3.7: Stem-and-Leaf Plots
Introduction
Ice Cream Counts
The “Add It Up Ice Cream Stand” has had an excellent summer. Mr. Harris told all of his employees that he is thrilled with the number of ice cream cones that were sold each day.
The last week of August was the most successful week of sales. Here are the counts that Mr. Harris collected on each day during this last week of August.
Mon - 78
Tues - 86
Wed - 52
Thurs - 67
Fri - 70
Sat - 75
Sun - 78
Julie wants to design a beautiful chart to give to Mr. Harris as a gift to show the best sales for the week.
“Why don’t you put those in a stem-and-leaf plot,” Jose suggests when Julie tells him the idea.
“Good idea,” Julie says and she gets to work.
Now it is your turn. You are going to make a stem-and-leaf plot to show Mr. Harris’ ice cream sales for his best week ever.
The title of the stem-and-leaf plot is “THE BEST WEEK EVER.”
Pay attention throughout this lesson so that you can build a stem-and-leaf plot to organize the data.
What You Will Learn
In this lesson you will learn the following skills.
- Organize a set of data in a stem-and-leaf plot.
- Use a stem-and-leaf plot to find the range of a set of data.
- Use a stem-and-leaf plot to find the mean, median and mode of a set of data.
Teaching Time
I. Organize a Set of Data in a Stem-and-Leaf Plot
A stem-and-leaf plot is a visual diagram where you organize numbers according to place value. The data is organized in either ascending or descending order. To build a stem-and-leaf plot, we use place value as our method of organizing data.
If we had a 15 as our number, the stem would be a ten since that is the tens place value. The leaf would be the 5.
To write it as a stem-and-leaf plot, here is what it would look like.
\begin{align*}1\ \bigg |\ 5\end{align*} This means 15.
A stem-and-leaf plot is most useful when looking at a series of data. When we have a series of data, we can organize them according to place value.
Let’s look at an example.
Example
22, 15, 11, 22, 24, 33, 45
Let’s say that we want to organize this data in a stem-and-leaf plot.
First, we organize them by the tens place since all of our numbers have tens places as the highest place value.
11, 15, 22, 22, 24, 33, 45
Next, we put each stem on the left side of our vertical line.
Notice that the largest of each place is on the left of the lines. Now we can put the ones or the stems on the right of the vertical line.
Each number in the data has been organized. The tens place is on the left for each number and the ones places that go with each ten are on the right side of the vertical bar.
This is our completed stem-and-leaf plot.
Helpful Hint 1
Notice that we list repeated values in the chart.
Let’s look at another example.
Example
33, 34, 36, 45, 40, 62, 67, 68
We start by organizing the stems separate from the leaves.
Notice that there isn’t a number in the fifties in the list of data.
We still need to include it in the stem-and-leaf plot. Because of this, we can leave the leaf empty, but we still include the stem.
Helpful Hint 2
List stems that are between numbers even if they don’t have leaves
Include zeros in the leaves for numbers that end in 0
Now that we know how to create a stem-and-leaf plot, how can we interpret the data?
Each stem and set of leaves creates an interval.
Let’s look at the intervals for the stem-and-leaf plot we just created.
The interval for the 30’s is 33 - 36.
The interval for 40’s is 40 - 45.
The interval for 60’s is 62 - 68.
Practice what you have learned. Go ahead and create a stem-and-leaf plot from the following data set.
1. 11, 10, 13, 22, 25, 30, 32, 44, 46, 47, 52, 53, 55, 72
Take a minute to check your work with a neighbor. Did you include a stem of 6?
II. Use a Stem-and-Leaf Plot to Find the Range of a Set of Data
Think back to our work on data. What is the range?
The range is the difference between the maximum score and the minimum score.
We can use a stem-and-leaf plot to find the range of a set of data.
Let’s look at the following example.
Example
The smallest number in the stem-and-leaf plot is 22. You can see that by looking at the first stem and the first leaf.
The greatest number is the last stem and the last leaf on the chart. In this case, the largest number is 55.
To find the range, we subtract the smallest number from the largest number. This difference will give us the range.
55 - 22 = 33
The range is 33 for this set of data.
Look at the following stem-and-leaf plot and answer these questions.
- What is the range for this data set?
- What is the smallest interval?
- What is the largest interval?
How did you do? Is the range accurate? Check your work with a friend.
III. Use a Stem-and-Leaf Plot to Find Mean, Median and Mode of a Set of Data
Remember back to our chapter on data?
We worked with data sets and found the mean, median and mode of each set of data.
The mean is the average of a set of data.
The median is the middle number of a set of data.
The mode is the number that occurs the most in a set of data.
We can use a stem-and-leaf plot to find the mean, median and mode of a set of data.
Let’s look at an example.
Example
Here we have a data set with numbers that range from 35 to 59.
The largest interval is from 55 to 59.
The smallest interval is from 35 to 38.
What is the mean for this set of data?
To find the mean, we add up all of the numbers in the set and divide by the number of values that we added.
35 + 36 + 37 + 38 + 40 + 40 + 41 + 42 + 43 + 55 + 55 + 55 + 56 + 57 + 58 + 59 = 747
We divide by the number of values, which is 16.
\begin{align*}\frac{747}{16} = 46.68\end{align*}
After rounding, our answer is 47.
What is the median for this set of data?
Well, remember that the median is the middle score. We just wrote all of the scores in order from the smallest to the greatest. We can find the middle score by counting to the middle two scores.
42 + 43 These are the two middle scores.
We can find the mean of these two scores and that will give us the median.
42 + 43 = 42.5
The median score is 42.5 for this data set.
What is the mode for this data set?
The mode is the value that appears the most.
In this set of data, 55 is the number that appears the most.
The mode is 55 for this data set.
Real Life Example Completed
Ice Cream Counts
The “Add It Up Ice Cream Stand” has had an excellent summer. Mr. Harris told all of his employees that he is thrilled with the number of ice cream cones that were sold each day.
The last week of August was the most successful week of sales. Here are the counts that Mr. Harris collected on each day during this last week of August.
Mon - 78
Tues - 86
Wed - 52
Thurs - 67
Fri - 70
Sat - 75
Sun - 78
Julie wants to design a beautiful chart to give to Mr. Harris as a gift to show the best sales for the week.
“Why don’t you put those in a stem-and-leaf plot,” Jose suggests when Julie tells him the idea.
“Good idea,” Julie says and she gets to work.
The first thing that we are going to do is to organize the data in a stem-and-leaf plot. The smallest stem is 5 and the largest stem is 8.
We can build the stem-and-leaf plot and fill in the stems and the leaves.
Now we have a stem and leaf plot with the data all arranged.
Use your notebook to answer the following questions on the data.
- What is the smallest number of ice cream cones sold?
- What is the largest number of ice cream cones sold?
- What is the range in the number of cones sold?
- What is the interval with the most values in it?
- What is the mode of this data set?
- What is the average number of cones sold?
Vocabulary
Here are the vocabulary words found in this lesson.
- Stem-and-leaf plot
- a way of organizing numbers in a data set from least to greatest using place value to organize.
- Data
- information that has been collected to represent real life information
- Ascending
- from smallest to largest
- Descending
- from largest to smallest
- Interval
- a specific period or arrangement of data
- Range
- the difference from the largest value to the smallest value
Technology Integration
Khan Academy Stem and Leaf Plots
Other Videos:
- http://www.mathplayground.com/howto_stemleaf.html – Great video on organizing, building and interpreting a stem and leaf plot.
Time to Practice
Directions: Build a stem-and-leaf plot for each of the following data sets.
1. 42, 44, 45, 46, 51, 52, 53, 60
2. 13, 11, 20, 21, 22, 30, 31, 32
3. 44, 45, 46, 48, 51, 53, 55, 67, 69
4. 10, 19, 19, 10, 11, 13, 14, 14, 15
5. 12, 13, 13, 21, 22, 23, 33, 34, 37, 40
6. 45, 46, 46, 46, 52, 52, 54, 77, 78, 79
7. 60, 60, 62, 63, 70, 71, 71, 88, 87, 89
8. 80, 81, 82, 90, 91, 92, 93, 93, 93, 94
9. 11, 12, 12, 13, 14, 14, 20, 29, 30, 32, 32, 52
10. 33, 45, 46, 47, 60, 60, 72, 73, 74, 88, 89
Directions: Use the stem-and-leaf plots that you created to answer the following questions.
11. What is the range of data in the stem-and-leaf plot in problem 2?
12. What is the mean of the set of data in problem 2?
13. If you round the mean to the nearest whole number, what is the mean now?
14. What is the mode of this data set in problem 2?
15. What is the median number in the data set in problem 2?
16. What is the range of the data in the stem-and-leaf plot in problem 6?
17. What is the mean of this set of data?
18. If we were to round this mean what would the new mean be?
19. What is the mode of this data set?
20. What is the median?
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