Question

Use the information given to find a convenient class width. Then list the class boundaries that can be used to create a relative frequency histogram. 8 classes for \(n=75\) measurements; minimum value \(=0\); maximum value \(=192\)

Short answer

Answer: The class boundaries are [0, 24), [24, 48), [48, 72), [72, 96), [96, 120), [120, 144), [144, 168), and [168, 192).

Step-by-step solution

Find the range

(Write the content here) First, find the range by subtracting the minimum value from the maximum value: Range = maximum - minimum Range = \(192 - 0 = 192\)

Calculate the class width

Next, divide the range by the number of classes to find the class width: Class width = \(\frac{Range}{number\:of\:classes}\) Class width = \(\frac{192}{8} = 24\)

Determine the class boundaries

Now, determine the class boundaries by adding the class width (24) to each lower boundary, starting from the minimum value (0): 1st class: \([0, 24)\) 2nd class: \([24, 48)\) 3rd class: \([48, 72)\) 4th class: \([72, 96)\) 5th class: \([96, 120)\) 6th class: \([120, 144)\) 7th class: \([144, 168)\) 8th class: \([168, 192)\) The class boundaries for creating a relative frequency histogram are: \([0, 24),[24, 48),[48, 72),[72, 96),[96, 120),[120, 144),[144, 168),[168, 192)\).

Key concepts

Frequency Histogram

A frequency histogram is an important tool in statistics that displays the frequency of data points within specified ranges or intervals, known as bins or classes. Imagine you needed to present the test scores of a class in a simple visual format. A frequency histogram would allow you to group scores into intervals and show how many students achieved scores within those intervals.

For instance, if we follow the original exercise and use eight bins to represent the number of measurements within each class, you would be able to see at a glance the distribution of measures across the range. This graphical representation makes it easier to understand patterns, such as skewness, modality, and spread within the data.

To be effective, the histogram should have an appropriate number of classes, which balances detail with the ability to discern patterns. Too many classes may result in a fragmented picture, while too few may oversimplify and obscure important characteristics of the data.

Statistical Class Boundaries

Statistical class boundaries are precise points that separate one class from another in a dataset. They're like lines on a football field, marking the start and end points of each interval or class. In our exercise, the boundaries are defined so that each measurement is included in only one class, using a non-overlapping range.

The specified boundaries ensure accuracy when tabulating or graphing data. In our solution, the boundaries of the first class might be [0, 24). The notation used here, with a square bracket on one end and a parenthesis on the other, indicates inclusion or exclusion. The square bracket means 0 is included in the class, while the parenthesis indicates that 24 is not included—it's the start of the next class.

Knowing these boundaries is crucial when creating histograms, as it ensures that each data point is associated with the correct class and prevents any data from being accidentally omitted or double-counted.

Range in Statistics

The range is one of the simplest measures of variability in statistics. It's the difference between the highest and lowest values in a dataset, giving us a quick sense of how spread out the data is. If we reported only the average score on a test, we might miss how wide or narrow the distribution of individual scores is.

In our example, we calculate the range by subtracting the minimum value from the maximum value in the dataset of 75 measurements. The textbook solution illustrates that with a minimum of 0 and a maximum of 192, the range is 192. It's essential to understand that the range does not consider how the data is distributed between the maximum and minimum, just the extent of the data.

Class Width Calculation

Class width plays a vital role in creating histograms, as it determines the size of each interval or class. It can impact the histogram's appearance and the conclusions you might draw from it. To calculate the class width, divide the range by the desired number of classes.

In our step by step solution, it's suggested that eight classes be used for presentation. By dividing the range (192) by the number of classes (8), we get a class width of 24. Each class will encompass a width of exactly 24 units, creating uniform intervals for the histogram. Remember that the classes must cover all possible values the data can take, and deciding on the right number of classes and width is more of an art than a science, requiring a balance between too much detail and not enough.