Master One-Variable Data with Descriptive Statistics, Dot Plots, and Box Plots

In this first lesson of the statistics unit, dot plots, box plots, and histograms will be used to analyze data.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Dot Plot
Frequency Table
Histogram
Box Plot

Explore

Matching Box Plots

Move the red points on the dot plot to change the corresponding blue box plot. Can the red dots be moved so that the two box plots match?

Interactive applet where points of the dot plot can be moved around.

Example

Drawing a Dot Plot

Izabella's favorite candy, Frutty, is sold in packs of thirty candies with three different flavors — apple, orange, and banana.

A candy, and three different flavors (apple, orange, banana)

Izabella wants to know how many banana-flavored candies there are in each pack, so she bought ten packs and counted the number of banana candies in each. Her results are as follows.

10, 8, 10, 9, 12, 9, 10, 10, 12, 10

Draw a dot plot to represent the data.

Answer

Hint

Begin by finding the range of the data, then draw a number line which covers this range.

Solution

The smallest number in the data set is $8$ and the largest is $12 .$ This means that the dot plot can be displayed above a horizontal number line that covers at least the numbers from $8$ to $12 .$ Here, a number line from $7$ to $13$ will be used.

The number of dots drawn on the dot plot above a certain number should match the frequency of that number in the data set.

10, 8, 10, 9, 12, 9, 10, 10, 12, 10

Given the data set compare the frequencies of the numbers.

The number $10$ appears five times in the data set, so there should be five dots above number $10$ on the number line.
The number $8$ appears once in the data set, so there should be one dot above number $8$ on the number line.
The number $9$ appears twice in the data set, so there should be two dots above number $9$ on the number line.
The number $12$ appears twice in the data set, so there should be two dots above number $12$ on the number line.

From here, the dot plot can be drawn as follows.

Example

Extracting Information From a Dot Plot

A multiple-choice test has ten questions. After grading the test, the teacher produced the following dot plot to show how many correct answers each student had on the test.

Number line showing the range from 0 to 10. There is one dot above 4, three dots above 5 and 9, two dots above 6 and 10, four dots above 7, and five dots above 8.

How many students are there in the class?

Hint

Each dot represents the performance of one student on the test.

Solution

Each dot represents the performance of a student on the test. For example, since there is one dot above the number $4,$ it means that one student answered four questions correctly. The rest of the dot plot can be interpreted similarly.

Number	Dots Above the Number	Conclusion
$0, 1, 2, 3$	$0$	There are no students who answered fewer than four questions correctly.
$4$	$1$	$One$ student answered four questions correctly.
$5$	$3$	$Three$ students answered five questions correctly.
$6$	$2$	$Two$ students answered six questions correctly.
$7$	$4$	$Four$ students answered seven questions correctly.
$8$	$5$	$Five$ students answered eight questions correctly.
$9$	$3$	$Three$ students answered nine questions correctly.
$10$	$2$	$Two$ students answered all ten questions correctly.

The number of students in the class is equal to the number of dots in the diagram.

1 + 3 + 2 + 4 + 5 + 3 + 2 = 20

There are

20

students in the class who took this test.

Pop Quiz

Dot Plots and Goals Scored

A college hockey team played $23$ games during a season. An enthusiastic fan made a dot plot of the number of goals the team scored in each game.

Example

Drawing a Histogram

The following data set shows the ages of the first

45

presidents of the United States when their presidencies began. The president's name and presidential period can be displayed by clicking on and holding down each point.

Ages=[57,61,57,57,58,57,61,54,68,51,49,64,50,48,65,52,56,46,54,49,51,47,55,55,54,42,51,56,55,51,54,51,60,62,43,55,56,61,52,69,64,46,54,47,70]. Points with the age of each president. When clicked, the president's name is displayed along with its presidential period

Starting at age

40,

group the data into

5 - year

intervals and draw a histogram of the results.

Answer

Hint

Group the data in a frequency table using the intervals asked in the prompt. The first interval will be the ages $40 - 44 .$

Solution

The frequency table below shows the grouping of the data starting at $40$ and using $5 - year$ intervals.

Interval	Frequency
$40 - 44$	$2$
$45 - 49$	$7$
$50 - 54$	$12$
$55 - 59$	$13$
$60 - 64$	$8$
$65 - 69$	$2$
$70 - 74$	$1$

Use these intervals and frequencies to draw the histogram.

Example

Analyzing a Histogram

In $1936,$ Sir Ronald Aymler Fisher published a paper entitled The Use of Multiple Measurements in Taxonomic Problems. Fisher investigated several measurements of three species of flowers.

The histogram below shows the summary of the data about the sepal length of the Iris virginica flowers.

A histogram with bar heights 1, 0, 6, 17, 14, 6, and 6.

How many Iris Virginica flowers did Fisher investigate in this paper?

Hint

Consider the height of the rectangles in the histogram.

Solution

In a histogram, the height of the rectangles shows the frequency of the data elements in the corresponding interval.

There is one flower with a sepal length between $45$ and $49$ millimeters.
There are no flowers with a sepal length between $50$ and $54$ millimeters.
There are six flowers with a sepal length between $55$ and $59$ millimeters.
There are seventeen flowers with a sepal length between $60$ and $64$ millimeters.
There are fourteen flowers with a sepal length between $65$ and $69$ millimeters.
There are six flowers with a sepal length between $70$ and $74$ millimeters.
There are six flowers with a sepal length between $75$ and $79$ millimeters.

The total number of flowers in the experiment is the sum of these counts.

1 + 0 + 6 + 17 + 14 + 6 + 6 = 50

Fisher investigated the data of about

50

Iris virginica flowers.

Pop Quiz

Histograms and Tree Heights

A ranger is surveying a forest. He randomly selected $40$ loblolly pines (Pinus taeda) and measured their heights. The histogram below is the summary of the data.

Example

Drawing a Box Plot

The following table shows the test scores of a class of

26

students.

8.5 15.5 5 8.5 13.5 13.5 1112156 7.5 16781213 12.5 13915 10.5 11 10.5 8 15.5 11.5

Draw a box plot of the data.

Answer

Hint

Rearrange the data in increasing order and find the five-number summary.

Solution

A box plot is a visual representation of the five-number summary of data. It is a scaled diagram that shows the relative positions of the

minimum

and

maximum

values,

the

median,

and the

first

and

third

quartiles .

The first step in finding these values is to arrange the data in increasing order.

58 10.5 12 13.5 16 6 8.5 11 12.5 15 7 8.5 111315 7.5 9 11.5 13 15.5 8 10.5 12 13.5 15.5

With an ordered data set, the minimum and maximum are easily identifiable. Here, the minimum is

5

and the maximum is

16 .

These are marked on a number line.

Since there are

26

values, the median is the mean of the numbers at the

13 th

and

14 th

positions.

58 10.5 12 13.5 16 6 8.5 11 12.5 15 7 8.5 111315 7.5 9 11.5 13 15.5 8 10.5 12 13.5 15.5

Now, the median can be determined by calculating the average of

11

and

11.5 .

\frac{1 1 + 1 1 . 5}{2} = 11.25

The median is

11.25 .

This is also marked on the number line.

The first quartile is the median of the first half of the data.

58 10.5 6 8.5 11 7 8.5 11 7.5 9 8 10.5

The third quartile is the median of the second half of the data.

12 13.5 16 12.5 15 1315 11.5 13 15.5 12 13.5 15.5

The first quartile is

8.5

and the third quartile is

13.5 .

These are also marked on the number line.

The box-plot is built using these points.

The quartiles mark the boundaries of the box.
The minimum and maximum values mark the end of the whiskers.
The median marks the position of the line that divides the box.

Putting all this together gives the box plot.

Example

Examining a Box Plot

In the $1994$ report The Population Biology of Abalone (Haliotis species) in Tasmania, the authors presented and investigated the measurements of $4177$ blacklip abalones.

The lengths of the shells in millimeters are summarized in the box plot below.

How many blacklip abalones' lengths were shorter than $90$ millimeters in this experiment?

Hint

Which part of the box plot is at $90 ?$

Solution

The left side of the box is at $90,$ so the first quartile of the lengths is $90$ millimeters.

The problem is now to find out how many data points are less than the first quartile. The first quartile is the median of the lower half of the data set. In this experiment there are

4177

data points, so by dividing this by

2,

the number of data points in the lower half can be found.

\frac{4 1 7 7}{2} = 2088.5

This means that in the lower half, there are

2088

data points. Now, by dividing

2088

2,

the placement of the lower quartile can be found.

\frac{2 0 8 8}{2} = 1044

The lower quartile is the average of the

1044 th

and the

1045 th

data points. Since the lower quartile is

90,

the

1045 th

data point is not less than

90 .

Therefore, the number of blacklip abalones that are shorter than

90

millimeters is less than

1045 .

The only option that meets this condition is

1007 .

Extra

Note that from the box plot, the only conclusion we can make is that the number of blacklip abalones shorter than $90$ millimeters is less than $1045 .$

It is possible for the $1044 th$ data to be $89,$ in which case the answer to the question would be $1044 .$
It is possible for the $1044 th$ data to be $90,$ in which case the answer to the question would be less than $1044 .$

In fact, there were $60$ blacklip abalones with a length of $90$ millimeters in the experiment. The answer option $1007$ reflects the actual answer to the question, but to get this value, the full data is needed — the box plot is not enough.

Pop Quiz

Analyzing Box Plots

The heights, in feet, of red alder (Alnus rubra) trees in a forest are summarized in the following box plot.

Closure

Stacked Histograms

In some cases, scientists use visual representations that go beyond the three types of plots discussed in this lesson. For example, the report about the blacklip abalones also contains data about their sex. This can be used to present a summary of the length in a stacked histogram.

A stacked histogram where the bars are split according to sex.

In addition to representing the lengths, this stacked histogram has colored bars that indicate the distribution of male, female, and infant blacklip abalones.

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Catch-Up and Review

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