# Cumulative Frequency Plots

A **cumulative frequency plot**
is a way to display cumulative information graphically.
It shows the number, percentage, or proportion of observations
that are less than or equal to particular values.

## Frequency vs. Cumulative Frequency

In a data set, the **cumulative frequency** for a
value *x* is the total number of scores that are less than
or equal to *x*. In this section, we use two charts to illustrate the
difference between frequency and cumulative frequency.
Both charts show scores for a test administered to 300 students.

In the first chart (shown below), column height indicates frequency - the number of students in each test score grouping. For example, about 30 students received a test score between 51 and 60.

In the next chart, column height shows cumulative frequency -
the number of students
*up to and including* each test score.
The chart below is
a cumulative frequency chart. It shows that 30 students received
a test score of at most 50; 60 students received a score of at
most 60; 120 students received a score of at most 70; and so on.

## Absolute vs. Relative Frequency

Frequency counts can be measured in terms of absolute numbers or relative numbers (e.g., proportions or percentages). The chart below duplicates the cumulative frequency chart above, except that it expresses the counts in terms of percentages rather than absolute numbers.

Note that the columns in the chart have the same shape, whether the Y axis is labeled with actual frequency counts or with percentages. If we had used proportions instead of percentages, the shape would remain the same.

## Discrete vs. Continuous Variables

Each of the previous cumulative charts have used a discrete variable on the X axix (i.e., the horizontal axis). The chart below duplicates the previous cumulative charts, except that it uses a continuous variable for the test scores on the X axis.

Let's work through an example to understand how to read this
cumulative frequency plot. Specifically, let's find the
median.
Follow the grid line to the right from the Y axis at 50%.
This line intersects the curve over the X axis at a test score of
about 73. This means that half of the students received a test score
of *at most* 73, and half received a test score of *at least* 73.
Thus, the median is 73.

You can use the same process to find the cumulative percentage associated with any other test score. For example, what percentage of students received a test score of 64 or less? From the graph, you can see that about 25% of students received a score of 64 or less.

## Test Your Understanding

**Problem 1**

Below, the cumulative frequency plot shows height (in inches) of college basketball players.

What is the interquartile range?

(A) 3 inches

(B) 6 inches

(C) 25 inches

(D) 50 inches

(E) None of the above

**Solution**

The correct answer is (B). The interquartile range is the middle range of the distribution, defined by Q3 minus Q1.

Q1 is the height for which the cumulative percentage is 25%. To find Q1 from the cumulative frequency plot, follow the grid line to the right from the Y axis at 25%. This line intersects the curve over the X axis at a height of about 71 inches. This means that 25% of the basketball players are at most 71 inches tall, so Q1 is 71.

To find Q3, follow the grid line to the right from the Y axis at 75%. This line intersects the curve over the X axis at a height of about 77 inches. This means that 75% of the basketball players are at most 77 inches tall, so Q3 is 77.

Since the interquartile range is Q3 minus Q1, the interquartile range is 77 - 71 or 6 inches.