Creating a Frequency Distribution with 5 Classes

Oct 13, 2025 by wbrum.org 106 views

Introduction

In the field of mathematics, frequency distribution is a crucial concept used to describe the distribution of data. It is a way to organize and summarize a large dataset by grouping the data into classes or categories. In this article, we will explore the concept of frequency distribution using a real-world example of patients' HDL cholesterol levels participating in a study.

What is Frequency Distribution?

Frequency distribution is a graphical representation of the distribution of data. It is a way to display the frequency or number of occurrences of each value in a dataset. The frequency distribution can be represented in various forms, including histograms, bar charts, and frequency polygons.

Creating a Frequency Distribution with 5 Classes

To create a frequency distribution with 5 classes, we need to first determine the range of the data. The range of the data is the difference between the highest and lowest values. In this case, the highest value is 70 and the lowest value is 28.

| Class | Frequency |
| --- | --- |
| 20-30 | 3 |
| 30-40 | 2 |
| 40-50 | 4 |
| 50-60 | 4 |
| 60-70 | 3 |

Step 1: Determine the Range of the Data

The range of the data is the difference between the highest and lowest values. In this case, the highest value is 70 and the lowest value is 28.

Range = Highest Value - Lowest Value
Range = 70 - 28
Range = 42

Step 2: Determine the Class Width

The class width is the difference between the upper and lower limits of each class. In this case, we want to create 5 classes, so we will divide the range by 5.

Class Width = Range / Number of Classes
Class Width = 42 / 5
Class Width = 8.4

Step 3: Determine the Class Limits

The class limits are the upper and lower limits of each class. In this case, we will round the class width to the nearest whole number and use it to determine the class limits.

Class Limits = Lower Limit + (Class Width \* (i - 1))
Class Limits = 20 + (8.4 \* (1 - 1))
Class Limits = 20 + 0
Class Limits = 20

Class Limits = Lower Limit + (Class Width \* i)
Class Limits = 20 + (8.4 \* 2)
Class Limits = 20 + 16.8
Class Limits = 36.8

Class Limits = Lower Limit + (Class Width \* (i - 1))
Class Limits = 36.8 + (8.4 \* (3 - 1))
Class Limits = 36.8 + 16.8
Class Limits = 53.6

Class Limits = Lower Limit + (Class Width \* i)
Class Limits = 53.6 + (8.4 \* 4)
Class Limits = 53.6 + 33.6
Class Limits = 87.2

Class Limits = Lower Limit + (Class Width \* (i - 1))
Class Limits = 20 + (8.4 \* (5 - 1))
Class Limits = 20 + 33.6
Class Limits = 53.6

Class Limits = Lower Limit + (Class Width \* i)
Class Limits = 53.6 + (8.4 \* 6)
Class Limits = 53.6 + 50.4
Class Limits = 104

Step 4: Determine the Frequency of Each Class

The frequency of each class is the number of data points that fall within that class. In this case, we will count the number of data points that fall within each class.

| Class | Frequency |
| --- | --- |
| 20-30 | 3 |
| 30-40 | 2 |
| 40-50 | 4 |
| 50-60 | 4 |
| 60-70 | 3 |

Conclusion

In this article, we explored the concept of frequency distribution using a real-world example of patients' HDL cholesterol levels participating in a study. We determined the range of the data, the class width, and the class limits, and then counted the frequency of each class. The frequency distribution is a useful tool for understanding the distribution of data and can be used in a variety of applications, including statistics and data analysis.

References

[1] "Frequency Distribution." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/topic/frequency-distribution.
[2] "Statistics." Wikipedia, Wikimedia Foundation, 22 Feb. 2023, en.wikipedia.org/wiki/Statistics.

Discussion

The frequency distribution is a useful tool for understanding the distribution of data. It can be used to identify patterns and trends in the data, and to make predictions about future data. In this case, the frequency distribution of the patients' HDL cholesterol levels shows that the majority of the data points fall within the range of 40-60. This suggests that the patients' HDL cholesterol levels are generally within a healthy range.

However, there are some limitations to the frequency distribution. For example, it does not take into account the actual values of the data points, only the frequency of each class. This can make it difficult to understand the underlying patterns and trends in the data.

In addition, the frequency distribution is sensitive to the choice of class limits. If the class limits are too narrow, the frequency distribution may not accurately reflect the distribution of the data. On the other hand, if the class limits are too wide, the frequency distribution may not provide enough detail about the distribution of the data.

Overall, the frequency distribution is a useful tool for understanding the distribution of data, but it should be used in conjunction with other statistical methods to get a more complete picture of the data.

Mathematical Formulation

The frequency distribution can be mathematically formulated as follows:

Let X be a random variable that takes on values in the range [a, b]. Let f(x) be the frequency function of X, which is defined as:

f(x) = P(X = x)

where P(X = x) is the probability that X takes on the value x.

The frequency distribution of X can be represented as:

F(x) = P(X ≤ x)

where F(x) is the cumulative distribution function of X.

The frequency distribution can be used to estimate the probability density function of X, which is defined as:

f(x) = dF(x)/dx

where dF(x)/dx is the derivative of F(x) with respect to x.

Code Implementation

The frequency distribution can be implemented in a variety of programming languages, including Python and R. Here is an example of how to implement the frequency distribution in Python:

import numpy as np
import matplotlib.pyplot as plt

# Define the data
data = np.array([70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62])

# Define the class limits
class_limits = np.array([20, 36.8, 53.6, 87.2])

# Define the frequency of each class
frequency = np.array([3, 2, 4, 3])

# Create a histogram of the data
plt.hist(data, bins=class_limits, weights=frequency, align='left', rwidth=0.8)
plt.xlabel('HDL Cholesterol Level')
plt.ylabel('Frequency')
plt.title('Frequency Distribution of HDL Cholesterol Levels')
plt.show()

Q: What is frequency distribution?

A: Frequency distribution is a graphical representation of the distribution of data. It is a way to display the frequency or number of occurrences of each value in a dataset.

Q: What are the steps to create a frequency distribution?

A: The steps to create a frequency distribution are:

Determine the range of the data.
Determine the class width.
Determine the class limits.
Count the frequency of each class.

Q: What is the range of the data?

A: The range of the data is the difference between the highest and lowest values.

Q: What is the class width?

A: The class width is the difference between the upper and lower limits of each class.

Q: What is the class limit?

A: The class limit is the upper and lower limit of each class.

Q: How do I determine the frequency of each class?

A: The frequency of each class is the number of data points that fall within that class.

Q: What is the purpose of frequency distribution?

A: The purpose of frequency distribution is to understand the distribution of data and to identify patterns and trends in the data.

Q: What are the limitations of frequency distribution?

A: The limitations of frequency distribution are:

It does not take into account the actual values of the data points, only the frequency of each class.
It is sensitive to the choice of class limits.

Q: How do I create a frequency distribution in Python?

A: You can create a frequency distribution in Python using the following code:

import numpy as np
import matplotlib.pyplot as plt

# Define the data
data = np.array([70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62])

# Define the class limits
class_limits = np.array([20, 36.8, 53.6, 87.2])

# Define the frequency of each class
frequency = np.array([3, 2, 4, 3])

# Create a histogram of the data
plt.hist(data, bins=class_limits, weights=frequency, align='left', rwidth=0.8)
plt.xlabel('HDL Cholesterol Level')
plt.ylabel('Frequency')
plt.title('Frequency Distribution of HDL Cholesterol Levels')
plt.show()

Q: How do I create a frequency distribution in R?

A: You can create a frequency distribution in R using the following code:

# Load the necessary libraries
library(ggplot2)

# Define the data
data <- c(70, 56, 48, 48, 53, 52, 66, 48, 36, 49, 28, 35, 58, 62)

# Define the class limits
class_limits <- c(20, 36.8, 53.6, 87.2)

# Define the frequency of each class
frequency <- c(3, 2, 4, 3)

# Create a histogram of the data
ggplot(data.frame(data, frequency), aes(x = data, y = frequency)) +
  geom_bar(stat = "identity") +
  labs(x = "HDL Cholesterol Level", y = "Frequency") +
  theme_classic() +
  theme(axis.text.x = element_text(angle = 90, hjust = 1))

Q: What are some real-world applications of frequency distribution?

A: Some real-world applications of frequency distribution include:

Understanding the distribution of customer purchases in a retail store.
Analyzing the distribution of exam scores in a classroom.
Identifying patterns and trends in stock prices.

Q: What are some common mistakes to avoid when creating a frequency distribution?

A: Some common mistakes to avoid when creating a frequency distribution include:

Not choosing the correct class limits.
Not counting the frequency of each class correctly.
Not taking into account the actual values of the data points.

Q: How do I troubleshoot common issues with frequency distribution?

A: Some common issues with frequency distribution include:

The frequency distribution does not accurately reflect the distribution of the data.
The frequency distribution is sensitive to the choice of class limits.

To troubleshoot these issues, you can try the following:

Check the class limits to ensure that they are correct.
Check the frequency of each class to ensure that it is correct.
Try using a different method to create the frequency distribution, such as using a histogram or a bar chart.