frequency distribution assignment

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base
• Frequency Distribution | Tables, Types & Examples

Frequency Distribution | Tables, Types & Examples

Published on June 7, 2022 by Shaun Turney . Revised on June 21, 2023.

A frequency distribution describes the number of observations for each possible value of a variable . Frequency distributions are depicted using graphs and frequency tables.

Table of contents

What is a frequency distribution, how to make a frequency table, how to graph a frequency distribution, other interesting articles, frequently asked questions about frequency distributions.

The frequency of a value is the number of times it occurs in a dataset. A frequency distribution is the pattern of frequencies of a variable. It’s the number of times each possible value of a variable occurs in a dataset.

Types of frequency distributions

There are four types of frequency distributions:

• You can use this type of frequency distribution for categorical variables .
• You can use this type of frequency distribution for quantitative variables .
• You can use this type of frequency distribution for any type of variable when you’re more interested in comparing frequencies than the actual number of observations.
• You can use this type of frequency distribution for ordinal or quantitative variables when you want to understand how often observations fall below certain values .

Here's why students love Scribbr's proofreading services

Discover proofreading & editing

Frequency distributions are often displayed using frequency tables . A frequency table is an effective way to summarize or organize a dataset. It’s usually composed of two columns:

• The values or class intervals
• Their frequencies

The method for making a frequency table differs between the four types of frequency distributions. You can follow the guides below or use software such as Excel, SPSS, or R to make a frequency table.

How to make an ungrouped frequency table

• For ordinal variables , the values should be ordered from smallest to largest in the table rows.
• For nominal variables , the values can be in any order in the table. You may wish to order them alphabetically or in some other logical order.
• Especially if your dataset is large, it may help to count the frequencies by tallying . Add a third column called “Tally.” As you read the observations, make a tick mark in the appropriate row of the tally column for each observation. Count the tally marks to determine the frequency.

How to make a grouped frequency table

• Calculate the range . Subtract the lowest value in the dataset from the highest.

• Create a table with two columns and as many rows as there are class intervals. Label the first column using the variable name and label the second column “Frequency.” Enter the class intervals in the first column.
• Count the frequencies. The frequencies are the number of observations in each class interval. You can count by tallying if you find it helpful. Enter the frequencies in the second column of the table beside their corresponding class intervals.

Round the class interval width to 10.

The class intervals are 19 ≤ a < 29, 29 ≤ a < 39, 39 ≤ a < 49, 49 ≤ a < 59, and 59 ≤ a < 69.

How to make a relative frequency table

• Create an ungrouped or grouped frequency table .
• Add a third column to the table for the relative frequencies. To calculate the relative frequencies, divide each frequency by the sample size. The sample size is the sum of the frequencies.

How to make a cumulative frequency table

• Create an ungrouped or grouped frequency table for an ordinal or quantitative variable. Cumulative frequencies don’t make sense for nominal variables because the values have no order—one value isn’t more than or less than another value.
• Add a third column to the table for the cumulative frequencies. The cumulative frequency is the number of observations less than or equal to a certain value or class interval. To calculate the relative frequencies, add each frequency to the frequencies in the previous rows.
• Optional: If you want to calculate the cumulative relative frequency , add another column and divide each cumulative frequency by the sample size.

Pie charts, bar charts, and histograms are all ways of graphing frequency distributions. The best choice depends on the type of variable and what you’re trying to communicate.

A pie chart is a graph that shows the relative frequency distribution of a nominal variable .

A pie chart is a circle that’s divided into one slice for each value. The size of the slices shows their relative frequency.

This type of graph can be a good choice when you want to emphasize that one variable is especially frequent or infrequent, or you want to present the overall composition of a variable.

A disadvantage of pie charts is that it’s difficult to see small differences between frequencies. As a result, it’s also not a good option if you want to compare the frequencies of different values.

A bar chart is a graph that shows the frequency or relative frequency distribution of a categorical variable (nominal or ordinal).

The y -axis of the bars shows the frequencies or relative frequencies, and the x -axis shows the values. Each value is represented by a bar, and the length or height of the bar shows the frequency of the value.

A bar chart is a good choice when you want to compare the frequencies of different values. It’s much easier to compare the heights of bars than the angles of pie chart slices.

A histogram is a graph that shows the frequency or relative frequency distribution of a quantitative variable . It looks similar to a bar chart.

The continuous variable is grouped into interval classes , just like a grouped frequency table . The y -axis of the bars shows the frequencies or relative frequencies, and the x -axis shows the interval classes. Each interval class is represented by a bar, and the height of the bar shows the frequency or relative frequency of the interval class.

Although bar charts and histograms are similar, there are important differences:

A histogram is an effective visual summary of several important characteristics of a variable. At a glance, you can see a variable’s central tendency and variability , as well as what probability distribution it appears to follow, such as a normal , Poisson , or uniform distribution.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s t table
• Student’s t distribution
• Quartiles & Quantiles
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Prevent plagiarism. Run a free check.

A histogram is an effective way to tell if a frequency distribution appears to have a normal distribution .

Plot a histogram and look at the shape of the bars. If the bars roughly follow a symmetrical bell or hill shape, like the example below, then the distribution is approximately normally distributed.

Categorical variables can be described by a frequency distribution. Quantitative variables can also be described by a frequency distribution, but first they need to be grouped into interval classes .

Probability is the relative frequency over an infinite number of trials.

For example, the probability of a coin landing on heads is .5, meaning that if you flip the coin an infinite number of times, it will land on heads half the time.

Since doing something an infinite number of times is impossible, relative frequency is often used as an estimate of probability. If you flip a coin 1000 times and get 507 heads, the relative frequency, .507, is a good estimate of the probability.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Turney, S. (2023, June 21). Frequency Distribution | Tables, Types & Examples. Scribbr. Retrieved August 12, 2024, from https://www.scribbr.com/statistics/frequency-distributions/

Is this article helpful?

Shaun Turney

Other students also liked, variability | calculating range, iqr, variance, standard deviation, types of variables in research & statistics | examples, normal distribution | examples, formulas, & uses, what is your plagiarism score.

Frequency Distribution Table: Examples, How to Make One

Contents (Click to skip to that section):

What is a Frequency Distribution Table?

• Using Tally Marks
• Including Classes

Types of Frequency Distribution

See also: Frequency Distribution Table in Excel

Frequency tells you how often something happened . The frequency of an observation tells you the number of times the observation occurs in the data. For example, in the following list of numbers, the frequency of the number 9 is 5 (because it occurs 5 times):

1, 2, 3, 4, 6, 9, 9, 8, 5, 1, 1, 9, 9, 0, 6, 9.

A frequency distribution is a summary of this type of data [1]. It gives us the number of observations within a specific interval, shown either graphically (usually with a bar chart or a histogram ) or as a f requency distribution table . Frequency in this context indicates the occurrence of a value within a specified interval, while distribution refers to the pattern of the variable’s frequency.

Tables can show either categorical variables (sometimes called qualitative variables ) or quantitative variables (sometimes called numeric variables). You can think of categorical variables as categories (like eye color or brand of dog food) and quantitative variables as numbers.

The following table shows what family planning methods were used by teens in Kweneng, West Botswana. The left column shows the categorical variable (Method) and the right column is the frequency — the number of teens using that particular method.

Frequency distribution tables give you a snapshot of the data to allow you to find patterns. A quick look at the above frequency distribution table tells you the majority of teens don’t use any birth control at all.

Back to Top

How to make a Frequency Distribution Table: Examples

Example 1: Tally marks are often used to make a frequency distribution table. For example, let’s say you survey a number of households and find out how many pets they own. The results are 3, 0, 1, 4, 4, 1, 2, 0, 2, 2, 0, 2, 0, 1, 3, 1, 2, 1, 1, 3. Looking at that string of numbers boggles the eye; a frequency distribution table will make the data easier to understand.

How to Draw a Frequency Distribution Table (Slightly More Complicated Example)

A frequency distribution table is one way you can organize data so that it makes more sense. For example, let’s say you have a list of IQ scores for a gifted classroom in a particular elementary school. The IQ scores are: 118, 123, 124, 125, 127, 128, 129, 130, 130, 133, 136, 138, 141, 142, 149, 150, 154. That list doesn’t tell you much about anything. You could draw a frequency distribution table , which will give a better picture of your data than a simple list.

How to Draw a Frequency Distribution Table: Steps.

Part 1: choosing classes.

Step 1: Figure out how many classes (categories) you need. There are no hard rules about how many classes to pick, but there are a couple of general guidelines:

• Pick between 5 and 20 classes. For the list of IQs above, we picked 5 classes.
• Make sure you have a few items in each category. For example, if you have 20 items, choose 5 classes (4 items per category), not 20 classes (which would give you only 1 item per category).

Note : There is a more mathematical way to choose classes. The formula is log(observations)\ log(2). You would round up the answer to the next integer . For example, log17\log2 = 4.1 will be rounded up to become 5. Another way to do this is with Sturges formula : Number of classes = 1 + 3.322 log N , where N is the number of items in the set.

Part 2: Sorting the Data

Step 2: Subtract the minimum data value from the maximum data value. For example, our IQ list above had a minimum value of 118 and a maximum value of 154, so:

154 – 118 = 36

Step 3: Divide your answer in Step 2 by the number of classes you chose in Step 1.

36 / 5 = 7.2

Step 4: Round the number from Step 3 up to a whole number to get the class width . Rounded up, 7.2 becomes 8 .

Step 5: Write down your lowest value for your first minimum data value:

The lowest value is 118

Step 6: Add the class width from Step 4 to Step 5 to get the next lower class limit:

118 + 8 = 126

Step 7: Repeat Step 6 for the other minimum data values (in other words, keep on adding your class width to your minimum data values) until you have created the number of classes you chose in Step 1. We chose 5 classes, so our 5 minimum data values are:

• 118 126 (118 + 8)
• 134 (126 + 8)
• 142 (134 + 8)
• 150 (142 + 8)

Step 8: Write down the upper class limits. These are the highest values that can be in the category, so in most cases you can subtract 1 from the class width and add that to the minimum data value. For example:

• 118 + (8 – 1) = 125
• 118 – 125
• 126 – 133
• 134 – 141
• 142 – 149 1
• 50 – 157

3. Finishing the Table Up

Step 9: Add a second column for the number of items in each class, and label the columns with appropriate headings:

Step 10: Count the number of items in each class, and put the total in the second column. The list of IQ scores are: 118, 123, 124, 125, 127, 128, 129, 130, 130, 133, 136, 138, 141, 142, 149, 150, 154.

That’s How to Draw a Frequency Distribution Table, the easy way!

Tip : If you are working with large numbers (like hundreds or thousands), round Step 4 up to a large whole number that’s easy to make into classes, like 100, 1000, or 10,000. Likewise with very small numbers — you may want to round to 0.1, 0.001 or a similar division.

There are a few variations of frequency distributions:

• Ungrouped frequency distribution : a table that shows the number of data points for each individual value. This is sometimes called just a “frequency distribution.” This is the type shown in example 1 above.
• Grouped frequency distribution : a table that shows the number of data points that fall within a range of values, called a class interval. This type is shown in example 2 above.
• Cumulative frequency distribution : shows the sum of all values up to the current class.
• Relative frequency distribution: shows the proportion of all values that fall within a particular class.
• Relative cumulative frequency distribution: shows the proportion of all values that are less than or equal to a particular value in a frequency distribution.

We can also create a relative frequency marginal distribution, which, shows relative frequencies rather than frequencies for marginal probability distributions [2].

• Blank, B. (2016). Elementary Statistics .
• Section 4.4: Contingency Tables and Association.

Frequency Distribution Table

A frequency distribution table displays the frequency of each data set in an organized way. It helps us to find patterns in the data and also enables us to analyze the data using measures of central tendency and variance. The first step that a mathematician does with the collected data is to organize it in the form of a frequency distribution table. All the calculations and statistical tests and analyses come later.

What is a Frequency Distribution Table?

A frequency distribution table is a way to organize data so that it makes the data more meaningful. A frequency distribution table is a chart that summarizes all the data under two columns - variables/categories, and their frequency. It has two or three columns. Usually, the first column lists all the outcomes as individual values or in the form of class intervals, depending upon the size of the data set. The second column includes the tally marks of each outcome. The third column lists the frequency of each outcome. Also, the second column is optional.

Do you know the meaning of "frequency?" Frequency indicates how often something occurs. For example, your heartbeat is 72 heartbeats/min under normal conditions. Frequency corresponds to the number of times a value occurs.

In our day-to-day lives, we come across a lot of information in the form of numerical figures, tables, graphs, etc. This information could be marks scored by students, temperatures of different cities, points scored in matches, etc. The information that is collected is called data . Once the data is collected, we have to represent it in a meaningful manner so that it can be easily understood. The frequency distribution table is one of the ways to organize data.

Here's a frequency distribution table example for you to understand this concept better. Jane is fond of playing games with dice. She throws the dice and notes the observations each time. These are her observations: 4, 6, 1, 2, 2, 5, 6, 6, 5, 4, 2, 3. To know the exact number of times she got each digit (1, 2, 3, 4, 5, 6) as the outcome, she classifies them into categories. An easy way is to draw a frequency distribution table with tally marks.

The table above is an example of a frequency distribution table. You can observe that all the data that was collected has been organized under three columns. Thus, a frequency distribution table is a chart summarizing the values and their frequencies. In other words, it is a tool to organize data. This makes it easy for us to understand the given set of information.

Thus, the frequency distribution table in statistics helps us to condense data in a simpler form so that it is easy for us to observe its features at a glance.

How to Construct a Frequency Distribution Table?

It is easy to make a frequency distribution table by using the steps given below:

• Step 1: Make a table with two columns - one with the title of the data you are organizing and the other column will be for frequency. [Draw three columns if you want to add tally marks too]
• Step 2: Look at the items written in the data and decide whether you want to draw an ungrouped frequency distribution table or a grouped frequency distribution table. If there are too many different values, then it is usually better to go with the grouped frequency distribution table.
• Step 3: Write the data set values in the first column.
• Step 4: Count how many times each item is repeating itself in the collected data. In other words, find the frequency of each item by counting.
• Step 5: Write the frequency in the second column corresponding to each item.
• Step 6: At last you can also write the total frequency in the last row of the table.

Let's look at an example. Ms. Jennifer is a teacher. She wants to look at the marks obtained by the students of her class in the last exam. She does not have the time to go through each test paper individually to see the marks. Thus, she asks Mr. Thomas to organize the data in a table so that it is easier for her to look at everyone's marks together. Ms. Jennifer suggests using a frequency distribution table to organize the data, so as to get a better picture of the data rather than using a simple list.

Using a frequency distribution table here is a good way to present the data as it will show Ms. Jennifer all the students' marks in one table. But how can a frequency distribution table be created? Mr. Thomas works hard to put together all the data. The following table shows the test scores of 20 students, i.e., for one class.

The frequency distribution table drawn above is called an ungrouped frequency distribution table . It is the representation of ungrouped data and is typically used when you have a smaller data set. Imagine how difficult it would be to create a similar table if you have a large number of observations, for example, the marks of students of three classes. The table we will get will be quite lengthy and the data will be confusing.

Hence, in such cases, we form class intervals to tally the frequency for the data that belongs to that specific class interval. To make such a frequency distribution table, first, write the class intervals in one column. Next, tally the numbers in each category based on the number of times it appears. Finally, write the frequency in the final column.

A frequency distribution table drawn above is called a grouped frequency distribution table .

What is Frequency Distribution Table in Statistics?

Frequency distribution in statistics is a representation of data displaying the number of observations within a given interval. The representation of a frequency distribution can be graphical or tabular. Now let us look at another way to represent data i.e., graphical representation of data. This is done using a frequency distribution table graph. Such graphs make it easier to understand the collected data.

• Bar graphs represent data using bars of uniform width with equal spacing between them.
• A pie chart shows a whole circle, divided into sectors where each sector is proportional to the information it represents.
• A frequency polygon is drawn by joining the mid-points of the bars in a histogram .

Frequency Distribution Table for Grouped Data

A frequency distribution table for grouped data is known as a grouped frequency distribution table. It is based on the frequencies of class intervals. As it is already discussed above that in this table, all the categories of data are divided into different class intervals of the same width, for example, 0-10, 10-20, 20-30, etc. And then the frequency of that class interval is marked against each interval. Look at an example of the frequency distribution table for grouped data given in the image below.

Cumulative Frequency Distribution Table

Cumulative frequency means the sum of frequencies of the class and all the classes below it. It is calculated by adding the frequency of each class lower than the corresponding class interval or category. An example of a cumulative frequency distribution table is given below:

Cumulative frequency distribution table calculators save a lot of time when tabulating the data. It makes calculations easy and leads to the organization of data in seconds.

Frequency Distribution Table Related Articles

Check these articles related to the concept of a frequency distribution table in math.

• Frequency Distribution
• Frequency Distribution Formula
• Cumulative Frequency
• How To Find Relative Frequency

Frequency Distribution Table Examples

Example 1: A school conducted a blood donation camp. The blood groups of 30 students were recorded as follows.

A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O

Represent this data in the form of a frequency distribution table.

Solution: The above data can be represented in a frequency distribution table as follow:

Example 2: Given below are the weekly pocket expenses (in $) of a group of 25 students selected at random.

37, 41, 39, 34, 41, 26, 46, 31, 48, 32, 44, 39, 35, 39, 37, 49, 27, 37, 33, 38, 49, 45, 44, 37, 36

Construct a grouped frequency distribution table with class intervals of equal widths, starting from 25 - 30, 30 - 35, and so on. Also, find the range of weekly pocket expenses.

Solution: The following table represents the given data:

In the given data, the smallest value is 26 and the largest value is 49. So, the range of the weekly pocket expenses = 49 - 26 = $23.

Example 3: Silvia and Ashley have a set of number cards with numbers from 1 to 10. They take out a number card and write the number that comes up. They continue doing the same at least 12 times. They get the following values:

5, 8, 9, 2, 3, 7, 3, 4, 5, 9, 3, 1

Construct a frequency table to arrange the data in better form.

go to slide go to slide go to slide

Book a Free Trial Class

Practice Questions on Frequency Distribution Table

go to slide go to slide

FAQs on Frequency Distribution Table

What is frequency distribution table.

A frequency distribution table is a tabular representation of the frequencies of the categories given. It represents the data in an organized manner that is useful for the graphical representation of data or to calculate mean, median, and mode , variance , etc. It has generally two columns, one is of the categories of data set, and the other one is of the frequency of each category. Sometimes, a tally marks column is also added before frequency that helps to count the frequency.

What is the Use of a Frequency Distribution Table?

A frequency distribution table is useful to perform calculations on the given data. It involves calculations involving measures of central tendency , variance, statistical tests, and analysis. Apart from that, a frequency distribution table is useful to represent the data in a neat manner that is easy to understand.

How to Make an Ungrouped Frequency Distribution Table?

To make an ungrouped frequency distribution table, follow the steps given below:

• Identify all the categories that are given in the data.
• Draw a table with two columns - one is of categories and another is of their respective frequencies. Draw three columns if you want to add tally marks too.
• Write each category in a separate row in column 1.
• Count the number of times they are occurring or repeating themselves in the collected data.
• Write those frequencies for each category in column 2.

What is Grouped Frequency Distribution Table?

A grouped frequency distribution table is a table that represents categories in the form of class intervals. It is mainly used with large data sets.

What is cf in Frequency Distribution Table?

In a frequency distribution table, cf means cumulative frequency. Cf represents the collective or total frequency of a category and all the categories lower or greater than that.

How to Interpret Frequency Distribution Table?

The following points must be kept in mind while interpreting a frequency distribution table:

• The first column is usually for the categories of the data set and the second or third column is usually for the frequency of each category.
• The number written on the right of each category is its frequency. It lies in the same row.
• There are no other category lies except the ones written in the first column of the table.

How to Draw Frequency Distribution Table?

There are mainly two or three columns in a frequency distribution table - column 1 for categories, column 2 for tally marks, and column 3 for frequency. So, to draw a frequency distribution table, we have to write data in this order only. First, we identify all the categories or class intervals, then we write them in separate rows in column 1. After that, we focus on each category one by one and count their frequencies. We write their respective frequency in the third column. This is how we can draw a frequency distribution table.

How to Get Class Boundary in Frequency Distribution Table?

A class boundary is a number that separates the class intervals without leaving any gaps. For example, if the two subsequent class intervals are given as 20-29 and 30-39. The class boundary is calculated as (upper limit of the first class interval + lower limit of the second class interval)/2. So, here class boundary = (29+30)/2, which is equal to 29.5.

What are the Types of Frequency Distribution Table?

There are mainly three types of frequency distribution table, which are given below:

• Ungrouped frequency distribution table
• Grouped frequency distribution table
• Cumulative frequency distribution table

• Math Article
• Frequency Distribution Table

Frequency Distribution Table Statistics

Frequency Distribution Table – Data Collection

In our day to day life, recording information is very crucial. A piece of information or representation of facts or ideas which can be further processed is known as data. The weather forecast, maintenance of records, dates, time, and everything is related to data collection.

The collection, presentation, analysis, organization and interpretation of observations or data is known as statistics. We can make predictions about the nature of data based on the previous data using statistics. Statistics are helpful when a large amount of data is to be studied and observed.

The collected statistical data can be represented by various methods such as tables, bar graphs, pie charts, histograms, frequency polygons, etc.

In the upcoming discussion, data collection through a frequency distribution table is discussed. 

What is Frequency Distribution Table in Statistics?

In statistics, a frequency distribution table is a comprehensive way of representing the organisation of raw data of a quantitative variable. This table shows how various values of a variable are distributed and their corresponding frequencies. However, we can make two frequency distribution tables:

(i) Discrete frequency distribution 

(ii) Continuous frequency distribution (Grouped frequency distribution)

How to Make a Frequency distribution table?

Frequency distribution tables can be made using tally marks for both discrete and continuous data values. The way of preparing discrete frequency tables and continuous frequency distribution tables are different from each other.

In this section, you will learn how to make a discrete frequency distribution table with the help of examples.

Suppose the runs scored by the 11 players of the Indian cricket team in a match are given as follows:

\(\begin{array}{l}25, 65, 03, 12, 35, 46, 67, 56, 00, 31, 17\end{array} \)

This type of data is in raw form and is known as raw data. The difference between the measure of highest and lowest value in a collection of data is known as the range. Here, the range is- \(\begin{array}{l}|67 – 00|, i.e. 67\end{array} \)

When the number of observations increases, this type of representation is quite hectic, and the calculations could be quite complex. As statistics is about the presentation of data in an organized form, the data representation in tabular form is more convenient.

Considering another example: In a quiz, the marks obtained by 20 students out of 30 are given as:

\(\begin{array}{l}12, 15, 15, 29, 30, 21, 30, 30, 15, 17, 19, 15, 20, 20, 16, 21, 23, 24, 23, 21\end{array} \)

This data can be represented in tabular form as follows:

Table 1: Frequency Distribution Table (Ungrouped)

​​The number of times data occurs in a data set is known as the frequency of data. In the above example, frequency is the number of students who scored various marks as tabulated. This type of tabular data collection is known as an ungrouped frequency table.

What happens if, instead of 20 students, 200 students took the same test. Would it have been easy to represent such data in the format of an ungrouped frequency distribution table? Well, obviously no. To represent a vast amount of information, the data is subdivided into groups of similar sizes known as class or class intervals, and the size of each class is known as class width or class size.

Frequency Distribution table for Grouped data

The frequency distribution table for grouped data is also known as the continuous frequency distribution table. This is also known as the grouped frequency distribution table. Here, we need to make the frequency distribution table by dividing the data values into a suitable number of classes and with the appropriate class height. Let’s understand this with the help of the solved example given below:

The heights of 50 students, measured to the nearest centimetres, have been found to be as follows: 

161, 150, 154, 165, 168, 161, 154, 162, 150, 151, 162, 164, 171, 165, 158, 154, 156, 172, 160, 170, 153, 159, 161, 170, 162, 165, 166, 168, 165, 164, 154, 152, 153, 156, 158, 162, 160, 161, 173, 166, 161, 159, 162, 167, 168, 159, 158, 153, 154, 159 

(i) Represent the data given above by a grouped frequency distribution table, taking the class intervals as 160 – 165, 165 – 170, etc. 

(ii) What can you conclude about their heights from the table?

(i) Let us make the grouped frequency distribution table with classes:

 150 – 155, 155 – 160, 160 – 165, 165 – 170, 170 – 175

Class intervals and the corresponding frequencies are tabulated as:

(ii) From the given data and above table, we can observe that 35 students, i.e. more than 50% of the total students, are shorter than 165 cm. 

Practice Problems

• The scores (out of 100) obtained by 33 students in a mathematics test are as follows: 69, 48, 84, 58, 48, 73, 83, 48, 66, 58, 84000 66, 64, 71, 64, 66, 69, 66, 83, 66, 69, 71 81, 71, 73, 69, 66, 66, 64, 58, 64, 69, 69 Represent this data in the form of a frequency distribution.
• The following are the marks (out of 100) of 60 students in mathematics. 16, 13, 5, 80, 86, 7, 51, 48, 24, 56, 70, 19, 61, 17, 16, 36, 34, 42, 34, 35, 72, 55, 75, 31, 52, 28,72, 97, 74, 45, 62, 68, 86, 35, 85, 36, 81, 75, 55, 26, 95, 31, 7, 78, 92, 62, 52, 56, 15, 63,25, 36, 54, 44, 47, 27, 72, 17, 4, 30. Construct a grouped frequency distribution table with width 10 of each class starting from 0 – 9.
• The value of π up to 50 decimal places is given below: 3.14159265358979323846264338327950288419716939937510 (i) Make a frequency distribution of the digits from 0 to 9 after the decimal point. (ii) What are the most and the least frequently occurring digits?

To keep learning please visit our website www.byjus.com and download BYJU’S-The Learning App from Google Play Store.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

• Frequency Distribution

By now we all the concept of frequency of data. But what is the meaning of frequency of a group of data and what is frequency distribution ? This lesson simplifies frequency distribution table for both, grouped and ungrouped data using simple examples.

Suggested Videos

Organization of data.

Statistics refers to the collection, organization , distribution, and interpretation of data or a set of observations. It is useful in understanding what a dataset reveals about a particular phenomenon. Trends can be studied and results can be drawn from data interpretation . Hence, statistics is a very useful tool to study data.

Download the Cheat Sheet of Statistics by clicking on the button below

Browse more Topics under Statistics

• Bar Graphs and Histogram
• Cumulative Frequency Curve
• Frequency Polygon
• Range and Mean Deviation
• Range and Mean Deviation for Grouped Data
• Range and Mean Deviation for Ungrouped Data
• Variance and Standard Deviation

Frequency Distribution: Introduction

To understand frequency distribution, let us first start with a simple example. We consider the marks obtained by ten students from a class in a test to be given as follows:

23, 26, 11, 18, 09, 21, 23, 30, 22, 11

This form of data is known as raw data . A statistical measure called range can be defined. It is the difference between the largest and smallest values of a data set. Here, range = 30 – 09 = 21.

Frequency Distribution Table

Now, imagine how difficult and cumbersome this process would get if there were a larger number of observations. If we were to include the test scores of all 20 students in this class, it would be very difficult to understand and interpret such data unless it is ‘organized’.

The objective of statistical interpretation is to organize data into a concise form so that interpretation and analysis become easy. It is for this reason that we organize larger data into a table called the frequency distribution table .

Ungrouped Data

Let the test scores of all 20 students be as follows:

23, 26, 11, 18, 09, 21, 23, 30, 22, 11, 21, 20, 11, 13, 23, 11, 29, 25, 26, 26

Note that the term frequency refers to the number of times an observation occurs or appears in a data set. Hence, in case of repetitions, the frequency increases. The table below will help you understand this better:

In the example above, the frequency refers to the number of students getting a particular mark in the test. Also, note that your frequency must always total the number of observations after tallying. Here, the total we have obtained after tallying the test scores of the students is 20 which is also the number of observations given.

Read more about Bar Graphs and Histograms here in detail.

A frequency distribution such as the one above is called an ungrouped frequency distribution table . It takes into account ungrouped data and calculates the frequency for each observation singularly.

Grouped Data

Now consider the situation where we want to collect data on the test scores of five such classes i.e. of 100 students. It becomes difficult to tally for each and every score of all 100 students. Besides, the table we will obtain will be very large in length and not understandable at once. In this case, we use what is called a grouped frequency distribution table .

Learn more about Range and Mean for Grouped Data here in detail.

Such tables take into consideration groups of data in the form of class intervals to tally the frequency for the data that belongs to that particular class interval.

Take a look at the table below to understand the concept better:

The first column here represents the marks obtained in class interval form. The lowest number in a class interval is called the lower limit and the highest number is called the upper limit . This example is a case of continuous class intervals as the upper limit of one class is the lower limit of the following class.

Note that in continuous cases, any observation corresponding to the extreme values of a class is always included in that class where it is the lower limit. For example, if we had a student who has scored 5 marks in the test, his marks would be included in the class interval 5-10 and not 0-5.

Analogous to continuous class intervals are disjoint class intervals . An example of such as case would be 0-4, 5-9, 10-14, and so on. The frequency distribution can be done for disjoint data as well, similar to how it is done above.

Solved Example for You

Question 1: The following is the distribution for the age of the students in a school:

• The lower limit of the first class interval.
• The class limits of the third class.
• The classmark for the interval 5-10.
• The class size.
• The lower limit of the first class interval i.e. 0-5 is ‘0’.
• The class limits of the third class, i.e. 10-15 are 10 (lower limit) and 15 (upper limit).
• The classmark is defined as the average of the upper and lower limits of a class. For 5-10, the classmark is (5+10)/2 = 7.5
• The class size is the difference between the lower and upper class-limits. Here, we have a uniform class size, which is equal to 5 (5 – 0, 10 – 5, 15 – 10, 20 – 15 are all equal to 5).

Question 2: Discuss the differences between the frequency table and the frequency distribution table?

Answer: The frequency table is said to be a tabular method where each part of the data is assigned to its corresponding frequency. Whereas, a frequency distribution is generally the graphical representation of the frequency table.

Question 3: What are the numerous types of frequency distributions?

Answer: Different types of frequency distributions are as follows:

• Grouped frequency distribution.
• Ungrouped frequency distribution.
• Cumulative frequency distribution.
• Relative frequency distribution.
• Relative cumulative frequency distribution, etc.

Question 4: What are some characteristics of the frequency distribution?

Answer: Some major characteristics of the frequency distribution are given as follows:

• Measures of central tendency and location i.e. mean, median, and mode.
• Measures of dispersion i.e. range, variance, and the standard deviation.
• The extent of the symmetry or asymmetry i.e. skewness.
• The flatness or the peakedness i.e. kurtosis.

Question 5: What is the importance of frequency distribution?

Answer: The value of the frequency distributions in statistics is excessive. A well-formed frequency distribution creates the possibility of a detailed analysis of the structure of the population. So, the groups where the population breaks down are determinable.

Customize your course in 30 seconds

Which class are you in.

• Statistics in Maths
• How to Find Range of Data Set with Examples

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Download the App

• School Guide
• Mathematics
• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Maths Formulas
• Class 8 Maths Notes
• Class 9 Maths Notes
• Class 10 Maths Notes
• Class 11 Maths Notes
• Class 12 Maths Notes

Frequency Distribution – Table, Graphs, Formula

Frequency Distribution is a tool in statistics that helps us organize the data and also helps us reach meaningful conclusions. It tells us how often any specific values occur in the dataset. A frequency distribution in a tabular form organizes data by showing the frequencies (the number of times values occur) within a dataset.

A frequency distribution represents the pattern of how frequently each value of a variable appears in a dataset. It shows the number of occurrences for each possible value within the dataset.

Let’s learn about Frequency Distribution including its definition, graphs, solved examples, and frequency distribution table in detail .

Frequency Distribution

Table of Content

What is Frequency Distribution in Statistics?

Frequency distribution graphs, frequency distribution table.

• Types of Frequency Distribution Table
• Frequency Distribution Table for Grouped Data
• Frequency Distribution Table for Ungrouped Data

Types of Frequency Distribution

Grouped frequency distribution, ungrouped frequency distribution, relative frequency distribution, cumulative frequency distribution, frequency distribution curve, frequency distribution formula, frequency distribution examples.

A frequency distribution is an overview of all values of some variable and the number of times they occur. It tells us how frequencies are distributed over the values. That is how many values lie between different intervals. They give us an idea about the range where most values fall and the ranges where values are scarce. 

To represent the Frequency Distribution, there are various methods such as Histogram, Bar Graph, Frequency Polygon, and Pie Chart.

A brief description of all these graphs is as follows:

A frequency distribution table is a way to organize and present data in a tabular form which helps us summarize the large dataset into a concise table. In the frequency distribution table, there are two columns one representing the data either in the form of a range or an individual data set and the other column shows the frequency of each interval or individual.

For example, let’s say we have a dataset of students’ test scores in a class.

Check: Difference between Frequency Array and Frequency Distribution

There are four types of frequency distribution :

In Grouped Frequency Distribution observations are divided between different intervals known as class intervals and then their frequencies are counted for each class interval. This Frequency Distribution is used mostly when the data set is very large.

Example: Make the Frequency Distribution Table for the ungrouped data given as follows:

23, 27, 21, 14, 43, 37, 38, 41, 55, 11, 35, 15, 21, 24, 57, 35, 29, 10, 39, 42, 27, 17, 45, 52, 31, 36, 39, 38, 43, 46, 32, 37, 25

As there are observations in between 10 and 57, we can choose class intervals as 10-20, 20-30, 30-40, 40-50, and 50-60. In these class intervals all the observations are covered and for each interval there are different frequency which we can count for each interval.

Thus, the Frequency Distribution Table for the given data is as follows:

In Ungrouped Frequency Distribution, all distinct observations are mentioned and counted individually. This Frequency Distribution is often used when the given dataset is small.

10, 20, 15, 25, 30, 10, 15, 10, 25, 20, 15, 10, 30, 25

As unique observations in the given data are only 10, 15, 20, 25, and 30 with each having a different frequency.

Thus the Frequency Distribution Table of the given data is as follows:

This distribution displays the proportion or percentage of observations in each interval or class. It is useful for comparing different data sets or for analyzing the distribution of data within a set.

Relative Frequency is given by:

Relative Frequency = (Frequency of Event)/(Total Number of Events)

Example: Make the Relative Frequency Distribution Table for the following data:

To Create the Relative Frequency Distribution table, we need to calculate Relative Frequency for each class interval. Thus Relative Frequency Distribution table is given as follows: Score Range Frequency Relative Frequency 0-20 5 5/50 = 0.10 21-40 10 10/50 = 0.20 41-60 20 20/50 = 0.40 61-80 10 10/50 = 0.20 81-100 5 5/50 = 0.10 Total 50 1.00

Cumulative frequency is defined as the sum of all the frequencies in the previous values or intervals up to the current one. The frequency distributions which represent the frequency distributions using cumulative frequencies are called cumulative frequency distributions . There are two types of cumulative frequency distributions:

Less than Type: We sum all the frequencies before the current interval. More than Type: We sum all the frequencies after the current interval.

• Cumulative Frequency
• How to Calculate Cumulative Frequency table in Excel

Let’s see how to represent a cumulative frequency distribution through an example, 

Example: The table below gives the values of runs scored by Virat Kohli in the last 25 T-20 matches. Represent the data in the form of less-than-type cumulative frequency distribution: 

Since there are a lot of distinct values, we’ll express this in the form of grouped distributions with intervals like 0-10, 10-20 and so. First let’s represent the data in the form of grouped frequency distribution.  Runs Frequency 0-10 2 10-20 2 20-30 1 30-40 4 40-50 4 50-60 5 60-70 1 70-80 3 80-90 2 90-100 1 Now we will convert this frequency distribution into cumulative frequency distribution by summing up the values of current interval and all the previous intervals.  Runs scored by Virat Kohli Cumulative Frequency Less than 10 2 Less than 20 4 Less than 30 5 Less than 40 9 Less than 50 13 Less than 60 18 Less than 70 19 Less than 80 22 Less than 90 24 Less than 100 25 This table represents the cumulative frequency distribution of less than type.  Runs scored by Virat Kohli Cumulative Frequency More than 0 25 More than 10 23 More than 20 21 More than 30 20 More than 40 16 More than 50 12 More than 60 7 More than 70 6 More than 80 3 More than 90 1 This table represents the cumulative frequency distribution of more than type. We can plot both the type of cumulative frequency distribution to make the Cumulative Frequency Curve .

A frequency distribution curve, also known as a frequency curve, is a graphical representation of a data set’s frequency distribution. It is used to visualize the distribution and frequency of values or observations within a dataset. Let’s understand it’s different types based on the shape of it, as follows:

Check: Grouping of Data

There are various formulas which can be learned in the context of Frequency Distribution, one such formula is the coefficient of variation. This formula for Frequency Distribution is discussed below in detail.

Coefficient of Variation

We can use mean and standard deviation to describe the dispersion in the values. But sometimes while comparing the two series or frequency distributions becomes a little hard as sometimes both have different units.

The coefficient of Variation is defined as, 

[Tex]\bold{\frac{\sigma}{\bar{x}} \times 100} [/Tex] Where, σ represents the standard deviation [Tex]\bold{\bar{x}}[/Tex]  represents the mean of the observations

Note: Data with greater C.V. is said to be more variable than the other. The series having lesser C.V. is said to be more consistent than the other.

Comparing Two Frequency Distributions with the Same Mean

We have two frequency distributions. Let’s say  [Tex]\sigma_{1} \text{ and } \bar{x}_1[/Tex]  are the standard deviation and mean of the first series and  [Tex]\sigma_{2} \text{ and } \bar{x}_2[/Tex]  are the standard deviation and mean of the second series. The Coefficeint of Variation(CV) is calculated as follows

C.V of first series =  [Tex]\frac{\sigma_1}{\bar{x}_1} \times 100 [/Tex]

C.V of second series =  [Tex]\frac{\sigma_2}{\bar{x}_2} \times 100 [/Tex]

We are given that both series have the same mean, i.e.,

[Tex]\bar{x}_2 = \bar{x}_1 = \bar{x} [/Tex]

So, now C.V. for both series are, 

C.V. of the first series =  [Tex] \frac{\sigma_1}{\bar{x}} \times 100[/Tex] C.V. of the second series =  [Tex]\frac{\sigma_2}{\bar{x}} \times 100[/Tex]

Notice that now both series can be compared with the value of standard deviation only. Therefore, we can say that for two series with the same mean, the series with a larger deviation can be considered more variable than the other one.

Example 1: Suppose we have a series, with a mean of 20 and a variance is 100. Find out the Coefficient of Variation. 

We know the formula for Coefficient of Variation,  [Tex]\frac{\sigma}{\bar{x}} \times 100 [/Tex] Given mean  [Tex]\bar{x}[/Tex]  = 20 and variance  [Tex]\sigma^2[/Tex]  = 100.  Substituting the values in the formula, [Tex]\frac{\sigma}{\bar{x}} \times 100 \\ = \frac{20}{\sqrt{100}} \times 100 \\ = \frac{20}{10} \times 100 \\ = 200 [/Tex]

Example 2: Given two series with Coefficients of Variation 70 and 80. The means are 20 and 30. Find the values of standard deviation for both series.

In this question we need to apply the formula for CV and substitute the given values.  Standard Deviation of first series.  [Tex]C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 70 = \frac{\sigma}{20} \times 100 \\ 1400 = \sigma \times 100 \\ 14 = \sigma  [/Tex] Thus, the standard deviation of first series = 14 Standard Deviation of second series.  [Tex]C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 80 = \frac{\sigma}{30} \times 100 \\ 2400 = \sigma \times 100 \\ 24 = \sigma  [/Tex] Thus, the standard deviation of first series = 24

Example 3: Draw the frequency distribution table for the following data: 

2, 3, 1, 4, 2, 2, 3, 1, 4, 4, 4, 2, 2, 2.

Since there are only very few distinct values in the series, we will plot the ungrouped frequency distribution.  Value  Frequency 1 2 2 6 3 2 4 4 Total  14

Example 4: The table below gives the values of temperature recorded in Hyderabad for 25 days in summer. Represent the data in the form of less-than-type cumulative frequency distribution: 

Since there are so many distinct values here, we will use grouped frequency distribution. Let’s say the intervals are 20-25, 25-30, 30-35. Frequency distribution table can be made by counting the number of values lying in these intervals.  Temperature Number of Days 20-25 2 25-30 10 30-35 13 This is the grouped frequency distribution table. It can be converted into cumulative frequency distribution by adding the previous values.  Temperature Number of Days Less than 25 2 Less than 30 12 Less than 35 25

Example 5: Make a Frequency Distribution Table as well as the curve for the data:

{45, 22, 37, 18, 56, 33, 42, 29, 51, 27, 39, 14, 61, 19, 44, 25, 58, 36, 48, 30, 53, 41, 28, 35, 47, 21, 32, 49, 16, 52, 26, 38, 57, 31, 59, 20, 43, 24, 55, 17, 50, 23, 34, 60, 46, 13, 40, 54, 15, 62}.

To create the frequency distribution table for given data, let’s arrange the data in ascending order as follows: {13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62} Now, we can count the observations for intervals: 10-20, 20-30, 30-40, 40-50, 50-60 and 60-70. Interval Frequency 10 – 20 7 20 – 30 10 30 – 40 10 40 – 50 10 50 – 60 10 60 – 70 3 From this data, we can plot the Frequency Distribution Curve as follows:

Statistics Data Handling Probability Distribution Variance and Standard Deviation

Conclusion – Frequency Distribution

The frequency distribution provides a clear summary of how often each value or category occurs within a dataset. It allows us to see the distribution of values and understand the pattern or spread of data. By organizing data into groups and displaying their frequencies, we gain insights into the central tendency, variability, and shape of the data distribution.

This facilitates better understanding and interpretation of the dataset, aiding in decision-making, analysis, and communication of findings.

Frequency Distribution- FAQs

Define frequency distribution in statistics.

A frequency distribution is a table or graph that displays the frequency of various outcomes or values in a sample or population. It shows the number of times each value occurs in the data set.

How Can I Construct a Frequency Distribution?

To construct a frequency distribution: Organize the data Decide on the number of classes Calculate the class width Create class intervals Count the frequencies for each interval Create a frequency table Optionally, visualize the data with graphs like histograms or bar charts

What are Types of Frequency Distribution?

There are four types of frequency distributions that are as follows: Grouped Frequency Distribution Ungrouped Frequency Distribution Relative Frequency Distribution Cumulative Frequency Distribution

What is Ungrouped Frequency Distribution?

An ungrouped frequency distribution is a distribution that shows the frequency of each individual value in a data set.

What is Grouped Frequency Distribution?

A grouped frequency distribution is a distribution that shows the frequency of values within specified intervals or classes.

What is Frequency Count Distribution?

Frequency count distribution is a way of organizing and displaying data to show how often each unique value (or range of values) appears in a dataset

What is Relative Frequency Distribution?

A relative frequency distribution is a distribution that shows the proportion or percentage of values within each interval or class.

What is Cumulative Frequency Distribution?

A cumulative frequency distribution is a distribution that shows the number or proportion of values that fall below a certain value or interval.

Please Login to comment...

Similar reads.

• School Learning
• Maths-Class-11

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

Grouped Frequency Distribution

Frequency is how often something occurs.

Example: Sam played football on:

• Saturday Morning,
• Saturday Afternoon
• Thursday Afternoon

The frequency was 2 on Saturday, 1 on Thursday and 3 for the whole week.

Frequency Distribution

By counting frequencies we can make a Frequency Distribution table.

Example: Newspapers

These are the numbers of newspapers sold at a local shop over the last 10 days:

22, 20, 18, 23, 20, 25, 22, 20, 18, 20

Let us count how many of each number there is:

It is also possible to group the values. Here they are grouped in 5s:

We just saw how we can group frequencies. It is very useful when the scores have many different values.

Example: Leaves

Alex measured the lengths of leaves on the oak tree (to the nearest cm):

9,16,13,7,8,4,18,10,17,18,9,12,5,9,9,16,1,8,17,1, 10,5,9,11,15,6,14,9,1,12,5,16,4,16,8,15,14,17

Let's try to group them, but what groups should we use?

To get started, put the numbers in order , then find the smallest and largest values in your data, and calculate the range (range = largest - smallest).

Example: Leaves (continued)

In order the lengths are:

1,1,1,4,4,5,5,5,6,7,8,8,8,9,9,9,9,9,9,10,10,11,12,12, 13,14,14,15,15,16,16,16,16,17,17,17,18,18

The smallest value (the "minimum") is 1 cm

The largest value (the "maximum") is 18 cm

The range is 18−1 = 17 cm

Now calculate an approximate group size, by dividing the range by how many groups you would like.

Then round that group size up to some simple value (like 2 instead of 1.83 or 5 instead of 4.26).

Let us say we want about 5 groups.

Divide the range by 5:

Then round that up to 4

Start Value

Pick a starting value that is less than or equal to the smallest value. Try to make it a multiple of the group size if you can.

In our case a start value of 0 makes the most sense.

Now calculate the list of groups. (We must go up to or past the largest value).

Starting at 0 and with a group size of 4 we get: 0, 4, 8, 12, 16

Write down the groups.

Include the end value of each group that must be less than the next group :

The last group goes to 19 which is greater than the largest value. That is OK: the main thing is that it must include the largest value.

(Note: If you don't like the groups, then go back and change the group size or starting value and try again.)

Upper and Lower Values For Each Group

Even though Alex only measured in whole numbers, the data is continuous , so "4 cm" means the actual value could have been anywhere from 3.5 cm to 4.5 cm. Alex just rounded the numbers to whole centimeters.

Here are the groups with the Lower and Upper limits shown:

Tally and Total

Now tally the results to find the frequencies. And do a total.

1,1,1,4,4,5,5,5,6,7,8,8,8,9,9,9,9,9,9,10,10,11,12,12, 13,14,14,15,15,16,16,16,16,17,17,17,18,18:

You might also like to make a Histogram of your data.

Assignment on frequency distribution PDF

Added on   2021-06-16

End of preview

Want to access all the pages? Upload your documents or become a member.

H16007 Statistics : Assignment lg ...

Symmetrically shaped graph assignment pdf lg ..., statistics hi6007 group - assignment lg ..., statistics group assignment lg ..., statistics: anova, regression analysis and hypothesis testing lg ..., statistics: examination scores, sample size, anova, regression model lg ....