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Algebra 1 View details
1. Representing One Variable Data
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Chapter 4
1. 

Representing One Variable Data

This lesson offers a deep dive into the world of one-variable data analysis, focusing on descriptive statistics. It teaches how to use dot plots and box plots for visualizing data. These tools are essential for understanding various types of data, from test scores in a classroom to the heights of trees in a forest. The lesson is designed to be practical, showing how these statistical methods can be applied in real-world scenarios such as diet analysis or student performance. By mastering these techniques, you can better interpret data, make more informed decisions, and even solve complex problems in your daily life.
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Lesson Settings & Tools
12 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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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.

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.
Draw a dot plot to represent the data.

Answer
A number line with one dot above 8, two dots above 9 and 12, and 5 dots above 10.

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 and the largest is This means that the dot plot can be displayed above a horizontal number line that covers at least the numbers from to Here, a number line from to will be used.

Number line showing the range from 7 to 13.
The number of dots drawn on the dot plot above a certain number should match the frequency of that number in the data set.
Given the data set compare the frequencies of the numbers.
  • The number appears five times in the data set, so there should be five dots above number on the number line.
  • The number appears once in the data set, so there should be one dot above number on the number line.
  • The number appears twice in the data set, so there should be two dots above number on the number line.
  • The number appears twice in the data set, so there should be two dots above number on the number line.

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

A number line with one dot above 8, two dots above 9 and 12, and 5 dots above 10.
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 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
There are no students who answered fewer than four questions correctly.
student answered four questions correctly.
students answered five questions correctly.
students answered six questions correctly.
students answered seven questions correctly.
students answered eight questions correctly.
students answered nine questions correctly.
students answered all ten questions correctly.
The number of students in the class is equal to the number of dots in the diagram.
There are students in the class who took this test.
Pop Quiz

Dot Plots and Goals Scored

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

Random question generator applet.
Example

Drawing a Histogram

The following data set shows the ages of the first 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 group the data into intervals and draw a histogram of the results.

Answer
Histogram representing the data

Hint

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

Solution

The frequency table below shows the grouping of the data starting at and using intervals.

Interval Frequency

Use these intervals and frequencies to draw the histogram.

Histogram representing the given data
Example

Analyzing a Histogram

In 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.

IrisVirginica.jpg

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 and millimeters.
  • There are no flowers with a sepal length between and millimeters.
  • There are six flowers with a sepal length between and millimeters.
  • There are seventeen flowers with a sepal length between and millimeters.
  • There are fourteen flowers with a sepal length between and millimeters.
  • There are six flowers with a sepal length between and millimeters.
  • There are six flowers with a sepal length between and millimeters.
The total number of flowers in the experiment is the sum of these counts.
Fisher investigated the data of about Iris virginica flowers.
Pop Quiz

Histograms and Tree Heights

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

Random histograms generator
Example

Drawing a Box Plot

The following table shows the test scores of a class of students.
Draw a box plot of the data.

Answer
Box-plot with minimum at 5, first quartile at 8.5, median at 11.25, third quartile at 13.5, and maximum at 16.

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 and the and the and The first step in finding these values is to arrange the data in increasing order.
With an ordered data set, the minimum and maximum are easily identifiable. Here, the minimum is and the maximum is These are marked on a number line.
Number line with minimum and maximum marked.
Since there are values, the median is the mean of the numbers at the and positions.
Now, the median can be determined by calculating the average of and
The median is This is also marked on the number line.
Number line with minimum, maximum, and median marked.
The first quartile is the median of the first half of the data.
The third quartile is the median of the second half of the data.
The first quartile is and the third quartile is These are also marked on the number line.
Number line with the five-number summary marked.

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.

Box plot above the number line.
Example

Examining a Box Plot

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

BlacklipAbalone.jpg

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

Box plot

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

Hint

Which part of the box plot is at

Solution

The left side of the box is at so the first quartile of the lengths is 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 data points, so by dividing this by the number of data points in the lower half can be found.
This means that in the lower half, there are data points. Now, by dividing by the placement of the lower quartile can be found.
The lower quartile is the average of the and the data points. Since the lower quartile is the data point is not less than Therefore, the number of blacklip abalones that are shorter than millimeters is less than The only option that meets this condition is

Extra

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

  • It is possible for the data to be in which case the answer to the question would be
  • It is possible for the data to be in which case the answer to the question would be less than

In fact, there were blacklip abalones with a length of millimeters in the experiment. The answer option 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.

Random box plot generator
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.



Level 1
Level 2
Level 3
Below is the five number summary for observations.
What is the lowest possible sum of the data set? Show that this is true for the general case.

Let's illustrate the data set in ascending order using the following points.

To obtain the lowest possible sum for the data set, we want the observations to be as low as possible.

Minimum and Maximum Value

Notice that we have been given the minimum and maximum value. This locks in the first and last observation of the data set.


Median

Next, we want to find the median. Since we have 12 observations, which is an even number, the median will be the average of the 6^(th) and 7^(th) value.

For the median to be 3, the sum of these numbers must be 6. Let's call the sixth number (3-m) and the seventh number (3+m). This gives an average of 3. (3-m)+(3+m)/2=6/2= 3

Quartiles

To find the quartiles, we must divide the observations on the left and right sides of the median in two equal halves. Since we have an even number of observations on both sides, the lower quartile will be the average of the 3^(rd) and 4^(th) observations. Similarly, the upper quartile is the average of the 9^(th) and 10^(th) observations.

If we call the third number (2-k) and the fourth number (2+k), we have a lower quartile of 2. Additionally, if we call the ninth and tenth numbers (4-l) and (4+l) respectively, we have an upper quartile of 4. (2-k)+(2+k)/2=4/2= 2 [1em] (4-l)+(4+l)/2=8/2= 4

Minimizing the Sum

Now we can illustrate the 12 observations. The dots represent numbers that have not been picked yet.

To minimize the sum, the black dots should assume the same value as the observation that comes before each of them. With this information, we can determine the remaining values.

Let's add all of the observations and simplify.

To minimize the sum, we should set k, m, and l all equal to 0. Therefore, the least value of the sum of the data set is 34. In this case, the data set is the following. 1,1,2,2,2,3,3,3,4,4,4,5

Representing One Variable Data

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