Question

when the data is quantitative, line graphs, histograms, stem-and-leaf plots, or box-and-whisker
plots are more appropriate choices.

• each of these types of data displays is used for different purposes.
• line graphs show how data changes over time.
• histograms, stem-and-leaf plots, and box-and-whisker plots show the frequency of data and

how it is distributed.
stem-and-leaf plots and box-and-whisker plots can give you a good picture of what the data is
doing. the stem-and-leaf plot provides detail and is extremely easy to construct, while the box-and-
whisker plot clearly shows the data’s median, quartiles, and outliers.
suppose that a teacher gives a math test and the scores are as
follows: 36, 54, 55, 55, 56, 56, 61, 62, 64, 64, 68, 70, 73, 74,
76, 76, 77, 77, 77, 77, 78, 80, 83, 85, 85, 85, 89, 91, 91, 92, 94,
and 97.

• use the list of test scores to finish constructing the stem-

and-leaf plot to the right. remember that in this type of
plot, the largest place value is listed to left and makes up
the stem, while the smaller place value is shown to the
right and is the leaf.
the same test scores are shown on the box-and-whisker
plot below, right.

• use your prior knowledge to label the first quartile,

median, and third quartile on the plot.

• why might you choose to use a box-and-

whisker plot over a stem-and-leaf plot? write
your answer in the box below.

box

box-and-whisker plot image: 35 40 45 50 55 60 65 70 75 80 85 90 95 100 with a box plot

ecap
provide an example of when each type of data display mentioned in this lesson should be used.

lines

Answer:

Constructing the Stem - and - Leaf Plot

First, we list out all the test scores: 36, 54, 55, 55, 56, 56, 61, 62, 64, 64, 68, 70, 73, 74, 76, 76, 77, 77, 77, 77, 78, 80, 83, 85, 85, 85, 89, 91, 91, 92, 94, 97.

Step 1: Stem = 8

We look for all the scores where the tens digit is 8. The scores are 80, 83, 85, 85, 85, 89. So the leaves (the units digits) are 0, 3, 5, 5, 5, 9. So for stem 8, the leaf is 0 3 5 5 5 9.

Step 2: Stem = 7

We look for all the scores where the tens digit is 7. The scores are 70, 73, 74, 76, 76, 77, 77, 77, 77, 78. The units digits (leaves) are 0, 3, 4, 6, 6, 7, 7, 7, 7, 8. So for stem 7, the leaf is 0 3 4 6 6 7 7 7 7 8.

Step 3: Stem = 6

We look for all the scores where the tens digit is 6. The scores are 61, 62, 64, 64, 68. The units digits (leaves) are 1, 2, 4, 4, 8. So for stem 6, the leaf is 1 2 4 4 8.

Step 4: Stem = 5

We look for all the scores where the tens digit is 5. The scores are 54, 55, 55, 56, 56. The units digits (leaves) are 4, 5, 5, 6, 6. So for stem 5, the leaf is 4 5 5 6 6.

Step 5: Stem = 4

We look for all the scores where the tens digit is 4. There are no scores with tens digit 4. So the leaf is empty.

Step 6: Stem = 3

We look for all the scores where the tens digit is 3. The score is 36. The units digit (leaf) is 6. So for stem 3, the leaf is 6.

Labeling the Box - and - Whisker Plot

1. First, we need to find the median (second quartile, \(Q_2\)), first quartile (\(Q_1\)) and third quartile (\(Q_3\)) of the data set.

• The number of data points \(n = 32\) (since we can count the number of scores: let's verify \(1 + 5+10 + 5+2+6 + 3=32\)? Wait, no, let's count the original list: 36 (1), 54,55,55,56,56 (5, total 6), 61,62,64,64,68 (5, total 11), 70,73,74,76,76,77,77,77,77,78 (10, total 21), 80,83,85,85,85,89 (6, total 27), 91,91,92,94,97 (5, total 32)).
• The median (for \(n = 32\), which is even) is the average of the \(\frac{n}{2}=16\)th and \((\frac{n}{2}+ 1)=17\)th values.
• Let's order the data (it's already ordered). The 16th value: let's count:
• First 6 values (up to 56s): 6 values.
• Next 5 values (61 - 68): 5 values (total 11).
• Next 10 values (70 - 78): 10 values (total 21). Wait, no, 6 (50s) + 5 (60s)=11, then 10 (70s) gives 21, then 6 (80s) gives 27, then 5 (90s) gives 32. Wait, the 16th value: 11 (up to 68)+\(x\) where \(x = 5\) (since 11+5 = 16). So the 16th value is the 5th value in the 70s group. The 70s group: 70,73,74,76,76,77,77,77,77,78. The 5th value is 76. The 17th value is the 6th value in the 70s group, which is 77. So the median \(Q_2=\frac{76 + 77}{2}=76.5\).
• The first quartile \(Q_1\): is the median of the first 16 values (since \(n = 32\), the first half is the first 16 values). The first 16 values: up to the 16th value (which is 76). The number of values in the first half \(n_1=16\) (even). The median of the first 16 values is the average of the 8th and 9th values.
• The first 6 values (50s): 6 values. Then the 60s group has 5 values (61,62,64,64,68). So the 7th value is 61, 8th is 62, 9th is 64. Wait, no, let's list the first 16 values:
• 36,54,55,55,56,56,61,62,64,64,68,70,73,74,76,76.
• The 8th value is 62, the 9th value is 64. So \(Q_1=\frac{62 + 64}{2}=63\).
• The third quartile \(Q_3\): is the median of the last 16 values (values from 17th to 32nd). The last 16 values: 77,77,77,77,78,80,83,85,85,85,89,91,91,92,94,97.
• The number of values \(n_2 = 16\) (even). The median is the average of the 8th and 9th values. The 8th value is 85, the 9th value is 85. So \(Q_3=\frac{85+85}{2}=85\).

• On the box - and - whisker plot, the left end of the box is \(Q_1 = 63\), the line inside the box is the median \(Q_2=76.5\), and the right end of the box is \(Q_3 = 85\).

Why choose Box - and - Whisker Plot over Stem - and - Leaf Plot?

A box - and - whisker plot is better for quickly identifying the spread of the data (range, inter - quartile range), the median, and the presence of outliers. It also provides a more concise visual summary of the data's distribution, especially when comparing multiple data sets. A stem - and - leaf plot shows more detailed information about each data point but can be more cluttered and less useful for quickly comparing the overall distribution characteristics (like spread and central tendency at a glance) compared to a box - and - whisker plot.

Example of when to use each data display

• Line Graph: Use a line graph when you want to show how a variable changes over time. For example, if you want to show the change in the average monthly temperature of a city over the course of a year.
• Histogram: Use a histogram when you want to show the frequency distribution of a continuous data set. For example, if you want to show the distribution of the heights of students in a school (heights are continuous data, and you can group them into intervals like 150 - 155 cm, 155 - 160 cm, etc., and show how many students fall into each interval).
• Stem - and - Leaf Plot: Use a stem - and - leaf plot when you want to see the detailed distribution of a data set and also be able to easily reconstruct the original data. For example, if you have the test scores of a small class and you want to see both the overall distribution and the individual scores.
• Box - and - Whisker Plot: Use a box - and - whisker plot when you want to quickly compare the distribution of multiple data sets (e.g., comparing the test scores of two different classes) or when you want to identify the spread, median, and outliers of a single data set at a glance.

Final Stem - and - Leaf Plot (Filled)

Stem	Leaf
8	0 3 5 5 5 9
7	0 3 4 6 6 7 7 7 7 8
6	1 2 4 4 8
5	4 5 5 6 6
4	(empty)
3	6

Final Labeling of Box - and - Whisker Plot

• First Quartile (\(Q_1\)): 63 (left end of the box)
• Median (\(Q_2\)): 76.5 (line inside the box)
• Third Quartile (\(Q_3\)): 85 (right end of the box)