Problem 10 - Basic Statistics for Business and Economics with Student

Question

A set of data contains 53 observations. The lowest value is 42 and the largest is \(129 .\) The data are to be organized into a frequency distribution. a. How many classes would you suggest? b. What would you suggest as the lower limit of the first class?

Step-by-Step Solution

Answer

a. Around 7 classes; b. Lower limit for the first class is 42.

Step 1: Understanding Class Numbers

To determine the number of classes, we use the formula for deciding the number of classes in a frequency distribution, called Sturges' rule: \( k = 1 + 3.322 \log_{10}(n) \), where \( n \) is the number of observations. Substituting in \( n = 53 \):\[ k = 1 + 3.322 \log_{10}(53) \approx 1 + 3.322 \times 1.724 \approx 6.726 \]. This suggests around 7 classes.

Step 2: Calculating Class Interval

To find a suitable class interval, we divide the range of the data by the number of classes. First calculate the range: \( 129 - 42 = 87 \). Then divide by the number of classes (approximated from Step 1): \( \frac{87}{7} \approx 12.43 \). A practical class interval is typically rounded, so we use 13.

Step 3: Determining the Lower Limit of First Class

The lowest value in the data set is 42. Thus, a suitable starting point (lower limit) for the first class is 42 as it represents the smallest value and can encompass all data points when arranged in classes with this limit.

Key Concepts

Sturges' rule

Sturges' rule is a handy guideline for determining the number of classes when organizing data into a frequency distribution. By using this rule, you can structure a dataset into a clear frequency setup that is easy to interpret. The rule proposes a formula: \( k = 1 + 3.322 \log_{10}(n) \), where \( k \) is the number of classes, and \( n \) is the total number of observations.
This formula helps ensure that the frequency distribution is neither too cluttered with too many classes nor too vague with too few. It strikes a balance by taking into account the size of the dataset. Let's see how it works in practice. Imagine you have 53 observations, as in our data set. By inserting this into the formula, you calculate \( k \approx 7 \), recommending seven classes for the optimal grouping.
Keep in mind that this is a guideline, not an absolute rule. It's always wise to consider the nature of your data and adjust if necessary to better fit the distribution.

class interval

The class interval is a crucial concept in frequency distributions. It is the segment width or the range each class covers in your data distribution. After using Sturges' rule to determine the number of classes, the next step is to find the class interval.
To find the class interval, you should calculate the range of your data first, which is the difference between the highest and lowest values. In our example, the range is calculated as \( 129 - 42 = 87 \).
Once you have the range, you divide it by the number of classes (rounded from Sturges' rule) to get the class interval. Therefore, \( \frac{87}{7} \approx 12.43 \). It's practical to slightly adjust this number to a more manageable figure, like rounding up to 13 in our case.

• The class interval helps evenly distribute data across each class.
• It provides the structure to the overall frequency distribution by determining the width of each segment.

This approach makes the data easier to analyze and interpret.

range of data

The range of data is the simplest measure of variability and an important aspect when arranging data into a frequency distribution. It is the difference between the maximum and minimum values in the dataset.
In our example, the range is determined by subtracting the smallest data value (42) from the largest one (129), giving us 87. This range informs us of the scale over which the observations are spread.
Knowing the range is essential for calculating the class interval. It helps in understanding how spread out the data is, but it's important to notice that the range doesn't provide insight into the distribution of individual data points.
With the range identified, you can now divide it by the number of classes to find a suitable class interval, a key step in designing a frequency distribution. Understanding the range offers a snapshot of how data varies from its min to max points, which can be a foundational part of summarizing and presenting data.