Problem 24 - Basic Statistics for Business and Economics

Question

A dataset consists of 145 observations that range from 56 to \(490 .\) What size class interval would you recommend?

Step-by-Step Solution

Answer

Use a class interval size of 55.

Step 1: Determine the Range

First, calculate the range of the data. The range is the difference between the maximum observation and the minimum observation. Range = Maximum value - Minimum value = 490 - 56 = 434.

Step 2: Determine the Number of Classes

The number of classes can be estimated using Sturge's formula, which is a guideline for the number of bins for a histogram: \[ k = 1 + 3.322 \, \log_{10}(n) \]Where \(n\) is the number of observations.In this case, \( n = 145 \). Thus, \[ k = 1 + 3.322 \, \log_{10}(145) \approx 8.46 \]You can round this to the nearest whole number, so use 8 intervals.

Step 3: Calculate the Class Interval Size

Now calculate the class interval size by dividing the range by the number of classes.Class Interval Size = \( \frac{\text{Range}}{\text{Number of classes}} = \frac{434}{8} \approx 54.25 \)Round this to a convenient number. A typical choice is usually a whole number, which would make it 55.

Key Concepts

Range Calculation

To calculate the range of a dataset, we need to find the maximum and minimum values within the data. The range provides a way to understand how spread out the data is. It is simply the subtraction of the smallest value from the largest one. In this particular exercise, the dataset ranges from 56 to 490. So, following the formula:

• Range = Maximum value - Minimum value

Plugging in the numbers, we calculate:

• Range = 490 - 56 = 434

The range gives us an initial idea of the distribution range of the data. It also lays the groundwork for determining how we break the data into smaller, more manageable parts such as class intervals.

Sturges' Formula

Sturges' Formula is a guideline used for determining the number of bins, or classes, for a histogram. It helps to organize data into intervals which makes it easier to see patterns or trends. The formula is expressed as:

• \[ k = 1 + 3.322 \log_{10}(n) \]

where \( k \) is the number of classes, and \( n \) is the number of observations. For this dataset, the number of observations \( n = 145 \). Using Sturges' formula:

• \[ k = 1 + 3.322 \log_{10}(145) \approx 8.46 \]

This calculation suggests using approximately 8 classes. By using a reasonable whole number, such as rounding to 8, we achieve a practical and visually informative histogram that represents the data effectively.

Histogram Bins

Histogram bins are the intervals into which we divide the entire range of the dataset. Choosing the right size for these bins is crucial as it affects how clearly the trends in the data are represented. Let's determine the width of each bin, also known as the class interval size. This is calculated using:

• Class Interval Size = \( \frac{\text{Range}}{\text{Number of classes}} \)

From the steps calculated earlier:

• Range = 434
• Number of classes = 8

Then the calculation is:

• Class Interval Size = \( \frac{434}{8} \approx 54.25 \)

We typically round the class interval size to a more straightforward number for ease of representation, which is often a whole number. In this instance, 55 is a suitable choice. By doing so, we make the histogram more accessible, aligning with better readability and analysis of the data.