Math & History: Frequency Polygons, and Time Series Graphs (Line Graphs)

Here we distinguish between two types of line graphs. While they both use points connected by line segments, they serve two completely different statistical purposes.

I. Frequency Polygons

A Frequency Polygon is used to show the distribution of a data set. It is essentially a "connected" version of a histogram.

A. Constructing a Frequency Polygon

1. Sort Data (Same as histogram) Arrange your data in ascending order. This helps you quickly identify the minimum and maximum values to determine the spread of the data.

2. Define Bins (Same as histogram) Decide on the number of bins and calculate the bin width.

Bin Width = (Maximum Value - Minimum Value) / Number of Bins.
Note: Ensure your bin widths are consistent throughout the graph.

3. Create a Frequency Table with Midpoints List your intervals and tally the frequencies as you would for a histogram. However, for a frequency polygon, you must also calculate the midpoint for each interval.

Formula: Midpoint = (Lower Boundary + Upper Boundary) / 2.
Example: If a bin range is 100–110, the midpoint is 105.

4. Draw and Label the Axes

Horizontal Axis (X-axis): Label this with the measured variable (e.g., "Time in Hours"). Instead of marking just the boundaries, mark the midpoints you calculated in Step 3.
Vertical Axis (Y-axis): Label this as "Frequency" or "Relative Frequency," ensuring the scale starts at zero.

5. Plot Points and Connect Instead of drawing vertical bars, you will create a line graph:

Plot the Points: For each bin, place a dot at the intersection of its midpoint on the X-axis and its frequency on the Y-axis.
Connect the Dots: Use a straight edge to connect the points in order from left to right.

Example:

One advantage of a frequency polygon is that it allows histogram-like data representation of two sets of data on the same graph. Two histograms on the same graph tend to shroud each other and make comparison more difficult, but two frequency polygons can be graphed together with much less interference

The figure below provides an example. The data come from a task in which the goal is to move a computer cursor to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial. The two distributions (one for each target) are plotted together. The figure shows that, although there is some overlap in times, it generally took longer to move the cursor to the small target than to the large one.

II. Time Series Graphs

Though time series graphs look similar to frequency polygons, they display. Frequency polygons display the distribution of a data set (how often values fall into class intervals). Time series graphs shows how a specific variable changes over time. The horizontal axis is always time (days, months, years, etc.), and the vertical axis is the recorded value at each time point (such as temperature or sales).

A. Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set using a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

1. Organize Data Arrange your data points into a table in chronological order.

2. Draw and Label the Axes

Horizontal Axis (X-axis): This is the time variable.
Vertical Axis (Y-axis): This is the quantity being measured.

3. Plot the data points

For each pair (time, value), find the position on the graph where that time on the x-axis meets that value on the y-axis.
Place a small point or dot at each of these positions.

4. Connect the points