Frequency Polygons

Learning Outcomes

  • Construct a frequency polygon

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the [latex]x[/latex]-axis and [latex]y[/latex]-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.

example

A frequency polygon was constructed from the frequency table below.

Frequency Distribution for Calculus Final Test Scores
Lower Bound Upper Bound Frequency Cumulative Frequency
[latex]49.5[/latex] [latex]59.5[/latex] [latex]5[/latex] [latex]5[/latex]
[latex]59.5[/latex] [latex]69.5[/latex] [latex]10[/latex] [latex]15[/latex]
[latex]69.5[/latex] [latex]79.5[/latex] [latex]30[/latex] [latex]45[/latex]
[latex]79.5[/latex] [latex]89.5[/latex] [latex]40[/latex] [latex]85[/latex]
[latex]89.5[/latex] [latex]99.5[/latex] [latex]15[/latex] [latex]100[/latex]

 

A frequency polygon was constructed from the frequency table below.

The first label on the [latex]x[/latex]-axis is [latex]44.5[/latex]. This represents an interval extending from [latex]39.5[/latex] to [latex]49.5[/latex]. Since the lowest test score is [latex]54.5[/latex], this interval is used only to allow the graph to touch the [latex]x[/latex]-axis. The point labeled [latex]54.5[/latex] represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point [latex]104.5[/latex] representing the interval from [latex]99.5[/latex] to [latex]109.5[/latex]. Again, this interval contains no data and is only used so that the graph will touch the [latex]x[/latex]-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.

Try It

Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in the table.

Age at Inauguration Frequency
[latex]41.5–46.5[/latex] [latex]4[/latex]
[latex]46.5–51.5[/latex] [latex]11[/latex]
[latex]51.5–56.5[/latex] [latex]14[/latex]
[latex]56.5–61.5[/latex] [latex]9[/latex]
[latex]61.5–66.5[/latex] [latex]4[/latex]
[latex]66.5–71.5[/latex] [latex]2[/latex]

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

example

We will construct an overlay frequency polygon comparing the scores with the students’ final numeric grade.

Frequency Distribution for Calculus Final Test Scores
Lower Bound Upper Bound Frequency Cumulative Frequency
[latex]49.5[/latex] [latex]59.5[/latex] [latex]5[/latex] [latex]5[/latex]
[latex]59.5[/latex] [latex]69.5[/latex] [latex]10[/latex] [latex]15[/latex]
[latex]69.5[/latex] [latex]79.5[/latex] [latex]30[/latex] [latex]45[/latex]
[latex]79.5[/latex] [latex]89.5[/latex] [latex]40[/latex] [latex]85[/latex]
[latex]89.5[/latex] [latex]99.5[/latex] [latex]15[/latex] [latex]100[/latex]
Frequency Distribution for Calculus Final Grades
Lower Bound Upper Bound Frequency Cumulative Frequency
[latex]49.5[/latex] [latex]59.5[/latex] [latex]10[/latex] [latex]10[/latex]
[latex]59.5[/latex] [latex]69.5[/latex] [latex]10[/latex] [latex]20[/latex]
[latex]69.5[/latex] [latex]79.5[/latex] [latex]30[/latex] [latex]50[/latex]
[latex]79.5[/latex] [latex]89.5[/latex] [latex]45[/latex] [latex]95[/latex]
[latex]89.5[/latex] [latex]99.5[/latex] [latex]5[/latex] [latex]100[/latex]

This is an overlay frequency polygon that matches the supplied data. The x-axis shows the grades, and the y-axis shows the frequency.

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.