.

Frequency Polygons

Important Questions Class 8 Maths Chapter 5 Data Handling

How to determine the class size?

In order to avoid confusion on the size of the class intervals that we need to take while grouping the data, one must follow the below steps.

Step 1: Identify the highest and the lowest (least) data values in the given observations.

Step 2: Find the difference between the highest and least value.

Step 3: Now, assume the number of class intervals we need (usually 5 to 20 classes are suggested to take based the number of observations).

Step 4: Divide the difference of highest and least value by the number of classes, this result in the size of the class interval.

Step 5: In case of any decimal number obtained as a class size take the nearest whole number greater than the obtained decimal as the class size.

We can show the above frequency distribution table graphically using a histogram. Consider class intervals on the horizontal axis and the frequency on the vertical axis.

The height of the bars represents the frequency of the class interval. There is no gap between the bars since there is no gap between the classes.

Frequently Asked Questions – FAQs

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• Key Differences

Know the Differences & Comparisons

Difference Between Ungrouped Data and Grouped Data

In simple words, ungrouped data or raw data is a mere list of numbers that does not convey anything. This is because no summarization or aggregation is possible. On the other hand, grouped data refers to the data which is bundled together in different classes or categories.

What is Data?

Data refers to the fresh information or a set of given facts, in numerical figures collected by individuals or obtained from various sources for a specified purpose.

Also Read : Difference Between Data and Information

This write-up will help you in understanding the difference between ungrouped data and grouped data.

Content: Ungrouped Data Vs Grouped Data

Comparison chart, prerequisites for the formation of classes.

Definition of Ungrouped Data

Numerical Data available in its authentic form is called ungrouped data. It is a collection of observations that are gathered initially during the research by the investigator or obtained from the specified source. It is gathered as individual observations, commonly as a table or range of jumbled values.

As the data is presented in its actual form, computation, analysis, and interpretation is cumbersome process. So, one cannot arrive at any conclusion, unless the data is arranged in a specific order, i.e. ascending or descending, when the observations are in numerical form.

For Example : Let us say there are 30 women in a colony, whose age group are as follows:

55, 35, 29, 35, 24, 77, 65, 45, 26, 29, 35, 66, 57, 59, 33, 31, 64, 28, 63, 55, 25, 69, 46, 38, 48, 61, 37, 55, 24, 64

Data available in such a form is called raw data . And each entry i.e. 55, 35, 296, and so forth, is the value or observation .

Suppose we want to analyze the women who are above 35 years, the data presented in the above form does not provide sufficient information, such as the highest or lowest age. So, let us arrange this data in increasing order.

24, 24, 25, 26, 28, 29, 29, 31, 33, 35, 35, 35, 37, 38, 45, 46, 48, 55, 55, 55, 57, 59, 61, 63, 64, 64, 65, 66, 69, 77

Now, you must be wondering – what do we call this type of data?

Well, data given in this form is called arrayed data . An array is nothing but an arrangement of numerical observations in the order of magnitude. Plus, the difference between the largest and smallest number is the range .

Here, the number of times each number occurs is called frequency .

What is Frequency Distribution?

A frequency distribution is the arrangement of data in tabular form which represents the frequency, with which each consecutive value of the variable occurs.

Also Read : Difference Between Qualitative and Quantitative Data

Definition of Grouped Data

When we assemble raw data into a number of classes, the resultant data is the grouped data. This is done to get a more condensed form of data. Data are grouped when the variable stretches over a wide range and there are a large number of observations, and it is not possible to arrange the data in any order, as it consumes a lot of time. Hence, it is pertinent to convert frequency into a class group called a class interval .

So, when the data are arranged in groups, it is called grouped data. A grouped frequency distribution indicates the number of times different values occur in different groups. When the data is arranged in this form, no information is lost.

Process of formation of Classes

• Ascertain the range of raw data, i.e. difference between the maximum and minimum observation.
• Decide the total number of classes into which raw data is grouped, generally 5 to 10.
• Next, the range is divided by the intended number of classes to calculate the approximate size of the class interval.
• Set up class limits with the help of class intervals.
• Now take one observation from data at a time and enter a tally mark against the class of which it is a part. Further, to make it more convenient we will write down the tally mark in a bundle of five. The fifth one will strike the other four in a diagonal pattern.
• In our next step, we will count the tally marks in each class, to get the frequency of each class. It must be noted that the total of frequency should be equal to the total number of observations.

Kinds of Grouped Data

• Discrete Data : Use of discontinuous class intervals, is called discrete data, such as 1-10, 11 – 20, etc.
• Continuous Data : It is characterized by the use of continuous class limits. such as 0 – 10, 10 – 20, etc.

Also Read : Difference Between Discrete and Continuous Data

Presentation of Grouped Data

For Example : With the help of class intervals 1 – 10, 11 – 20, 21 – 30,…. Prepare the frequency distribution for the given data:

22, 65, 32, 36, 28, 17, 15, 64, 86, 52, 36, 55, 9, 48, 65, 96, 14, 6, 37, 69, 76, 49, 37, 46, 21, 67, 92, 98

Here, the class represents 1 – 10, wherein both 1 and 10 are included.

Also Read : Difference Between Primary and Secondary Data

Key Differences Between Grouped Data and Ungrouped Data

After talking about the meaning and types of the two forms of data, let us understand the difference between grouped data and ungrouped data:

• Ungrouped data implies the data expressed in the form as it is gathered. On the other hand, when the displaced data is arranged in a condensed form by creating groups of the observations collected, is called grouped data.
• Ungrouped data is commonly preferred when data is collected by the researcher, as it is a simple list of numbers. As against, grouped data is preferred at the time of data analysis.
• Ungrouped data is not classified or organized into different classes, whereas grouped data is organized into a number of classes.
• Ungrouped data is presented in the form of lists, whereas, frequency tables are used to express, grouped data.
• Ungrouped data is not summarized, it is the original form of data, as it is collected by the researcher. Contrastingly, grouped data is summarized in a frequency distribution.
• There should be non-overlapping and continuous classes.
• There should not be any gaps amidst classes.
• The size of the classes should be the same.
• Avoid open-ended classes, such as less than 5 or more than 9
• Limits of each class have to be selected in a way that no confusion regarding which class a certain observation of the given data is a part of.
• A proper title should be given to the table so that it can convey exactly what the table is about.

Ungrouped data, as already said, is the data collected from the respondents in its actual form or as it is. The division of data into different classes is called grouped data.

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Frank James Narra says

April 3, 2022 at 7:33 am

Thank you very much. It helps a lot

Bindeshwar S. Kushwaha says

January 7, 2023 at 3:26 pm

Good article for understanding types of grouped data.

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Introduction to Grouped Data

In this page, grouped data is introduced and explained with some examples. lesson outline

We studied about data recording and presentation with tally marks and tabular form. Then we studied about bar-graphs to represent the data in a spatially laid-out form which helps to compare data or find the minimum or maximum, etc.

it does not work

make it work

make bargraph work too

The bar graph with continuous range is shown in the figure. This form of bar-graph is called " histogram ".

The word "histogram", means written or recorded information in the form of vertical bars". "-gram" : written or recorded information + "histo" : in the form of vertical bars.

different possibilities in grouping

Histogram of the same is given in the figure.

The outline of material to learn "basics of statistics and probability" is as follows. Note: Click here for detailed outline of "Basics of Statistics and Probability" •   Introduction     →   Introduction to Statistics     →   Organizing Data : Tally Table     →   Pictograph     →   Bargraph •   Data Analysis     →   Cumulative Frequency     →   Representative Values of Data     →   Central Tendencies     →   Bargraphs & Piecharts •   Probability Fundamentals     →   Predicting Future     →   Random Experiment     →   Probability     →   Standard Experiments •   Statistics Grouped Data     →   Grouped Data     →   Probability in Grouped Data     →   Class Parameters of Grouped Data     →   Methods to find Mean of Grouped data     →   Mode of Grouped data     →   Median of Grouped Data next