Grade 8 Workbook — Chapter 16: The Data Cycle and Boxplots

SOL 8.PS.2 · Companion to Textbook Chapter 16

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 150.

PAGE 1 — Chapter opener

Chapter 16 · The Data Cycle and Boxplots

Standard 8.PS.2

In this chapter you will:

Run all four stages of the data cycle with boxplots as the display
Write a question that needs data, and decide how to collect it
Explain how statistical bias keeps a sample from representing the population
Represent a data set of at most 20 items with a boxplot
Find the lower extreme, lower quartile, median, upper quartile, upper extreme, range, and interquartile range
Describe what an extreme data point (outlier) does to shape and spread
Compare two data sets shown in boxplots
Justify which display fits a situation, and spot a misleading display

Words to know: data cycle · statistical question · population · sample · statistical bias · random selection · five-number summary · lower extreme · upper extreme · median · lower quartile · upper quartile · range · interquartile range · boxplot · whisker · outlier · resistant · skewed

Quartile rule used all chapter. Order the data. The median splits it into a lower half and an upper half; when the count is odd the median belongs to neither half. $Q_1$ is the median of the lower half, $Q_3$ is the median of the upper half.

A boxplot does not show individual data values, how many values there are, or the mean. Every data set here has at most 20 items.

PAGE 2 — The four stages

16.1 The Data Cycle, Pointed at Boxplots

FIGURE: fig1-data-cycle-boxplots.png (full width)

Label the loop. Write the four stage names in order.

Stage one ____________________ Stage two ____________________

Stage three ____________________ Stage four ____________________

Boxplot or not? Check the questions a boxplot could answer.

☐ What is a typical value? ☐ What is everyone's favorite sport?

☐ How spread out are the values? ☐ How many students said exactly 20?

☐ How do two groups compare?

Name the four stages of the data cycle in order.
Which stage includes computing the five-number summary? __________
Which stage includes deciding whom to survey? __________
A student says the cycle is finished the moment the boxplot is drawn. Correct the statement in one sentence.
Name two things a boxplot shows and two things it does not show.

Shows: ______________________ ______________________

Does not show: ______________________ ______________________
Which stage does "acquiring existing data from a website" belong to? __________

PAGE 3 — Practicing the cycle

Independent practice

Classify each activity by stage.

a) writing the survey question ______ b) sorting 14 numbers ______

c) emailing your conclusion ______ d) recording 18 measurements ______
Explain in one sentence why the data cycle is drawn as a loop.
Write a statistical question about your class that a boxplot could summarize.
Write a question about your class that a boxplot could not answer, and say why not.
For your question in item 9, state exactly what one piece of data you would record from each person.
A boxplot is drawn from 16 test scores. Can you tell how many students scored exactly $85$ ? Explain.
Can you find the mean of a data set from its boxplot? Explain.
Application. A school nurse wants to know whether eighth graders get enough sleep. State the question, the data needed, and how you would collect it from at most 20 students.
Reasoning. Name one thing a boxplot does better than a histogram, and one thing a histogram does better.

Exit ticket 16.1

Name the four stages in order. ____________________________________
In which stage is the boxplot drawn? __________
One question about center: ______________________ One about spread: ______________________
Give one reason a finished boxplot might send you back to stage 1.

PAGE 4 — Questions, data, and bias

16.2 Asking the Question, Getting the Data, and Statistical Bias

A sharp question names four things. Fill in the blanks.

Who — the ____________ you want to describe

What — the exact ____________ measured, with units

When — the ____________ frame

Why — a reason to expect the answers to ____________

Four kinds of bias. Match each to its description.

Bias	Description
selection
response
nonresponse
voluntary response

Guided practice

Define population and sample in your own words.

Population: ____________________________________________

Sample: ________________________________________________
Statistical and numeric? Circle yes or no.

a) pets per student — yes / no b) school mascot — yes / no c) minutes per bus run — yes / no
Name the population and the sample: a coach measures the vertical jump of 15 of the 60 students who tried out.

Population: ______________________ Sample: ______________________
Name the bias: a lunch survey given only to students who buy school lunch. __________________
Name the bias: "Wouldn't you agree our library needs more computers?" __________________
Explain in one sentence how random selection reduces bias.

PAGE 5 — Bias practice

Independent practice

Sharpen each question so a boxplot could answer it.

a) Do students sleep enough? _________________________________

b) Is the walk to school long? _______________________________

c) Do eighth graders read? __________________________________
For each question in item 26, name the quantity recorded and its units.

a) ______________ b) ______________ c) ______________
For each question in item 26, name the collection method and why.

a) ______________ b) ______________ c) ______________
Name the bias and the direction it pushes results.

a) Asking about exercise only at a gym. ________________________

b) Emailing a survey about internet speed. ______________________

c) Asking students their grades out loud. _______________________

d) A form anyone may fill out about school spirit. ________________
A student says a sample of 20 is biased just because 20 is small. Correct the statement.
A student records bus-ride times for the 12 riders on her own bus to describe all riders. Representative? Explain.
Describe a way to choose 20 students at random from a roster of 340.
Give an example of acquiring existing data about temperatures, and one thing to check about the source.
Application. Homework minutes on a school night: write the sharpened question, the population, a random sampling plan for at most 20 students, and one bias it avoids.
Error analysis. "I asked 20 friends, so my sample represents the school." What is wrong?

Exit ticket 16.2

Define statistical bias. ______________________________________
Population ______________________ Sample ______________________ (18 of 200 band members are timed)
Bias and fix: a travel-distance survey handed out in the car-rider line. ______________________
Why can bias not be detected from the list of numbers alone? ______________________

PAGE 6 — The five-number summary

16.3 The Five-Number Summary

Fill in the table.

Statistic	What it is
lower extreme (minimum)
lower quartile ( $Q_1$ )
median
upper quartile ( $Q_3$ )
upper extreme (maximum)

$\text{range} = \underline{\hspace{2cm}} - \underline{\hspace{2cm}} \qquad \text{IQR} = \underline{\hspace{2cm}} - \underline{\hspace{2cm}}$

FIGURE: fig2-quartiles-split-the-data.png (full width)

Guided practice

Put in order and find the median: $9, 4, 12, 7, 15$ .

Ordered: ______________________ Median: ______
For item 40, find $Q_1$ and $Q_3$ . $Q_1 =$ ______ $Q_3 =$ ______
For item 40, find the range and the IQR. Range: ______ IQR: ______
Five-number summary of $2, 4, 6, 8, 10, 12$ : ______, ______, ______, ______, ______
Range and IQR for item 43. Range: ______ IQR: ______
Five-number summary of $3, 5, 8, 10, 12, 15, 18$ : ______, ______, ______, ______, ______
Why is $10$ in neither half when you find the quartiles in item 45?
State the quartile rule used in this book in your own words.

PAGE 7 — Summary practice

Independent practice

Show your ordered list and both halves for each.

$45, 50, 52, 55, 58, 60, 62, 65, 70, 75$

Summary: ______, ______, ______, ______, ______ Range: ______ IQR: ______
$11, 13, 14, 16, 19, 21, 23, 28$

Summary: ______, ______, ______, ______, ______ Range: ______ IQR: ______
$58, 60, 63, 65, 66, 68, 70, 72, 74, 77, 80, 84$

Summary: ______, ______, ______, ______, ______ Range: ______ IQR: ______
$2, 3, 3, 5, 7, 8, 9, 9, 10, 14$ — find the summary and explain how repeats are handled.

Summary: ______, ______, ______, ______, ______
$55, 57, 60, 60, 63, 64, 66, 70$

Summary: ______, ______, ______, ______, ______
Median of $1, 3, 4, 4, 6, 8, 9$ : ______ Median of $10, 12, 13, 15, 16, 18, 19, 20, 22$ : ______
Range $30$ , lower extreme $12$ . Upper extreme: ______
$Q_1 = 18$ , IQR $= 11$ . $Q_3 =$ ______
Why can the IQR never be larger than the range?
Two data sets of 5 values with the same median and different IQRs:

Set 1: ______________________ Set 2: ______________________
Application. Dash times: $6.2, 5.8, 7.1, 6.5, 6.0, 6.9, 7.4, 6.3, 5.9$ seconds.

Summary: ______, ______, ______, ______, ______ Range: ______ IQR: ______
Error analysis. A student reads $Q_1 = 3$ and $Q_3 = 11$ straight off $8, 3, 15, 6, 11$ . Find the error and the correct summary.

Exit ticket 16.3

Summary of $6, 9, 11, 14, 20$ : ______, ______, ______, ______, ______
Range: ______ IQR: ______
Summary of $22, 25, 28, 30, 31, 34$ : ______, ______, ______, ______, ______
What does the IQR measure that the range does not? ______________________

PAGE 8 — Building a boxplot

16.4 Building and Reading a Boxplot

The five steps. Number them 1 to 5.

____ Draw the whiskers out to the extremes. ____ Order the data and find the summary.

____ Draw the median line inside the box. ____ Draw a labeled number line.

____ Draw the box from $Q_1$ to $Q_3$ .

FIGURE: fig3-anatomy-of-a-boxplot.png (full width)

FIGURE: fig4-boxplot-hides-values.png (full width)

FIGURE: fig5-bus-ride-boxplot.png (full width)

Guided practice

Reading-times boxplot — name all five plotted statistics.

______, ______, ______, ______, ______
Same plot: range ______ IQR ______
Same plot: which whisker is longer, and what does that tell you?
Bus-ride boxplot — name all five plotted statistics.

______, ______, ______, ______, ______
Bus-ride boxplot: about what fraction of rides took longer than $24$ minutes? ______
Describe the boxplot for $3, 5, 8, 10, 12, 15, 18$ .

Box: ______ to ______ Median line: ______ Whiskers: ______ and ______
Describe the boxplot for $2, 4, 6, 8, 10, 12$ .

Box: ______ to ______ Median line: ______ Whiskers: ______ and ______

PAGE 9 — Drawing and reading

Independent practice

Draw each boxplot on the axis provided.

$21, 24, 25, 27, 30, 33, 35, 38, 40$ — scale $20$ to $40$ , ticks every $2$ .

(blank axis)
$45, 50, 52, 55, 58, 60, 62, 65, 70, 75$ — scale $40$ to $80$ , ticks every $5$ .

(blank axis)
$58, 60, 63, 65, 66, 68, 70, 72, 74, 77, 80, 84$ — scale $55$ to $85$ , ticks every $5$ .

(blank axis)
Whiskers at $10$ and $46$ , box edges $18$ and $34$ , median line $22$ . List all seven statistics.
In item 74, which part of the box is longer, below or above the median? What does that say?
In item 74, about what fraction of the data lies between $18$ and $34$ ? ______
Sketch two boxplots with the same range and different IQRs.

(work space)
Why can a boxplot not tell you how many values are in the data set?
The median line is exactly centered in the box. What does that say about the middle half?
Application. Customers in the first hour on 8 days: $19, 11, 28, 14, 23, 16, 13, 21$ .

Summary: ______, ______, ______, ______, ______

Observation: ______________________ Conclusion: ______________________
Application. Highs for 12 days: $58, 60, 63, 65, 66, 68, 70, 72, 74, 77, 80, 84$ °F. Two observations and one conclusion for a gardener.
Error analysis. "The median line is always halfway between the box edges." Explain the error with your own data set.

Exit ticket 16.4

$10, 12, 13, 15, 16, 18, 19, 20, 22$ : box ______ to ______, median ______, whiskers ______ and ______
Whiskers at $5$ and $29$ , box edges $12$ and $20$ . Range ______ IQR ______
About what fraction lies between the box edges? ______
Two statistics you cannot read from a boxplot: ______________________

PAGE 10 — Extreme data points

16.5 Extreme Data Points

FIGURE: fig6-outlier-effect.png (full width)

Complete the table from the figure.

Statistic	11 games	With the 76-point game
lower extreme
$Q_1$
median
$Q_3$
upper extreme
range
IQR

Which changed the most? ______________ Which barely changed? ______________

Guided practice

Define extreme data point (outlier) in your own words.
Upper extreme in panel 1: ______ In panel 2: ______
Compare the IQR before and after the $76$ -point game, and explain why it barely changed.
Compare the range before and after, and explain the big change.
In panel 3, why does the whisker stop at $36$ ? ______________________
Which is more resistant to an outlier, the range or the IQR? Why?

PAGE 11 — Outlier practice

Independent practice

$5, 6, 7, 8, 9, 10, 11, 12, 13$ : range ______ IQR ______. Add $40$ : range ______ IQR ______
Describe how the boxplot in item 93 changes shape.
$71, 74, 75, 77, 78, 80, 82, 84, 86$ : median ______ IQR ______. Add $12$ : median ______ IQR ______
In item 95, which statistic changed most? ______ Which did not change at all? ______
With a high outlier, which whisker is longer? Explain.
Why does the median move so little when one very large value is added?
Mean of $20, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36$ : ______ With $76$ added: ______

Compare that change to the median's change from $28$ to $29$ : ______________________
Application. Delivery times: $18, 20, 21, 22, 24, 25, 26, 28, 30, 62$ minutes.

Summary: ______, ______, ______, ______, ______ Range: ______ IQR: ______

Which would you quote to a customer, and why? ______________________
Error analysis. "The outlier ruins the graph, so I deleted it." What is wrong, and what should you do?

Exit ticket 16.5

Adding one very large value: range changes ______________, IQR changes ______________. Why?
How does a high outlier change the shape of a boxplot? ______________________
One situation to remove an outlier: ______________ One to keep it: ______________

PAGE 12 — Comparing two boxplots

16.6 Comparing Boxplots, Choosing a Display, Spotting a Misleading One

FIGURE: fig7-two-classes-comparison.png (full width)

FIGURE: fig8-same-median-different-spread.png (full width)

Compare in three passes: center → spread → overlap.

Guided practice

Two-class figure: Class A median ______ IQR ______ Class B median ______ IQR ______
Which class is more consistent, and how do you know? ______________________
Which class contains the single highest score? ______
Two-routes figure: medians ______ and ______. Why are the routes not equivalent?
Which route guarantees arriving within an hour? Justify with two statistics.
Name one thing the dot plot shows that the boxplot does not. ______________________
How can the same data look so different on two scales? ______________________

PAGE 13 — Choosing a display, spotting a misleading one

FIGURE: fig9-choosing-a-representation.png (full width)

FIGURE: fig10-misleading-different-scales.png (full width)

Match the display to the question.

Display	Best when the question is
dot plot
histogram
boxplot
circle graph
line graph

Independent practice

$P = 12, 14, 15, 17, 18, 20, 22, 24, 26, 30$ and $Q = 16, 17, 18, 19, 20, 21, 22, 23, 24, 25$ .

$P$ : ______, ______, ______, ______, ______ $Q$ : ______, ______, ______, ______, ______

Compare center, spread, and overlap: __________________________________
Same median, very different IQRs — what does that mean in a context of your choice?
Same range, different medians — what can and cannot you conclude?
Justify the best display, naming the deciding feature.

a) shares choosing four electives ______________________

b) city population 2015–2024 ______________________

c) morning vs afternoon bus routes, and reliability ______________________

d) which of 18 quiz scores occurred most often ______________________
Two house-price boxplots on $0$ – $300$ and $150$ – $250$ axes. Why is the comparison invalid?
Three components to check on any boxplot: ______________, ______________, ______________
Very wide box, very short whiskers. Describe the data. ______________________
Why is "the wider section of the box has more data values" a misreading?
A boxplot with no sample size given. Give a 6-value set and a 20-value set, and say why the omission matters.
Application. Team X: $4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 20, 26$ . Team Y: $8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 22$ .

X: ______, ______, ______, ______, ______ Y: ______, ______, ______, ______, ______

More reliable team, justified by IQR and range: ______________________
Application. Comparing grade 7 and grade 8 homework time for the school board: which display, why, and what would you label?
Error analysis. "Group 1's box is wider, so Group 1 has more students." Explain and correct.

Exit ticket 16.6

Group 1: median $40$ , IQR $6$ . Group 2: median $40$ , IQR $22$ . Compare in one sentence.
Display to compare the spread of two groups, and why: ______________________
Two misleading components: ______________________, ______________________
Why must two boxplots share one axis? ______________________

PAGE 14 — Review, Part A

Chapter 16 Review

Part A — The data cycle, collecting data, and bias

Four stages in order, one phrase each.
Sharpen "Do eighth graders carry heavy backpacks?" Name the population and the data recorded.
A collection method for at most 20 students, and why.
Name the bias and the direction it pushes results.

a) weighing only on locker-cleanout Fridays ______________________

b) asking for volunteers ______________________

c) weighing only in the athletics hallway ______________________
What does representative sample mean, and why does bias prevent it?

PAGE 15 — Review, Part B

Part B — Organizing, representing, and describing

$1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 14, 18$

Summary: ______, ______, ______, ______, ______ Range: ______ IQR: ______
$100, 102, 105, 108, 110, 115, 120, 125$

Summary: ______, ______, ______, ______, ______ Range: ______ IQR: ______
State the quartile rule, then apply it to $3, 5, 8, 10, 12, 15, 18$ .

FIGURE: fig11-blank-boxplot-axes.png (full width)

On the middle axis above, draw a boxplot for $11, 13, 14, 16, 19, 21, 23, 28$ .
On the top axis above, draw a boxplot for $6, 8, 10, 12, 14, 15, 18, 20, 22, 26, 30, 36$ .

FIGURE: fig12-practice-boxplot.png (full width)

From the boxplot above: lower extreme ______ $Q_1$ ______ median ______ $Q_3$ ______ upper extreme ______
Range ______ IQR ______ What does each measure? ______________________

PAGE 16 — Review, Parts C and D

Part C — Outliers, analysis, and comparison

Two observations and one conclusion a librarian could act on, from the boxplot on the previous page.
What does the upper whisker's length tell you compared with the lower whisker's?
Range goes $8 \to 35$ but IQR stays $5$ when $40$ is added. Explain both, from how each is computed.
How does one very large value change the shape of a boxplot? (two sentences)

FIGURE: fig13-review-comparison.png (full width)

Summary of each team from the plot, then compare center and spread.

X: ______, ______, ______, ______, ______ Y: ______, ______, ______, ______, ______
More reliable team ______ Single best game ______ Justify both: ______________________

Part D — Choosing a display and spotting a misleading one

Justify the best display for each.

a) spread of scores in two classes ______________________

b) fraction choosing each of three field trips ______________________

c) every one of 15 times, including repeats ______________________

d) daily sales across two weeks ______________________
Why can categorical data never go in a boxplot? ______________________
Three questions to ask about a poster boxplot with no axis labels and no sample size.
Two boxplots printed on different scales: the false impression, and the fix.
Reasoning. A club claims its members read more than the average student, using a boxplot from 12 volunteers. Name the bias, name one misleading feature, and describe a plan that would settle the claim.