Problem 2 - Basic Statistics for Business and Economics

Question

A set of data consists of 45 observations between \(\$ 0\) and \(\$ 29 .\) What size would you recommend for the class interval?

Step-by-Step Solution

Answer

Use a class interval of size 5.

Step 1: Understanding Data Range

First, identify the range of the data set, which is the difference between the maximum and minimum values. Here, the range is \( 29 - 0 = 29 \) dollars.

Step 2: Choosing Number of Classes

A common rule of thumb for choosing the number of classes is the Sturges' formula: \( k = 1 + 3.322 \, \log_{10}(n) \), where \( n \) is the number of observations. Substituting \( n = 45 \) gives \( k = 1 + 3.322 \, \log_{10}(45) \). Calculating further, \( \log_{10}(45) \approx 1.653 \), so \( k \approx 6.491 \). Since the number of classes must be an integer, we can round this to \( 7 \) classes.

Step 3: Calculating Class Interval Size

To find the class interval size, divide the range by the number of classes. Thus, the interval size \( h = \frac{29}{7} \approx 4.14 \). Rounding to a convenient number, a class interval of \( 5 \) is recommended.

Key Concepts

Sturges' Formula

Sturges' formula is a handy tool used in statistics for determining the appropriate number of classes, or intervals, when listing observational data. This formula helps in creating a histogram or frequency distribution. It strikes a balance between under-simplifying and over-complicating the data representation.

This valuable formula suggests that the number of classes, denoted as \(k\), can be calculated with:

• \( k = 1 + 3.322 \, \log_{10}(n) \)

Where \(n\) represents the number of observations in the dataset.

In our exercise, with 45 observations, we apply Sturges' formula as follows:

• \( k = 1 + 3.322 \, \log_{10}(45) \)
• Calculating, \( \log_{10}(45) \approx 1.653 \), which implies \( k \approx 6.491 \)
• Since \(k\) must be an integer, we round it to 7 classes

Using this method ensures that the data is neither too sparsely nor too finely divided, making analysis and interpretation more accurate and insightful.

Data Range

The data range is a fundamental concept in statistics that provides an overview of the data spread by indicating the difference between the maximum and minimum values. Understanding this concept is crucial for determining how to distribute your data into classes or bins.

In this scenario, we looked at a data set with values ranging from \\(0\ to \\)29\. To find the range, perform a simple subtraction:

• \(29 - 0 = 29\)

This value, 29 dollars, tells us the total extent of our data, essentially how widespread or concentrated the values are across the dataset.

A broad range suggests diverse data, while a smaller range hints at data points very close in value. This information is crucial for the next steps, such as finding the number of classes and the class interval.

Number of Classes

After finding the data range and using Sturges' formula to determine the number of classes, it's essential to consider how these classes influence data readability and analysis.

The number of classes directly affects how we group data points, influencing how a histogram appears or how a frequency distribution is interpreted:

• Having too few classes can oversimplify the data, hiding important variations and trends.
• Conversely, too many classes can complicate the data, making patterns and general tendencies hard to discern and interpret.

Using our calculation, we established that it's ideal to divide our dataset into 7 distinct classes. This gives a balanced representation, making it easier for anyone analyzing the data to notice trends and make informed conclusions.

Choosing the right number of classes is crucial for effective data visualization and accurate interpretation, as it impacts how shared or varied the data points appear within the distribution.