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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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Representing One Variable Data
Slide of 12
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

The following box plot represents a data set of observations.

Box plot with a minimum value of 20, a maximum value of 90, a lower quartile of 60, an upper quartile of 80 and a median of 70
Which of the following statements about the box plot are true?

Let's consider the statements one at a time.

Statement A

From the diagram, we see that the maximum value is 90. Therefore, this value is definitely in the data set which means the statement A is true.

Statement B

Be careful not to confuse the median and mean of a data set. The vertical line inside of a box plot shows the median of a data set. The box plot does not show the mean.

We do not have enough information to calculate the mean. It could be 70, but it does not have to be. That means statement B is not true.

Statement C

Notice that there are 107 observations in the data set. This means that the 54^(th) observation is the median when the data set is ordered from least to greatest.

Similarly, since we have an odd number of observations on the left and right side of the median, the quartiles are represented by the 25^(th) and 81^(th) observation, respectively. That is, the set contains the first and third quartiles.

From the box plot, we see that the upper quartile is 80. Consequently, the data set contains the value 80. Statement C is true.

Statement D

From the diagram, we see that the lower quartile is 60. In any data set, at most 25 % of the observations are below the lower quartile Q_1.

Therefore, statement D is true.

Statement E

Let's first count the number of observations that are in each part of the box. Let's zoom in on the box and illustrate the number of observations in it.

Calculate the sum of the observations. 1+26+1+26+1=55 As we can see, 55 observations are somewhere in the box including the start and end of the box. In theory, it is possible that all values in the box only fall on the quartiles and median. Let's calculate what percentage of the set these 55 observations represent. 55/107≈ 51.4 % The 55 observations represent more than 50 % of the set. Therefore, statement E is false.

In a statistics class, all the students had their heights measured. The values are presented in the following histogram.

Histogram: [3,150-155],[2,155-160],[5,160-165],[6,165-170],[4,170-175],[2,175-180],[1180-185],[2,185-190]
 How many of the students are at least centimeters tall?
What is the percentage of students who are at least centimeters tall but less than centimeters tall?

How many students are at most centimeters tall? Use the following dot plot to solve the question.

Dot plot showing the number of students of different lengths

The students who are at least 175 centimeters tall are those in the last three bars of the histogram.

These bars show 2, 1, and 2 students respectively, for a total of 5 students.

Let's mark the part of the histogram showing students who are at least 160 centimeters tall but less than 180 centimeters tall.

Next, we will add the lengths of these bars to obtain the number of students within this height range. 5+6+4+2=17 To determine the percentage of students who are in this interval, we must also find the total number of students in the class. To do that, we will sum the number of students in each interval of the histogram. 3 + 2 + 5 + 6 + 4 + 2 + 1 + 2 = 25 Now we can determine the percentage of students in this interval.

Therefore, 68 % are at least 160 centimeters tall but less than 180 centimeters tall.

Notice that the measure 163 centimeters is in the 160-165 bar. However, a histogram does not tell us anything about the distribution of values within one of the bars. That means we cannot use the histogram to determine how many students are less than 163 centimeters tall. Here is where the dot plot comes in handy.

A dot plot shows the individual observations. As we can see, all students in the histogram's 160-165 bar are, in fact, 160 centimeters tall. We now can determine the total number of students who are at most 163 centimeters tall. 1+2+2+5=10

The following box plot represents a data set of observations.

A gray box plot with a minimum value of 0, a lower quartile of 3, a median of 5 a upper quartile of 8 and a maximum value of 10

Which of the following dot plots could represent the box plot?

Six different dot plots showing different data sets.

Examining the box plot, we notice that it has a minimum value of 0 and a maximum value of 10. Of the six box plots, dot plot B has a minimum value of 1 and dot plot E has a maximum value of 9. Therefore, they cannot describe the box plot.

To determine which of the remaining options could fit the box plot, we should consider the number of observations in the data set. Since we have 15 observations, the median and quartiles are represented by observations in the data set.

As we can see, the lower and upper quartiles are the 4^(th) and 12^(th) observations, respectively and the median is the 8^(th) observation.

Let's mark the 4^(th), 8^(th), and 12^(th) observations in the remaining dot plots. If a dot plot reflects the box plot, these observations should be at the tick-marks corresponding to 3, 5, and 8, respectively.

As we can see, A and D have the correct quartiles and median while C and F do not. Therefore, we also have to discard C and F.

Of the given dot plots, only A and D could be used to describe the box plot.

Below we see the five number summary for a data set containing observations.
What is the greatest possible sum of the data set if all of the observations are integers?

Let's first draw a diagram that illustrates the data set in ascending order.

To get the highest possible sum for this data set, we want each observation to be as high as possible.

Minimum and Maximum Value

From the exercise, we have been given the minimum and maximum value. This means the first and last observations are locked in.


Median

To find the median, we have to consider the number of observations in the data set. The data set contains 7 observations which is an odd number. Therefore, the median must be the 4^(th) observation.


Quartiles

To find the quartiles, we divide the observations that are on the left and right sides of the median in two equal halves. Because the number of observations on both sides is odd, the lower and upper quartile will be the 2^(nd) and 6^(th) observations, respectively. This locks in two more values.


Remaining Values

We have two remaining values which we can pick arbitrarily as long as they are not less than the previous observation and not greater than the observation that comes after. Since we want the sum of to be the greatest possible, we will let these observations take the same value as the one that comes after.


Greatest Possible Sum

By adding all of the observations, we get the greatest possible sum of the data set. 1+2+3+3+4+4+5=22

Representing One Variable Data

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