Problem 28 - Basic Statistics for Business and Economics with Student

Question

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

Step-by-Step Solution

Answer

A class interval size of 55 is recommended.

Step 1: Determine the Range

The range of the data set is the difference between the maximum and minimum values. We calculate it as follows: \[ \text{Range} = 490 - 56 = 434 \]

Step 2: Decide on the Number of Classes

A common rule of thumb for deciding the number of classes is to use the Sturges' formula, which suggests the number of classes \( k \) as: \[ k = 1 + 3.322 \log_{10}(n) \] where \( n \) is the number of observations. Substituting the given number of observations, \( 145 \): \[ k = 1 + 3.322 \log_{10}(145) \] Calculating the logarithm and simplifying: \[ \log_{10}(145) \approx 2.161 \] \[ k \approx 1 + 3.322 \times 2.161 \approx 8.18 \] So, we round this result to \( k = 8 \) classes.

Step 3: Calculate the Class Interval Size

The class interval size \( c \) can be calculated by dividing the range of the data set by the number of classes \( k \): \[ c = \frac{\text{Range}}{k} = \frac{434}{8} \approx 54.25 \] Since class intervals are typically whole numbers, we can round this to \( 55 \).

Key Concepts

Range

The range of a data set is a measure of how spread out the data is. It tells us the difference between the maximum and minimum values in our data set. For the given data with values from 56 to 490, the range is calculated by subtracting the smallest number from the largest. Thus, the range is 434.

• Range = Maximum value - Minimum value
• In this context, Range = 490 - 56
• Resulting in a Range = 434

Understanding the range is crucial because it gives an initial idea of the dispersion of your data.
A larger range indicates more variability while a smaller range suggests that the data points are more closely clustered.

Number of Classes

Deciding on the number of classes for a data set is an important step in creating a frequency distribution. The number of classes (or bins) determines how the data is grouped. A common approach to determine this number is to use Sturges' formula. This helps in choosing a number that is neither too detailed nor too generalized.According to Sturges' formula:

• The number of classes \( k \) is calculated as \( k = 1 + 3.322 \log_{10}(n) \).
• Here, \( n \) represents the number of observations — in this case, 145.

This formula uses the base-10 logarithm to take into account the number of observations, ensuring that the number of classes grows logically with larger data sets.
After calculating the logarithm, we get approximately 8 classes, suggesting a balanced grouping of data that will summarize it effectively without losing critical information.

Sturges' Formula

Sturges' formula is a widely-used method in statistics to help determine the optimal number of classes for a frequency distribution table. It provides a systematic way to decide how detailed your data representation should be.Here's the formula again for clarity:

• \( k = 1 + 3.322 \log_{10}(n) \)
• Where \( k \) = number of classes, and \( n \) = number of observations.

For this scenario where we have 145 observations, calculating with Sturges’ formula gives us about 8 classes. This method works well especially when dealing with data sets where you want to ensure that your histogram or frequency distribution chart captures trends accurately while avoiding over-complication with too many classes.
This balance is essential in meaningful data analysis, providing a clear picture without overpowering details.