2.4 Frequency polygons

Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose ashistograms, but are especially helpful in comparing sets of data. Frequency polygons are also a good choice fordisplayingcumulative frequency distributions.

To create a frequency polygon, start just as for histograms, by choosing a class interval. Then draw an X-axis representing thevalues of the scores in your data. Mark the middle of each class interval with a tick mark, and label it with the middle valuerepresented by the class. Draw the Y-axis to indicate the frequency of each class. Place a point in the middle of eachclass interval at the height corresponding to its frequency. Finally, connect the points. You should include one classinterval below the lowest value in your data and one above the highest value. The graph will then touch the X-axis on bothsides.

A frequency polygon for 642 psychology test scores is shown in . The first label on the X-axis is 35. This represents an interval extending from 29.5 to39.5. Since the lowest test score is 46, this interval has a frequency of 0. The point labeled 45 represents the intervalfrom 39.5 to 49.5. There are three scores in this interval. There are 150 scores in the interval that surrounds85.

You can easily discern the shape of the distribution from . Most of the scores are between 65 and 115. It is clear that the distribution is not symmetric inasmuchas good scores (to the right) trail off more gradually than poor scores (to the left). In the terminology of Chapter 3 (where wewill study shapes of distributions more systematically), the distribution is skewed .

A cumulative frequency polygon for the same test scores is shown in . The graph is the same as before except that the $Y$ value for each point is the number of students in the corresponding classinterval plus all numbers in lower intervals. For example, there are no scores in the interval labeled "35," three in theinterval "45,"and 10 in the interval "55."Therefore the $Y$ value corresponding to "55" is 13. Since 642 students took the test, the cumulative frequencyfor the last interval is 642.

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn fordifferent data sets. provides an example. The data come from a task in which the goal is to movea computer mouse 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 in . The figure shows that although there is some overlap in times, it generally took longer to move the mouse tothe small target than to the large one.

It is also possible to plot two cumulative frequency distributions in the same graph. This is illustrated in using the same data from the mouse task. The difference in distributions for the two targets is againevident.