Frequency Distribution

A frequency distribution is a method for organizing data and determining how often each value occurs.

• It shows how many times each value or range of values occurs in a dataset.
• Instead of examining raw, scattered numbers, this approach presents data in a structured table or graph, making patterns, trends, and comparisons easy to identify.

Frequency distributions allow even nontechnical readers, such as founders or investors, to interpret large datasets efficiently. Today, they are considered a core component of data analysis, statistics, and data visualization.

Frequency Distribution Graphs

To represent a frequency distribution, various methods are available, including histograms, Bar Graphs, Frequency Polygons, and Pie Charts.

Frequency Distribution Table

A frequency distribution table is a way to organize and present data in a tabular form, which helps us summarize a 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 showing the frequency of each interval or individual.

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

Types of Frequency Distribution

There are four types of frequency distribution:

1. Grouped Frequency Distribution

In a 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:

Class Interval	Frequency
10 - 20	5
20 - 30	8
30 - 40	12
40 - 50	6
50 - 60	3

2. Ungrouped Frequency Distribution

In an 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:

Value	Frequency
10	4
15	3
20	2
25	3
30	2

3. Relative Frequency Distribution

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:

\text{Relative Frequency} =\frac{\text{Frequency of Event}}{\text{Total number of events}}

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

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

4. Cumulative Frequency Distribution

A cumulative frequency distribution shows the total number of observations up to and including a certain value or class. It helps in understanding how values accumulate across a dataset:

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

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 a less-than-type cumulative frequency distribution: 

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.

Frequency Distribution 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.

1. Normal Distribution: Symmetrical and bell-shaped; data concentrated around the mean.

2. Skewed Distribution: Not symmetric; can be positively skewed (right-tailed) or negatively skewed (left-tailed).

3. Bimodal Distribution: Two distinct peaks or modes in the frequency distribution, suggesting data from different populations.

4. Multimodal Distribution: More than two distinct peaks or modes in the frequency distribution.

5. Uniform Distribution: All values or intervals have roughly the same frequency, resulting in a flat, constant distribution.

6. Exponential Distribution: The rapid drop-off in frequency as values increase, resembling an exponential function.

7. Log-Normal Distribution: The logarithm of the data follows a normal distribution, often used for multiplicative data, positively skewed.

Frequency Distribution Formula

Various formulas 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.

Frequency (f) = Number of times a value occurs

Coefficient of Variation

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

The coefficient of Variation is defined as, 

\bold{\frac{\sigma}{\bar{x}} \times 100}

Where,

• σ represents the standard deviation
• \bold{\bar{x}} represents the mean of the observations

Note: Data with a 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 that \sigma_{1} \text{ and } \bar{x}_1are the standard deviation and mean of the first series, and \sigma_{2} \text{ and } \bar{x}_2are the standard deviation and mean of the second series. The Coefficient of Variation(CV) is calculated as follows

C.V. of the first series = \frac{\sigma_1}{\bar{x}_1} \times 100

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

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

\bar{x}_2 = \bar{x}_1 = \bar{x}

So, now CVs for both series are, 

C.V. of the first series =  \frac{\sigma_1}{\bar{x}} \times 100

C.V. of the second series = \frac{\sigma_2}{\bar{x}} \times 100

Notice that now, both series can be compared with the value of the 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.

Related Articles:

• Probability Distribution
• Variance and Standard Deviation
• Cumulative vs Relative Frequency
• Grouping of Data
• How to Calculate Cumulative Frequency Table in Excel

Solved Examples of Frequency Distribution

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

Solution: 

We know the formula for Coefficient of Variation, 

\frac{\sigma}{\bar{x}} \times 100

Given mean \bar{x} = 20 and variance \sigma^2 = 100. 

We know ,

Standard Deviation \sigma=\sqrt{varience}=\sqrt{100}

Standard Deviation \sigma=10

Substituting the values in the formula,

\frac{\sigma}{\bar{x}} \times 100 \\ = \frac{10}{20} \times 100 \\ = \frac{10}{20} \times 100 \\ = 50

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

Solution: 

In this question we need to apply the formula for CV and substitute the given values. 

Standard Deviation of first series. 

C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 70 = \frac{\sigma}{20} \times 100 \\ 1400 = \sigma \times 100 \\ \sigma=14

Thus, the standard deviation of first series = 14

Standard Deviation of second series. 

C.V = \frac{\sigma}{\bar{x}} \times 100 \\ 80 = \frac{\sigma}{30} \times 100 \\ 2400 = \sigma \times 100 \\ \sigma=24

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

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

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 a less-than-type cumulative frequency distribution: 

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

Example 5: Make a Frequency Distribution Table 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}

Solution:

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