PoW-TER Problem Packet
Mr. Pearson’s Statistics Lesson (Author: Peggy McCloskey)
1. The Problem: Mr. Pearson’s Statistics Lesson [Problem #1586]
After a recent test, Mr. Pearson created the following stem-and-leaf plot to display the
scores of the students in his two math classes.
This was Mr. Pearson's introduction to a statistics unit. Since you are not in Mr. Pearson's
class, you may need to do some research before you will understand the meaning of this
type of graph.
Once you understand the information provided on the graph, use your knowledge to find
the range, mean, median, and mode of the data. Be sure that your explanation includes a
definition of each of these terms and explains how you used the data to find the values.
Bonus: Represent the given data on a histogram.
Note: This problem, Mr. Pearson’s Statistics Lesson, is one of many from the Math
Forum @ Drexel's Problems of the Week Library. Are you interested in having access to
more and also the many teacher resources that the Math Forum provides? View
information about the different levels of membership including a free Trial Account from
this page: http://mathforum.org/pow/productinfo.html
2. About the Problem
Exploring data familiar and meaningful to students will encourage them to discover
mathematics in everyday situations. Using scores from an assessment provides a realistic
approach to representing data and analyzing the data for a middle school child. Finding a
factious set of data from Mr. Pearson’s class protects the middle school ego. One can
find such data in the Math Forum library.
The above problem Mr. Pearson’s Statistics Lesson problem #1586 in the Math Forum
library can be used when teaching a unit on data analysis. This problem presents a stem
and leaf display of grades and asks students to find the measures of central tendency for
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the given data of test scores. The initial goals of the problem would be to have students
demonstrate the ability to read and comprehend a stem and leaf plot, to identifying the
median and mode and to compute the range and mean of a recent test scores in two
classrooms. This Math Forum problem asks the students to include the definitions of
statistical landmarks range, mode, median and mean, and the process used by the students
to compute and to analyze the given data. The writing prompts will give the students an
opportunity to reflect on their learning and make connections to data used in their lives.
An added bonus to using this problem in the classroom is a similar problem in the Math
Forum POW library, Organization of Data # 986, could be assigned as an individual
assessment of the learning goals. Has the student demonstrated an ability to read a stem
and leaf plot? Is the student able to identify and compute the statistical measures of
central tendency? Either problem could also be used as a portfolio entry. Both problems
fit perfectly in a middle school unit on collecting and analyzing data. Mr. Pearson’s
Statistics Lesson and Organization of Data are aligned to the NCTM Curriculum
Standards: Standard 5: Collect, organize and display data; Use proper statistical methods
to analyze data; Develop inferences that are based on that data.
A data set of scores is familiar to most students but the organization of data into a stem
and leaf plot might not be as well known to some students. Since the problem states the
type of representation and suggests that one might need to do some research to
understand the data allow students the time to explore stem and leaf plots or provide them
with tools to discover how to read the data. (index, glossary, reference book and links).
These same resources could be used to review or discover the meaning of the four
statistical landmarks: range, mean, median and mode. If possible assign the research as
homework with the goal of having the students prepared with working definitions to
begin the discussion.
It is important to set the stage of making sure students are able to read and interpret the
data presented in a stem and leaf plot before they can demonstrate a working knowledge
of the statistical landmarks by finding the range, mean, median and mode. Using this
problem in a classroom the discussion could enhance the initial goal of this lesson by
discovering different methods for finding the median (goal 2). An optional goal 3 could
be to introduce the concept of the number of a set, N(x), and use this information as
another method to find the median of a set. Students need the N(x) to find the mean.
When presenting this problem as part of a classroom discussion or as a written
assignment one needs to be aware of common misconceptions students might reveal.
Acknowledge their thinking process and pose channeling question to ensure conceptual
understanding in the discussion. As teacher we channel the discussion through
questioning to achieve the learning goals.
Having studied fifty submissions to Mr. Pearson’s Statistics Lesson at the Math Forum
and used the problem in the classroom I am aware of the misconceptions students reveal
in written and oral work. The children are given a copy of the problem and asked to
begin to solve this problem. The problem is also projected so that once you begin the
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discussion the data is easily referenced. Roam among the students and look for those who
can and cannot read the data. As you do, think about the following questions and try to
determine if they’re appropriate (and for whom) and, if not, other questions you might
pose…
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What do you notice about the data?
What does a typical score in a math test look like?
I wonder if you can find that score in the stem and leaf plot.
What other scores do you notice in the stem and leaf plot?
I wonder what the word stem might represent?
What does the word leaf mean in this context?
I wonder if anyone got a perfect score. How would that be represented?
How could one record a score less than 10 in the display?
If a key to reading the graph was found during individual discovery through research, you
can call on that student to share the discovery. Otherwise another option could be to ask
leading questions to guide the students in reading the plot and creating a key such as 7│9
= 79 to read the data.
3. Sample Student Solutions and Discussion
The following samples were taken from work submitted to the math forum. The six
student generated solutions presented here are what one can envision as student responses
in a classroom discussion where students were given time to tackle this problem.
Selection #1: Reading a Stem and Leaf and a Finger Technique
The range is 54, the mean is 81, the median is 82.5, and the mode is 83 and 98.
Range is maximum minus minimum. To find the maximum, I went 9 l 9 = 99, the highest
number. To find the minimum, I went 4 l 5 = 45. Then, I did 99 - 45 = 54. Then, I
needed to find out the mean. Mean is the average, which I have to add up all the
numbers, and then divided by how many numbers there are on the plot. I added up all the
numbers, and it came up to be 4,293. Then, I counted how many numbers there are in the
plot, which it came out to be 53 numbers. Then, I did 4,293 / 53 = 81. Next, I needed to
find the median.
Median is the middle number of the plot. I used the finger technique to find my median.
My left finger started at 4 l 5, and my right finger started at 9 l 9. When my fingers met, it
was at 82 and 83. I did 83 - 82 = 1 / 2 = 0.50 + 82 = 82.5 because I needed to find the
half of 82 and 83, so the median is 82.5. Finally, I needed to find the mode. A mode is the
most numbers in the plot. I looked at the plot, and saw six 83s and six 98s, which was the
most numbers I saw. The mode is 83 and 98.
Reading the plot is crucial to solving this problem. Ask this student to share a key to
reading the stem and leaf plot (9|9 =99). Ask another to providing a different example
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using data from the problem. The key was reinforced when using the data to compute the
range and when explaining the finger technique of finding the median. The definition of
median was correct and this student explained how his fingers covered data from the left
and from the right till he came to the middle value. The key to reading the graph is
necessary to continue with the learning goals.
Since one of the goals is to find different ways to find the median, call on a student who
mentioned the finger method or who in your roaming the room you saw using this
technique. This would be an ideal time for this student to demonstrate the finger
technique on a stem and leaf plot stressing that one moves increasing order from the left
and decreasing order from the right. Stress that the procedure of crossing out or covering
up works when data is in order. The student made a mistake in the count, hopefully the
mistake will be corrected during the demonstration of the finger technique.
Although not needed in the given data set, the explanation of what to do when two data
items remain is correct. The student knew the median would be the average of the two
middle data pieces. Have ten scores on the board and ask another student to find the
median. Have a small set of scores randomly recorded and ask for a demonstration of the
finger method. It is important to stress that the set be in increasing or decreasing order
before using the finger technique (or cover up process). In the classroom we could use
this student’s response to understand a key and show others how to read the graph. The
finger technique could be used to find the correct median. Conversation on the ideas in
this solution first addresses the goal of reading the plot and provides one method for
finding the median.
Selection #2: Another Method – finding the Median
The Range of this problem is 9, the Mode is 5 and 8, the Median is 2, and the Mean is
77.7.
I got the Range by taking the biggest number in the list and subtracting it by the smalest
number in the list. I got the Mode by seeing wht number appears he most and i happen to
be 5 and 8 because 5 appears 11 times and so do 8. I got the Median by dividing the total
of numbers by 2 and got 26.5 so i rounded the number abd got 27 so what ever was he 27
number was the Median. And last for not least the Mean. I got he mean by adding up all
he scores and divided it by 53 and got the answer 77.7
While roaming you might see a range or a mode found by using the leaf section only.
Although the answers are incorrect, the mistakes are due to not knowing how to read the
graph. It is important to praise what was done correctly. This student knew how to find
the range, mode, and median but applied the knowledge to the leaf section of the plot
only. Now that we have established a key to reading stem and leaf plots we can revise the
answers as we discuss finding the range and the mode according to the definitions stated.
Students who are unfamiliar with the reading of the stem and leaf plot can demonstrate a
working knowledge of the range being the highest value minus the lowest value.
Reinforce this knowledge and review the reading of the data using a key. Ask the class
how the range of 9 was computed. Did anyone report a range of 5? A report of the range
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as 9 (9-0=9) used the highest digit in the leaf minus the lowest digit in the leaf and
another with a range of 5 used the digits in the stem (9-4=5). Ask the question again what
is a typical math score? Does the range of 9 or 5 make sense?
If a key were added to this plot how would you report your range and mode? What was
the maximum test score? What was the minimum test score? What is the difference?
What scores appear the most in the data list? How would you revise your statement
concerning the mode?
Point out that this student was not aware of reading the stem and leaf plot but the
definitions were correct and correctly applied using the data in the leaf section of the plot
in finding the range, mode and median.
Ask if anyone knows another way to find the median. Hopefully when you were roaming
you might have found a student who was dividing by the number of data items. Calling
on this student we could continue the discussion. Goal 2 is looking for another way to
find the median and this selection can reinforce the reading of a stem and leaf plot.
The valuable piece in this response is another method to find the median, goal two in the
lesson. Call on this student to explain the process used. This counting of data items gives
one a lead into introducing the number of a set N(x) - a third or optional goal. This
student divided the number of data items 53 by 2. Getting 26.5 the student knew that the
27th piece of data was the median of the set. The 27 piece of data can be counted
increasingly or decreasingly. Remember again to stress that this works when the data is in
order. Call on another to demonstrate this method using the random data sets you have on
the board.
Discussion on the ideas drawn from the above student response address the initial goal of
reading the plot and demonstrating a working knowledge of range, median and mode.
Goal 2 is addressed and the teacher has the opportunity to address the idea of the Number
of a set.
Selection #3: Median with even and odd number in data set
The range is 54, the mean is 79.3, the median is 82, and there are no mode.
Range, the difference between the highest and lowest numbers in a list, to find the range
you have to do 99-45=54. Mean, (average) add and divide by how many numbers are in
the list. I add all the numbers that were given, and I divide it by 53=79.3.
Median, the middle number, but sometimes it may be two middle numbers, so the first
thing you do is to put the numbers in order you add both numbers and divide it by 2, but
if there's is an odd number you don't do anything. My median is 82 because it was an odd
number.
Mode, the numbers that occurs the most in a list of numbers.But I didn't have no mode for
my answer, because no numbers occurs the most.
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Using ideas expressed in Selection #2 and #3 together the class could come up with
another way of finding the median and eliminate the mistakes of crossing out or the
finger method. Look at the number of the set; that is the number of items in the set.
(notation: N(x) ). This value the number of a set is also needed to find the average.
Selection #2 told us to divide the number by 2. Selection #3 goes further and brings the
concepts of even and odd into the discussion.
What is the median if N(x) is odd? Will the median be an element of the data set? If N(x)
is even is the median an element of the data set? When must one compute the average of
the middle data pieces? In Selection #2 the student knew that the 27th data item was the
median. If the student read the stem and leaf as 8│2 as 82, the median was found
correctly.
Selection #2 and #3 can also lead us into the discussion on mode. Although students
communicate that the mode was the number that occurred the most; in the reporting of
the mode one finds the “most” mistakes. Mode is the number(s) that occur most often.
This is a “happy mistake” in the discussion when one learns what the student means by
“most”. When asked, what is the mode in the given data? Some students only found one
mode; others reported no mode not realizing that there can be multiple modes. Incorrectly
students think if 83 and 98 were each reported six times and no data item was reported 7
times then there is no mode. The highest count or frequency of a data item is the mode.
No mode means each piece of data has the same frequency. Remember students who did
not know how to read the data reported 5 and 8 as the digits appearing most in the leaf
portion. Reiterate that there can be multiple modes and that no mode means each value
occurs in the data set the same number of times
Ideas from selection #3 can continue the discussion on goal 1, demonstrating a working
knowledge of the measures of central tendency. Conversation about the median extends
the thinking on goal 3 to analyze the data with respect to the N(x) being even or odd. This
response with respect to the mode provides opportunity to discuss the frequency of the
data values.
Selection #4: Range Phrasing of Subtraction
The range is 54, the mean is 81, the median is 82, and the modes are 83 and 98.
First I solved for the range if the data set. To find the range I subtracted the maximum
(the greatest number) from the minimum (the smallest number). The maxcimum was 99
and the minimum was 45 so i subracted 45 from 99 (99-45), and that equals 54, so the
range is 54. Then I solved for the mean, or average. The first thing i did was add up all of
the numbers, which equals 4293, and since there were 53 numbers, I had to then divide
4293 by 53, to get the mean of 81.
The next thing I had to solve for was the median. The median is the middle number of a
data set. To solve for the median I simply crossed out the largest number, then the
smallest number, then the second largest number, then the second smallest number, and
so on until there was only one number left, this number is the median.
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The final number that I had to find was the mode. To find the mode you must look at the
set of numbers and then see which numbers are the most reoccurring number. In this case
there were two numbers that reoccurring the most, these numbers were 83 and 98.
To continue developing goal 1 in demonstrating a working knowledge of the statistical
landmarks point out the process of fining the range and the self correction in the given
example. Subtraction is not commutative; students know that intuitively for no one
reports a negative range.
Also encourage the use of the vocabulary of maximum and minimum and average as
another name for mean. In selection #4 one can see the student knows the definition of
landmarks and used the data to show the subtraction model used to find the range. The
student could correct the second sentence using the example given.
Ask how one can compute the average? What is the mean? Many students communicate
that they know the process of finding the mean. They would find the sum of data items
and divide by the number of items. Ask, what a name can we give to the number of
items?. (N(x) the number of the set).
The majority of students make a mistake in computing the sum or counting when solving
for the mean. Note how in finding the mean the student gives us the sum (4293) and the
number of the set (53). If they did not report a sum or the divisor one can not discover
their mistakes easily. Asking what was the divisor used to find the mean, can reinforce
the discussion on the Number of a set. There are two places we use the number of a set;
computing the mean and identifying the median. (Goal 2 and 3)
Selection #5: Correct for testing not for Communication
The mode is 83 and 98, the median is 82, the range is 54, the mean is 81.
45,55,59,61,61,65,65,
Selection #6: Wrap up – the value of communication and example
The range is 54. The mean is 82. The median is 82. The mode is 83 or 98.
To find the range, I remembered the definition of the range. The definition was "The
maximum subtracted by the minimum". I found the minimum was 45 and the max 99. 9945 = 54
To find the mean, I also remembered the definition. The mean is the average of all of the
numbers. I then went on to add all of the numbers and then divide them by the total of the
numbers so that I could find the average.
(45+55+59+61+61+65+65+65+68+68......+99)/53 = 82
To find the median, I looked back in my notes to find that the median means, "The middle
number". I then went on covering up numbers so that I could find the middle number and
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see the median. I found that the middle number was 82. To find the mode, I refreshed my
memory by looking it up in my book. It meant, "The most reoccurring number." I then
went on to look at the most repeated number so that I could see the mode. Both nubmers
83 and 92 were repeated 6 times.
To summarize and model what is expected in a written communication to explain process
and examples to verify solutions, write the values for the landmarks as in Selection #5.
This response is perfect in terms of answering the question. This is what would be needed
in formal testing but this not what was ask for or needed in a collaborative setting. Can
you learn anything from selection # 5?
Selection #6 though is what one is looking for as a response to the initial learning goal of
Math Forum’s Mr. Pearson’s Statistics Lesson. Communication is a key objective of
the problem. Many can follow the explanations and verify the answers. Each definition is
correct; the examples used to illustrate the definitions perfect. The mean is an incorrect
value but the definition and the example are perfect which will give others a chance to
compute the mean when their job is collaboration in the problem solving area and
verification. Since the student showed us what was done we can say the computation
error, or data entry error (calculator or spreadsheet) came from the sum. Asking
questions to elicit responses consistent with selection #6 would summarize the goals of
reading the plot, defining range, mode, median and mean and demonstrate the process by
providing an example. After this last discussion I would assign the revision of an
individual written response to finish in the remaining class time or as a homework
assignment.
4. Conclusion
The given set of data is full of opportunities for the teacher to differentiate instruction
further and introduce analysis of the data. Pose questions to describe Mr. Pearson’s
classes. Use the measures of central tendency in the data to support the description of
Mr. Pearson’s classes or to characterize and describe the test itself. Which measure more
accurately represents and supports their description of the two classes, and or the test.
Another approach to analysis of the data could be a discussion on how changes in
different pieces of data would change the measures of central tendency and how these
changes could be reflected in their descriptions of the classes and or the test. We could
wonder was the test easy or hard? Do the students know the material? Are the classes
homogenous or heterogeneous? How would the measures of central tendency support the
students ideas or descriptions?
An extension to Mr. Peasons’s Statistics Lesson could lead the students to organize a
double stem and leaf plot separating the data into two classes. Comparisons of the
landmarks of each class can be made and inferences described based on their creations
and supported by the new landmarks of each class. Other extensions could be to create a
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frequency table, a histogram, and a box and whisker plot. Technology can be integrated
using Excel, and TI-84 to represent the data.
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