Constructing a Frequency Distribution and a Frequency Polygon In Exercises 35 and 36, construct a frequency distribution and a frequency polygon for the data set using the indicated number of classes. Describe any patterns.
Declaration of Independence
Number of classes: 5
Data set: Number of children of those who signed the Declaration of Independence (Source: The U.S. National Archives & Records Administration) 5 2 12 18 7 4 10 8 16 3 3 7 3 1 2 7 13 0 8 3 7 5 2 6 0 6 7 9 0 11 9 10 7 8 13 5 8 3 5 0 3 13 3 15 5 6 3 2 5 2 0 3 7 12 4 1

Verified step by step guidance

1. Step 1: Organize the data by identifying the minimum and maximum values in the data set. This will help determine the range, which is calculated as \(\text{Range} = \text{Maximum value} - \text{Minimum value}\).
2. Step 2: Calculate the class width by dividing the range by the number of classes (which is 5). Use the formula \(\text{Class width} = \frac{\text{Range}}{\text{Number of classes}}\). If the result is not a whole number, round up to the next whole number to ensure all data points fit into the classes.
3. Step 3: Construct the class intervals starting from the minimum value, adding the class width to create each subsequent class. Make sure the classes do not overlap and cover the entire range of data.
4. Step 4: Tally the data points into each class interval to find the frequency for each class. This will give you the frequency distribution, which lists each class interval alongside its frequency.
5. Step 5: To create the frequency polygon, plot the midpoints of each class interval on the x-axis and their corresponding frequencies on the y-axis. Connect these points with straight lines. Finally, analyze the shape of the polygon to describe any patterns such as skewness, peaks, or uniformity.

Key Concepts

Frequency Distribution

A frequency distribution organizes data into classes or intervals and shows how many data points fall into each class. It helps summarize large data sets by grouping values, making patterns easier to identify. For example, counting how many people have 0-3 children, 4-7 children, etc., provides a clear overview of the data.

Frequency Polygon

A frequency polygon is a graphical representation of a frequency distribution, created by plotting class midpoints against their frequencies and connecting the points with straight lines. It helps visualize the shape and trends of the data, such as peaks or gaps, and is useful for comparing distributions.

Class Intervals and Number of Classes

Class intervals divide the data range into equal segments, and the number of classes determines how many intervals are used. Choosing an appropriate number of classes (like 5 in this question) balances detail and clarity, avoiding too many or too few groups, which affects the interpretability of the frequency distribution and polygon.