7.4.1 Mean, Median & Range
Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw conclusions about the data, compare different data sets, and make predictions.
For example: By looking at data from the past, Sandy calculated that the mean gas mileage for her car was 28 miles per gallon. She expects to travel 400 miles during the next week. Predict the approximate number of gallons that she will use.
Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet to examine this impact.
For example: How does dropping the lowest test score affect a student's mean test score?
Overview
Standard 7.4.1 Essential Understandings
Prior to 7th grade, students have calculated the mean, median and range of a set of data. Students working in this standard will expand their understanding to calculating measures of center and spread given a data display. They will draw conclusions from the displays, including the effects that inserting a data value or deleting a data value will have on the mean, median and range. Students will also use spreadsheets and other technology to display these representations. Students will move from these displays in 7th grade to displaying bivariate data using scatterplots in 8th grade.
In 7th grade, students will be able to compare and draw conclusions about 2 data sets using the calculations of mean, median, and range.
All Standard Benchmarks
7.4.1.1
Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw conclusions about the data, compare different data sets, and make predictions.
7.4.1.2
Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet to examine this impact.
7.4.1.1
Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw conclusions about the data, compare different data sets, and make predictions.
For example: By looking at data from the past, Sandy calculated that the mean gas mileage for her car was 28 miles per gallon. She expects to travel 400 miles during the next week. Predict the approximate number of gallons that she will use.
7.4.1.2
Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet to examine this impact. For example: How does dropping the lowest test score affect a student's mean test score?
What students should know and be able to do [at a mastery level] related to these benchmarks
- Collect and organize data in a variety of displays.
- Compare data sets.
- Predict patterns and trends for the data; for example, if one knows how much 12 cans of juice cost, how can they use that to help them predict how much 20 cans would cost
- Be able to find a missing value if the mean is known, know how many data values there are, and what all but one of the data values is. For example, there are 5 scores. If the mean on the 5 scores is 17, and four of the scores are 16, 19, 17, and 14, what is the 5th score; answer is 19.
- Understand that inserting or deleting an additional data point may impact the mean and median differently.
- Design, create, and display simple experiments to collect data.
- Analyze the data collected and make general statements about the data, including comparing different data sets to one another.
Work from previous grades that supports this new learning includes:
- know and use definitions of mean, median and range.
- calculate mean, median and range
- know how to create and analyze data in a variety of ways: tables, bar graphs, pictographs, number line plots, double bar graphs, line graphs and spreadsheets.
- collecting data
- arithmetic with fractions, percentages and decimals
- graphing relationships between variables
- representing quantities with variables
- doing prob/stat provides a valuable setting for practicing other math skills.
- division
- ordering numbers
- using frequency tables
- reading picture graphs, bar graphs, double bar graphs, line graphs, Venn diagrams, and number line plots
NCTM Standards
Data Analysis and Probability Standard for Grades 6-8
Select and use appropriate statistical methods to analyze data.
- find, use, and interpret measures of center and spread, including mean and interquartile range.
- discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots and scatterplots.
Common Core State Standards (CCSS)
Statistics and Probability
6.SP: Develop understanding of statistical variability.
6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.
6.SP.2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
6.SP.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
- 6.SP.3.a. Summarize and describe distributions.
6.SP.4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.5. Summarize numerical data sets in relation to their context, such as by:
- 6.SP.5.a. Reporting the number of observations.
- 6.SP.5.b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
- 6.SP.5.c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
- 6.SP.5.d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
Misconceptions
Student Misconceptions and Common Errors
- When finding the median of a set of even numbered data (e.g., 10 pieces of data), they sometimes struggle to see that the median may NOT be listed, and that it could be the 'average' (mean) of the middle two numbers, or any number between the middle two numbers. However, they should use mental math in some instances; e.g., if the two middle numbers are 40 and 42, they should be able to see that the number halfway between is 41.
- When students are asked to find the mean of a data set and have access to their calculator, they may forget to enter the calculations using correct order of operations; i.e., entering 2 + 3 + 4 + 5+ 6 / 5 in one line.
- When trying to see what deleting a low value does to the original mean, students think it lowers the new mean because they took out a low value.
- If students are making graphs to display the data they have collected, it is important that they use the same scales on the graphs so that it is easier to compare the graphs/data.
- If given a set of data and asked to find the median, students sometimes will forget to put the data in order from least to greatest first before trying to find the middle number.
Vignette
In the Classroom
Vignette.1
I heard a nice story a few years back. I don't know if it's true, but it's fun anyway. The story was that the University of North Carolina surveyed their graduates asking them how much money they made, and calculated the average salary for each major. Can you guess which major had the highest average salary?
Answer:
Geography |
Explanation:
Michael Jordan majored in Geography. |
Geography faculty at the University of North Carolina like to point out that in 1986, those who graduated with a major in Geography had the highest average starting salaries in the class - $250,000. The punchline to this joke is that basketball legend, Michael Jordon, graduated from UNC with a major in Geography in 1986. In that particular data set, Michael Jordan is clearly an outlier whose astronomical earnings skew the results and obscure the real market for geography majors.
Vignette.2
How tall is an average person?
The students listened with delight as I read aloud a portion of Chapter 1 from Harry Potter and the Sorcerer's Stone. They were all curious as to how I would tie this book into a math lesson.
I stopped reading after I read the passage that describes Hagrid's size. Then I asked, "How big is Hagrid?" Students repeated the information from the story, that he was twice as tall and five times as wide as a usual person.
I then asked, "How do we find out how tall a 'usual' person is?"
Hands shot up. One student suggested we find the average height of the students in the class. We had recently worked with measures of central tendency (mean, median, and mode). When I asked what an average was, Lisa said, "We add up the heights of everyone and divide by the number of people."
Mike said, "It can also be the height that happens the most."
Tim added, "We could also find the one in the middle."
We decided as a class to measure everyone's height, add up all of the heights and then divide by the number of students in the class. Then I brought up the issue of measuring accurately. I told them we needed to come up with a consistent way to measure everyone. Anna said that everyone would need to take off his or her shoes before we measured. We decided that was a good idea and that the best way to measure would be to stand up against a wall and use yardsticks or meter sticks.
Then I brought up another issue: Units. I asked the students whether they all wanted to use centimeters or inches. There was some disagreement on which to use, but when we put it to a vote, inches won. I created a large table on the chalkboard that had columns for the students' names, their heights, and their shoulder spans (see below).
Name | Height (inches) | shoulder span (inches) |
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I arranged the students into groups of three. I explained to the students that each group needed to create a similar table for its data and also record its information on my table once the group had finished measuring. I asked them to first measure the height of each person in the group and then we would start over and measure shoulder spans. One member from each group came up and took two yardsticks; each group then found its own area in the room or in the hall where there was a wall that could be used to lean up against for the measuring.
As the students began to work, I observed many different strategies for measuring. One group decided it would be easiest and most accurate to tape the yardsticks to the wall and just have the person stand in front of them. This proved to be more difficult than the students had planned, for they discovered that one end of each yardstick had an extra quarter inch on it. Melissa said, "Just cover up that last one-fourth inch."
Jessie argued, "You can't do that, it wouldn't be accurate that way!"
Students were concerned about getting accurate measures. As Timothy held the yardstick up to Nathan, he said, "The yardstick is unreliable because it isn't long enough to measure someone."
In another group, students argued about how to hold the yardstick. Sean said, "Hold it straight! Wait, it's not straight, it won't be accurate if it's not straight. You'll be shorter if it's not straight!"
A fourth group had their yardsticks taped to the wall and had each student put his or her back up to the wall to measure. Andrea asked me, "What if it falls in between two inches?" I asked all of the groups to pause and posed the same question to them. As a class, we decided to round to the nearest quarter inch.
Students recorded their heights on the board. As the table filled with measurements, students began to look at the data and notice discrepancies.
Michael said, "Hey, I know I'm taller than Elizabeth, but her number on the board is bigger than mine."
I asked how we should remedy this, and everyone agreed that we should re-measure the heights that were in question. This initial measuring process took longer than expected as students took time to decide how to measure accurately and then had to re-measure in some cases.
The next step was to measure shoulder spans. I told the class that the easiest way would be to have one student stand next to the wall with his or her shoulder touching the wall and have another student measure out from the wall across to the other shoulder with the yardstick. We discussed the importance of having the yardstick perpendicular to the wall, just as the yardstick for measuring height had to be perpendicular to the floor. The second round of measuring went more quickly. Students copied the class data onto their own tables so that they would be easier to analyze and manipulate. Once the data were complete in the class table, several students began to calculate the mean of each measurement (height and shoulder span) with their calculators.
I asked the students to stop calculating for a few moments so we could have a discussion. "What is a mode" I asked.
Amy said, "Um, mode is the most."
I asked, "You mean the number that occurs the most?"
She replied, "Yes."
Then I asked, "What's a median?"
Charlotte said, "The median is the one in the middle." I asked her if she had any special way of remembering that.
She said no, but Michael immediately raised his hand, saying, "I remember it because the median is the middle of the road!"
"Oh, that's a good way to think about it," I replied. Then I asked, "What's a mean?"
Lisa explained, "That's when you add them all up and divide."
Next I asked them how they were going to enter a number like 12¼ into their calculators. Nathan said, "One-fourth is point two five on a calculator." I asked him "Why?" and he said, "Because twenty-five is one-fourth of a hundred."
Next I asked, "Then what is twelve and three-fourths?"
Several students replied in unison that it would be 12.75.
I explained that we were going to find each of these measures of central tendency for our class data about heights. I asked students if they had a good strategy for finding the median. Kendra replied that it would be better if the data were in order from smallest to largest. I said, "OK, so please use a piece of paper to rewrite each person's height in order from smallest to largest."
After students had time to find the class median height (57.75 inches) I asked, "Is there any way we can check to make sure that the median of fifty-seven point seventy-five is accurate?"
Angela said, "We could line everyone up by height." The class agreed this was a good idea, so everyone lined up from shortest to tallest.
I asked, "If there are twenty people in class, where is the middle person?"
Everyone replied, "Ten!" So, we counted in ten from one side and ten from the other, which left two students, Alex and Michael, in the middle. I asked each what his height was. Alex was 57.5 inches and Michael was 58 inches. I asked if that information agreed with the results we had gotten for the median. Elizabeth said, "Yes, because fifty-seven point seventy-five is between fifty-seven and a half and fifty-eight. So the median is right between them."
Alex chimed in with, "Yeah, because twenty-five down from seventy-five is fifty."
Students worked in their groups to find the mean, median, and mode for the shoulder span data, just as they had done for the height data.
To conclude this part of the investigation, I asked students to think back to the Harry Potter book. Hagrid was five times as wide and twice as tall as a typical person. Could we now figure out about how big Hagrid was? We discussed which of the measures to use: mode, mean, or median. Students decided that either the median or the mean would be good choices because they more accurately represented the middle of their data. Using a median of 57.75 and mean of 57.61 for height, and a median of 14.5 and a mean of 14.275 for shoulder span, they figured that Hagrid must be about 9 ½ feet tall and about 6 feet wide!
Follow-up Activity
I next asked students, "Do you think there is a relationship between a person's height and a person's shoulder span?" They looked at the class table of data and shared various ideas. Most students thought that a taller person would have a wider shoulder span. Alonso said, "If you divide the mean for the height by the mean for the shoulder span, it's about four times, so maybe the shoulder span is one-fourth of a person's height."
Some mentioned that the relationship wasn't always the same, pointing to some specific names on the chart that didn't fit that rule. Elizabeth said, "Why don't we divide the height by the shoulder span for every person and see if those are the same?" I said this was a good idea, but we were going to look at the data in a new way to help us see if there was a relationship.
I posted a piece of inch-squared flip-chart paper on the board. I drew two axes and labeled the horizontal (x) axis Shoulder Span and the vertical (y) axis Height. As a class we decided how to scale the graph. We decided to count each square on the x-axis as a half inch so the data wouldn't be too crowded. We also decided not to start the scales at zero, but to start at 10 inches for shoulder span and 50 inches for height, so that the data would fit on the graph. Each student came up to the chart and slotted a point to represent his or her measurements. As I created the scatter plot on the chart paper, students created their own scatter plots on centimeter-squared paper.
This was the students' first experience with a scatter plot and it prompted excellent discussion about what it means to have a relationship between two variables, in this case height and shoulder span. Nathan said, "The people who were the tallest were also the widest." Others noted that in general, the taller people had the wider shoulder spans. I asked a student to come up and place the yardstick so that it went through the middle of the data points, showing the trend. Markita said, "Maybe some of the points are off because we didn't measure exactly right."
Alex added, "That's true, but even so, each person isn't going to be exactly on the line."
I posed a question to the class: "So, even though many students don't fall on the line, do you think there is a relationship between the two variables?" Students shared various ideas with the class and then I asked them to each write an explanation of their thinking.
In conclusion, we tried to plot Hagrid's estimated measurements on the scatter plot. His data was off the chart, but it was clear that his data point wouldn't fit along the line we had created. Students reasoned that if both his height and shoulder span were twice as big (or five times as wide), then it would be on the line they created. As it is, Hagrid's data point would be significantly below the line, since his width was five times larger and his height was just twice as big. This exploration enabled students to get experience in measuring accurately and in analyzing statistics in order to draw conclusions. Students were highly engaged and reluctant to stop at the end of the investigation.
Taking directly from: Bay-Williams, J. M. (2004). Math and literature 6-8, "Harry Potter and the Sorcerer's Stone." Math Solutions. (1st ed.). 38-46.
Resources
Teacher Notes
Students may need support in further development of previously studied concepts and skills. Students may know how to find average, but not know WHY they are doing what they are doing. By defining mean as "average/equal distribution" or a "balance point" of a set of data, students can better understand why they are doing what they are doing when they find mean.
- Give students a list of numbers and ask them to calculate the mean, allowing them to use their calculator. More than likely at least one student will add and divide in the same step on the calculator without parentheses , therefore causing an incorrect mean. Have the discussion with students about correct order of operations and always asking themselves if the answers make sense in the original context.
Ex: Find the mean of 5, 4, 3, 2, 1. If done incorrectly on the calculator they would get 14.2. Students need to realize if they get that answer that they did something wrong because the average of that list could not be 14.2.
- When calculating the mean of a data set on graphing calculator, students may make an order of operations error. For example, students may inaccurately calculate the mean of 18, 15, and 27 to be 42 by typing 18+15+27÷3 in their calculators. Be sure to review correct order of operations and make sure the students either type the '=' sign before dividing, or put parenthesis around the addends to avoid this mistake.
- Given the following test scores, a student wants to earn the highest grade possible. The teacher has told the students they can select what they want for their final grade. Which one, mean or median, would give the student the best possible score?
Test scores: 88, 93, 84, 91
Median: 89.5
Mean: 89
Median is the higher score, so it is the better option.
Test Scores: 88, 97, 84, 91
Median: 89.5
Mean: 90
Mean is the higher score, so it is the better option
- A 'tip' to finding the median can be presented after working with median for many examples. In an ordered set of numbers, the median can be found by the following: (n+1)/2th position, where n is the number of data in the set. For example, if there are 15 data values, the position of the median would be (15 + 1) / 2 = 8, so the number in the 8th position is the median. If there are 20 data values, the position of the median would be (20 + 1) / 2 = 10.5, so the "number" (not really a number in the data set) in between the 10th and 11th number in the ordered data is the median.
- In this benchmark students are required to calculate mean, median and range from a given graph. The possible graphs include all graphs from previous grades: bar graph, double bar graph, and line plots. The graphs given may also include stem and leaf plot. Students will not be expected to make a stem and leaf, but should be able to read the data within the plot; calculate and interpret the mean, median, or range; and make predictions and conclusions based on the data.
Note: Mode is not listed in the standards, but this benchmark might be a natural place to talk about mode and the term "measures of center." Since mode is one of the measures of center, this would be a natural place to expand on this topic. In many resources, where you find mean and median, you will also find mode.
- Data Grapher
- Averages and The Phantom Tollbooth Students participate in activities in which they focus on connections between mathematics and children's literature. Using The Phantom Tollbooth as a literature basis, students explore the concept of averages.
- Mean and Median This applet allows the user to investigate the mean, median, and box-and-whisker plot for a set of data that they create. The data set may contain up to 15 integers, each with a value from 0 to 100.
- State Data Map Information can be represented in many ways, and this applet allows the user to represent data about the states using colors. The state with the highest data value is darkest; other states are shaded proportionally. Investigate any of the data provided-or enter data of your own (population, letters in states, number of representatives, etc.).
Additional Instructional Resources
- How to Determine the Mean A Youtube video on 3 different problems of finding mean.
- Simulation on mean and median This Java applet shows how the relationship between the mean and the median and illustrates several aspects of these measures of central tendency.
- Mean, Median & Mode Learning Upgrade. A Youtube video that explains mean, median, and mode in a song.
- Mean, Median, Mode & Range A BrainPOP video (will open but need subscription to review).
- How do you read a Stem and Leaf Plot? A very basic step by step video.
- Bay-Williams, J. M. (2004). Math and literature 6-8, "Harry Potter and the Sorcerer's Stone," Math Solutions (1st ed.). 38-46.
stem and leaf plot: A plot where each data value is split into a "leaf" (usually the last digit) and a "stem" (the other digits). For example "32" would be split into "3" (stem) and "2" (leaf).
The "stem" is used to group the scores and each "leaf" indicates the individual scores within each group.
quantitative data: Data which can be measured
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- Can students predict the mean given a set of data?
- Can students predict what will happen to a mean or median if higher or lower data values are added?
- What does the shape of the display tell us about the locations of the mean and median?
- What patterns do they see in the line plot?
- Can students define mean in more than one way (as an equal distribution and as a balance point)?
Assessment
- A packing company mailed six packages with a mean weight of 4.8 pounds. Suppose the mean weight of five of these packages is 5.1 pounds. What is the weight of the sixth package?
A. 4.95 B. 5.1 C. 3.3 D. 4.85
Answer: C
- The mean of six numbers is 18. If one number is 23, what is the mean of the other five numbers?
A. 16 B. 20 C. 25 D. 17
Answer: D
- Given a data set consisting of 8 numbers, describe the effect increasing each data value by one would have on the mean, median, and range. Use examples to support your answers.
Effect on mean:
Effect on median:
Effect on range:
- Which set has the greater range?
Set T: {3, 11, 7, 18, 2, 9}
Set W: {6, 11, 21, 19, 17, 5}
How much greater is that set's range than the other set's range?
Answer: Set W, (range of 16, compared to range of 15 for Set T), 1 more
● Charlie's scores for eight assignments are given:
21 27 31 23 28 18 23 x
If the median for his scores is 25, what is the possible value for x?
A. 23 B. 25 C. 28 D. 26
Answer: C
Taken from Grade 7 Unit 7 - Data Distributions MCAS Released Items
http://cpsdmath.wikispaces.com/file/view/Grade+7+Unit+7+MCAS+Items.pdf
Correct answer: C
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"Non-Routine Mean" Problems
Problem: | The mean of 29 test scores is 77.8. What is the sum of these test scores? | |
Solution: | To find the mean of n numbers, we divide the sum of the n numbers by n. |
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| If we let n = 29, we can work backwards to find the sum of these test scores. |
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| Multiplying the mean by the 29 we get: 77.8 x 29 = 2,256.2 |
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Answer: | The sum of these test scores is 2,256.2
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- The mean of a set of numbers is 54. The sum of the numbers is 1,350. How many numbers are in the set?
To find n, we need to divide 1,350 by 54.
n = 25, so there are 25 numbers in the set.
- Gini's test scores are 95, 82, 76, and 88. What score must she get on the fifth test in order to achieve an average of 84 on all five tests?
Solution: 79
- The Lachance family must drive an average of 250 miles per day to complete their vacation on time. On the first five days, they travel 220 miles, 300 miles, 210 miles, 275 miles and 240 miles. How many miles must they travel on the sixth day in order to finish their vacation on time?
Solution: 255 miles
http://www.mathgoodies.com/lessons/vol8/non_routine_mean.html
Differentiation
- Put numbers in order for the students.
- Have no more than 6 pieces of data.
- This is a good standard to meet the needs of kinesthetic learners by collecting data (to be used for calculating measures of center) that require movement. For example, students could do jumping jacks for 2 minutes and measure their pulse.
- Suggest to students when finding median to write them on a strip of paper in order and then physically fold the paper in half to help them find the median; also can have them cross off one at each end of the number line to gradually get to the median, or middle number.
- Memory tricks can be helpful in so many areas. Mean, median, (and mode) are good examples of that fact.
- Whenever possible provide ELL students with a list of essential vocabulary (for example: mean, median, range) a day or two before the introduction of a new lesson so that the students can use a simplified English or bilingual dictionary to learn the meanings and familiarize themselves with the words. This approach also helps students identify prior knowledge about the topic from their native languages.
- Teach students to use graphic organizers such as webs, Venn diagrams, and charts to help them better comprehend these texts. These are visual tools that help ELL students understand and organize information. They are like mind maps which promote active learning. Graphic organizers can also help students develop higher level thinking skills and promote creativity.
- Memory tricks can be very helpful in so many areas. Mean, median, (and mode) are good examples of that fact.
- Students can examine outliers in data sets. They could look at what happens to mean, median, and range when an outlier is either added to or taken away from a data set.
- Introduce box and whisker plots to students. This seems a natural transition as an extension for some students. A box-and-whisker plot can be useful for handling many data values. They allow people to explore data and to draw informal conclusions when two or more variables are present. It shows only certain statistics rather than all the data. Five-number summary is another name for the visual representations of the box-and-whisker plot. The five-number summary consists of the median, the quartiles, and the smallest and greatest values in the distribution. Immediate visuals of a box-and-whisker plot are the center, the spread, and the overall range of distribution.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: | Teachers are: |
collecting meaningful data (this standard is a good place to review measurement skills by collecting data about student heights, shoe length or arm span). | giving students multiple data sets for students to compare. |
calculating mean, mean (measures of center) and range; displaying data in line plots. | asking "what would happen if" questions about data sets and displays. For example: What would happen to the mean, median and range if you added a data item to the set that is above the mean? |
using graphing technologies to calculate measures of center and display data. | helping students visualize the mean of a data set as the balance point. |
looking for and describing patterns in a line plots shape and clusters in the data. | asking questions about the shape of displaying data and clusters of data. |
creating data sets given specific means, medians, and ranges. | allowing students to design their own experiments, collect data, and analyze results. |
calculating mean, median and range of a data set given only a line plot. | reminding students to pay attention to correct order of operations on a calculator when calculating the mean. |
Parent Resources
- This website explains the concepts of mean, median, and range.
- Review of line plots: a website explaining the data display of stem and leaf plot.
- Another presentation of stem and leaf plots.