Frequency Polygons1.32003/05/302003/07/18 11:04:37.462 GMT-5DavidLanelane@rice.eduDavidLanelane@rice.eduMatthewJeanesmjeanes@rice.edu
Frequency polygons are a graphical device for understanding the
shapes of distributions. They serve the same purpose as
histograms, but are especially helpful in comparing sets of
data. Frequency polygons are also a good choice for displaying
cumulative frequency distributions.
To create a frequency polygon, start just as for histograms, by choosing a class
interval. Then draw an X-axis representing the values of
the scores in your data. Mark the middle of each class interval
with a tick mark, and label it with the middle value represented
by the class. Draw the Y-axis to indicate the frequency of each
class. Place a point in the middle of each class interval at the
height corresponding to its frequency. Finally, connect the
points. You should include one class interval below the lowest
value in your data and one above the highest value. The graph
will then touch the X-axis on both sides.
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 to
39.5. Since the lowest test score is 46, this interval has a
frequency of 0. The point labeled 45 represents the interval
from 39.5 to 49.5. There are three scores in this
interval. There are 150 scores in the interval that surrounds
85.
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
inasmuch as good scores (to the right) trail off more gradually
than poor scores (to the left). In the terminology of
(where we will study shapes of distributions more
systematically), the distribution is
skewed.
Frequency polygon for the psychology test scores.
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 class
interval plus all numbers in lower
intervals. For example, there are no scores in the interval
labeled "35," three in the interval "45,"and 10 in the interval
"55."Therefore the Y value corresponding to "55" is 13. Since
642 students took the test, the cumulative frequency for the
last interval is 642.
Cumulative frequency polygon for the psychology test
scores.
Frequency polygons are useful for comparing distributions. This
is achieved by overlaying the frequency polygons drawn for
different data sets. provides an example. The data come
from a task in which the goal is to move a 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 . shows that although there is some overlap in times, it
generally took longer to move the mouse to the small target than
to the large one.
Overlaid frequency polygons.
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
again evident.
Overlaid cumulative frequency polygons.
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cumulative frequency distributions
A distribution showing the number of observations less than or
equal to values on the X-axis.
class interval
Also known as bin width, the class interval is a division of
data for use in a histogram. For instance, it is possible to
partition scores on a 100 point test into class intervals of
1-25, 26-49, 50-74 and 75-100.
cumulative frequency polygon
A frequency polygon whose vertices represent the sum of all
previous class frequencies of the data. For example, if a
frequency polygon had vertices of 5, 8, 3, 7, and 10, the
cumulative frequency polygon on the same data would have
vertices of 5, 13, 16, 23, and 33.