Problem 3 - Basic Statistics for Business and Economics

Question

A set of data consists of 230 observations between \(\$ 235\) and \(\$ 567 .\) What class interval would you recommend?

Step-by-Step Solution

Answer

Use a class interval of 23.

Step 1: Determine the range

First, we calculate the range of the data set. The range is the difference between the highest and lowest values.\[\text{Range} = 567 - 235 = 332\]

Step 2: Decide on the number of classes

The number of classes can be determined using the formula \( k = \sqrt{n} \), where \( n \) is the number of observations. In this case, there are 230 observations.\[k = \sqrt{230} \approx 15\] So, we will use 15 classes.

Step 3: Calculate the class interval width

To find the class width, divide the range by the number of classes and round up to the nearest whole number.\[\text{Class Interval} = \frac{332}{15} \approx 22.13\]Rounding up, we recommend a class interval of 23.

Step 4: Final recommendation

Based on our calculations, we recommend using a class interval of 23 for organizing the data set into a frequency distribution. This will efficiently use the range of the data and provide a meaningful number of classes.

Key Concepts

Understanding Frequency Distribution

Frequency distribution is a crucial concept in statistics, particularly in the context of organizing data in a meaningful way. At its core, frequency distribution refers to the way in which data is aggregated and presented to show how often each value or range of values occurs. This is typically done by grouping data into classes and showing the frequency of data points within each class.

A frequency distribution table consists of a series of classes, each with a specific interval or range, allowing us to quickly understand the spread and pattern of the data. This way, large data sets can be summarized effectively. When choosing class intervals, it's important to ensure they comprehensively represent all potential values in the data set.

• Classes should be mutually exclusive (no overlap between classes).
• Class intervals should be equal in size to maintain consistency.

Breaking down data into suitable frequency distributions helps in spotting trends and variations. For example, in a data set describing income levels, the frequency distribution helps highlight the most common income brackets, providing valuable insights.

Steps in Range Calculation

The first step in organizing data into classes is calculating the range, which is the difference between the highest and lowest values in the data set. This provides crucial information about the spread of the data, enabling better decisions regarding class width.

In our example problem, we calculated the range using the formula:
\[\text{Range} = \text{Highest value} - \text{Lowest value} = 567 - 235 = 332\]
Calculating the range provides insight into the variability of the data, and it's essential for determining the width of class intervals. A larger range suggests a wider spread in data points, potentially requiring more classes to effectively capture this dispersion.

• Range calculation is a straightforward yet vital step in histogram analysis and frequency distribution.
• It sets the foundation for subsequent steps in data classification, including determining class intervals.

Understanding the range can profoundly impact how data is perceived and analyzed, shaping insights and decisions derived from statistical analysis.

Calculating the Number of Classes

Determining the number of classes is a pivotal step in creating a frequency distribution. Choosing the correct number of classes helps ensure that the entire data set is represented adequately and meaningfully.

One common method to estimate the number of classes is to use Sturges' Rule or the square root of the number of observations. In the context of the provided exercise, we employed the square root formula: \[k = \sqrt{n} \approx \sqrt{230} \approx 15\]
This calculation provides a balance between oversimplifying the data with too few classes and overcrowding it with too many. A well-chosen number of classes aids in creating a clear, interpretable graph or table.

• It ensures all data points are categorized, preventing loss of information.
• The chosen number of classes should provide a clear and understandable overview of data frequencies.

Choosing the right number of classes enables you to see patterns and trends more clearly, thus making statistical analysis more insightful and accurate.