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 .
Table of Content
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:
| Graph Type | Description | Use Cases |
|---|---|---|
| Histogram | Represents the frequency of each interval of continuous data using bars of equal width. | Continuous data distribution analysis. |
| Bar Graph | Represents the frequency of each interval using bars of equal width; can also represent discrete data. | Comparing discrete data categories. |
| Frequency Polygon | Connects midpoints of class frequencies using lines, similar to a histogram but without bars. | Comparing various datasets. |
| Pie Chart | Circular graph showing data as slices of a circle, indicating the proportional size of each slice relative to the whole dataset. | Showing relative sizes of data portions. |
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.
Test Score
Frequency
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
Solution:
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.
Example: Make the Frequency Distribution Table for the ungrouped data given as follows:
10, 20, 15, 25, 30, 10, 15, 10, 25, 20, 15, 10, 30, 25
Solution:
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:
| Score Range | 0-20 | 21-40 | 41-60 | 61-80 | 81-100 |
|---|---|---|---|---|---|
| Frequency | 5 | 10 | 20 | 10 | 5 |
Solution:
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:
Check:
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:
| 45 | 34 | 50 | 75 | 22 |
| 56 | 63 | 70 | 49 | 33 |
| 0 | 8 | 14 | 39 | 86 |
| 92 | 88 | 70 | 56 | 50 |
| 57 | 45 | 42 | 12 | 39 |
Solution:
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:
Frequency Distribution Curve Types
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.
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,
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.
We have two frequency distributions. Let’s say [Tex]\sigma_ \text < and >\bar_1[/Tex] are the standard deviation and mean of the first series and [Tex]\sigma_ \text < and >\bar_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_1> \times 100 [/Tex]
C.V of second series = [Tex]\frac_2> \times 100 [/Tex]
We are given that both series have the same mean, i.e.,
[Tex]\bar_2 = \bar_1 = \bar [/Tex]
So, now C.V. for both series are,
C.V. of the first series = [Tex] \frac> \times 100[/Tex]
C.V. of the second series = [Tex]\frac> \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.
Solution:
We know the formula for Coefficient of Variation,
[Tex]\frac> \times 100 [/Tex]
Given mean [Tex]\bar[/Tex] = 20 and variance [Tex]\sigma^2[/Tex] = 100.
Substituting the values in the formula,
[Tex]\frac> \times 100 \\ = \frac> \times 100 \\ = \frac \times 100 \\ = 200 [/Tex]
Solution:
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> \times 100 \\ 70 = \frac \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> \times 100 \\ 80 = \frac \times 100 \\ 2400 = \sigma \times 100 \\ 24 = \sigma [/Tex]
Thus, the standard deviation of first series = 24
Solution:
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 |
| 37 | 34 | 36 | 27 | 22 |
| 25 | 25 | 24 | 26 | 28 |
| 30 | 31 | 29 | 28 | 30 |
| 32 | 31 | 28 | 27 | 30 |
| 30 | 32 | 35 | 34 | 29 |
Solution:
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 |
Solution:
To create the frequency distribution table for given data, let’s arrange the data in ascending order as follows:
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:
Read More:
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.
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.
An ungrouped frequency distribution is a distribution that shows the frequency of each individual value in a data set.
A grouped frequency distribution is a distribution that shows the frequency of values within specified intervals or classes.
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
A relative frequency distribution is a distribution that shows the proportion or percentage of values within each interval or class.
A cumulative frequency distribution is a distribution that shows the number or proportion of values that fall below a certain value or interval.
