In Exercises 15 and 16, construct the frequency polygons.


Chicago Commute Times Use the frequency distribution from Exercise 13 in Section 2-1 to construct a frequency polygon. Does the graph suggest that the distribution is skewed? If so, how?

Verified step by step guidance

1. Step 1: Recall that a frequency polygon is a graphical representation of a frequency distribution. It is created by plotting points corresponding to the midpoints of each class interval on the x-axis and their respective frequencies on the y-axis, and then connecting these points with straight lines.
2. Step 2: Identify the class intervals and their corresponding frequencies from the frequency distribution provided in Exercise 13 of Section 2-1. Calculate the midpoint of each class interval using the formula: $\mathrm{Midpoint}=\frac{\mathrm{Lower}+\mathrm{Upper}}{2}$. These midpoints will serve as the x-coordinates for the frequency polygon.
3. Step 3: Plot the midpoints on the x-axis and their corresponding frequencies on the y-axis. For each class interval, place a point at the intersection of the midpoint and its frequency.
4. Step 4: Connect the plotted points with straight lines. To complete the frequency polygon, extend the lines to the x-axis at the midpoints of the intervals immediately before the first class and after the last class, with frequencies of zero.
5. Step 5: Analyze the shape of the frequency polygon. If the graph is not symmetric and has a longer tail on one side, it suggests that the distribution is skewed. For example, a longer tail on the right indicates positive skewness, while a longer tail on the left indicates negative skewness.

Key Concepts

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. It organizes data into categories or intervals, showing the number of observations within each category. This foundational concept is crucial for visualizing data patterns and understanding the overall distribution of values.

Frequency Polygon

A frequency polygon is a graphical representation of a frequency distribution, created by plotting points for the frequency of each category and connecting them with straight lines. This type of graph helps to visualize the shape of the distribution, making it easier to identify trends, peaks, and potential skewness in the data.

Skewness

Skewness refers to the asymmetry of a distribution, indicating whether data points are concentrated on one side of the mean. A distribution is positively skewed if it has a long tail on the right, while a negatively skewed distribution has a long tail on the left. Understanding skewness is essential for interpreting the shape of the frequency polygon and assessing the nature of the data.