Problem 7 - Basic Statistics for Business and Economics with Student

Question

A set of data consists of 38 observations. How many classes would you recommend for the frequency distribution?

Step-by-Step Solution

Answer

The recommended number of classes is 6.

Step 1: Understand the Sturges' Rule

To determine the number of classes for a frequency distribution, Sturges' Rule is often used. Sturges' Rule provides a formula to calculate the recommended number of classes: \( k = 1 + 3.322 \log_{10} n \). Here, \( n \) represents the number of observations.

Step 2: Apply Sturges' Formula

Given that the number of observations \( n = 38 \), substitute 38 into the formula: \( k = 1 + 3.322 \log_{10} 38 \).

Step 3: Calculate Logarithm of 38

Use a calculator to find \( \log_{10} 38 \). The decimal logarithm of 38 is approximately 1.580.

Step 4: Calculate the Number of Classes

Now substitute \( \log_{10} 38 \approx 1.580 \) into the Sturges' formula: \( k = 1 + 3.322 \times 1.580 \).

Step 5: Solve for k

Calculate \( 3.322 \times 1.580 \approx 5.247 \). Then, add 1: \( k = 1 + 5.247 = 6.247 \).

Step 6: Round to the Nearest Whole Number

The final step is to round 6.247 to the nearest whole number, resulting in \( k = 6 \).

Key Concepts

Sturges' Rule

Sturges' Rule is a simple guideline used to determine the number of classes in a frequency distribution, particularly when dealing with a relatively small sample size. It helps create a balance between having too many or too few classes.
This rule gives a formula as:

• \[ k = 1 + 3.322 \log_{10} n \]
• Where \( k \) is the number of classes, and \( n \) is the total number of observations in your dataset.

Using this rule ensures that your frequency distribution is neither too detailed nor too generalized, offering a clearer picture of data distribution. Sturges' Rule is commonly used because it adapts to different sizes of data.
Keep in mind that it’s mainly beneficial for data with fewer than 1000 observations. Larger datasets might require more sophisticated methods to set the number of classes appropriately.

Number of Classes

The number of classes \( k \) is a crucial element in creating a frequency distribution. In essence, classes are subdivisions that help condense data, enabling us to interpret the dataset better.

• Having the right number of classes is key to an effective representation of data.
• Too many classes can make the dataset needlessly complex, while too few can oversimplify and distort patterns.
• Following Sturges' Rule, the calculated number of classes ensures each class interval is neither too wide nor too narrow.

Determining the number of classes involves the calculation: \( k = 1 + 3.322 \log_{10} n \).By ensuring the number of classes is well-adjusted, you can achieve a frequency distribution that accurately represents your data. This process is foundational in statistical analysis.

Logarithm Calculation

In the context of Sturges' Rule, logarithm calculations are indispensable for determining the number of classes. Specifically, we use the base 10 logarithm (logarithm base 10).

• The formula \( \log_{10} n \) needs to be evaluated when applying Sturges' Rule.
• In calculation, \( \log_{10} 38 \approx 1.580 \) is obtained, which becomes part of the formula calculation.
• Precise logarithm values can usually be computed with calculators or logarithm tables.

These calculations help scale the size of data by converting it to a smaller, more manageable form. Understanding how to find these values accurately ensures the correct application of the rule and helps in producing a valid frequency distribution.