Grade 6 Workbook — Chapter 17: Mean as a Balance Point; Outliers

SOL 6.PS.2 · Companion to Textbook Chapter 17

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/.

PAGE 1 — Chapter opener

Chapter 17 · Mean as a Balance Point; Outliers

Standard 6.PS.2

In this chapter you will:

• Show the mean of a data set as the balance point on a line plot (dot plot)
• Find the mean, median, mode, and range of a data set
• Determine what happens to the measures when a value is added, removed, or changed
• Identify outliers by observing patterns and describe their effect

Words to know: mean · line plot (dot plot) · balance point · median · mode · range · outlier

Tool note: all arithmetic in this chapter can be done by hand. Keep a place to record sums.

PAGE 2 — The balance point

17.1 Mean as a Balance Point

FIGURE: fig1-balance-point.png (full width) [Number line 0 to 8 drawn as a board on a triangular fulcrum at 4, with one dot above 2, one above 4, one above 6, labeled "mean = 4 (balance point)".]

$$\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}$$

Data: 2, 4, 6. Sum: ______ Number of values: ______ Mean: ______

Distances from the mean.

Left total: ______ Right total: ______ Do they match? ______

At the mean, the total distance on the left equals the total distance on the right.

PAGE 3 — Balance practice

Find the Balance Point

FIGURE: fig2-balance-deviations.png (half width) [Number line 0 to 7 on a fulcrum at 3, dots above 1, 2, 3, and 6, with labeled distances 2, 1, 0, and 3, and totals "left 3" and "right 3".]

Find each mean, then check the left and right totals.

Find the missing value.

$3$, $4$, ______, $8$ has a mean of $5$.

Four values have a mean of $7$. Three are $5$, $6$, $8$. The fourth is ______.

Explain. Why can the balance point sit where there is no dot?

PAGE 4 — Exit ticket 17.1

Exit Ticket · Lesson 17.1

Name: ________________________ Date: ____________

1. Mean of $3$, $3$, $4$, $6$: ______

2. For that data, total distance left: ______ right: ______

3. $2$, $6$, ______ has a mean of $5$.

4. Explain what "balance point" means for a dot plot.

PAGE 5 — Three centers and one spread

17.2 Measures of Center and Range

Complete the table.

Careful. Before finding a median you must always ____________________ the data.

PAGE 6 — Reading a dot plot

Measures from a Line Plot

FIGURE: fig3-dot-plot-siblings.png (full width) [Line plot "Number of Siblings, 20 Students": number line 0 to 5 with 3 dots above 0, 7 above 1, 5 above 2, 3 above 3, 1 above 4, 1 above 5; fulcrum marker at 1.75.]

Use the line plot.

Total number of students: ______

Sum of all the values: ______

Mean: ______ Median: ______ Mode: ______ Range: ______

More practice.

PAGE 7 — Exit ticket 17.2

Exit Ticket · Lesson 17.2

Name: ________________________ Date: ____________

1. For $4$, $7$, $7$, $10$: mean ______ median ______ mode ______ range ______

2. Median of $9$, $2$, $7$, $4$, $6$: ______

3. Range of $15$, $8$, $22$, $11$: ______

4. Explain how a data set can have no mode.

PAGE 8 — Before and after

17.3 What Happens When Data Changes

Predict first, then compute.

Complete both rows of each table.

A. Start with $4$, $6$, $8$, $10$, $12$; then add $20$.

B. Start with $6$, $8$, $10$, $12$, $14$; then remove $14$.

C. Start with $15$, $18$, $21$, $26$; then change $26$ to $30$.

PAGE 9 — Change practice

Adding, Removing, Changing

1. $2$, $4$, $6$, $8$, $10$ → add $12$. New mean ______ median ______ range ______

2. $20$, $25$, $30$, $35$ → remove $20$. New mean ______ median ______ range ______

3. $8$, $8$, $9$, $10$, $15$ → change $15$ to $10$. New mean ______ median ______ mode ______ range ______

4. $4$, $6$, $8$ has a mean of $6$. Add ______ to keep the mean at $6$.

5. $5$, $10$, $15$ has a mean of $10$. Add ______ to make the new mean $11$.

6. Kara bowls $90$, $95$, $100$, $105$, $110$, then a $130$.

   Before: mean ______ median ______ range ______

   After: mean ______ median ______ range ______

Explain. How can the mean change while the median stays the same?

PAGE 10 — Exit ticket 17.3

Exit Ticket · Lesson 17.3

Name: ________________________ Date: ____________

1. $3$, $5$, $7$ has a mean of $5$. Add $9$. New mean: ______

2. Median of $4$, $6$, $8$, $10$: ______ After removing $10$: ______

3. For $6$, $8$, $10$, $12$, change $12$ to $20$. New mean: ______

4. What happens to the mean when you add a value below the mean? Why?

PAGE 11 — Spotting outliers

17.4 Outliers

FIGURE: fig4-outlier-dot-plot.png (full width) [Line plot "Minutes Late to Practice": number line 10 to 60, dots above 12, 14, 15, 15, 16, and one lone dot above 60 with a wide gap, arrow labeled "outlier".]

An outlier is a value far from the rest. Look for the gap, not just the biggest number.

Circle the outlier, or write "none."

With and without. Data: $12$, $14$, $15$, $15$, $16$, $60$.

Which two measures changed the most? ____________ and ____________

PAGE 12 — Outlier effects

What Outliers Do

1. $4$, $5$, $5$, $6$, $30$ → outlier ______

   With: mean ______ median ______ mode ______ range ______

   Without: mean ______ median ______ mode ______ range ______

2. $1$, $45$, $47$, $48$, $49$ → outlier ______

   With: mean ______ median ______ range ______

   Without: mean ______ median ______ range ______

3. Allowances: $5$, $5$, $6$, $8$, $36$ → outlier ______

   With: mean ______ median ______ Without: mean ______ median ______

   Which measure better describes a typical allowance? ____________ Why?

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Decide. A stopwatch records $600$ seconds when every other runner finished near $60$. What should the class do, and what must they report either way?

PAGE 13 — Exit ticket 17.4

Exit Ticket · Lesson 17.4

Name: ________________________ Date: ____________

Data: $8$, $9$, $10$, $11$, $52$

1. Outlier: ______

2. Mean with the outlier: ______ Mean without it: ______

3. Median with: ______ without: ______ How much did it change? ______

4. Explain how you identify an outlier by observing patterns in the data.

PAGE 14 — Chapter 17 review, part 1

Chapter 17 Review

Part A · Mean as a balance point

1. Mean of $3$, $5$, $7$, $9$: ______ Left total: ______ Right total: ______

2. Balance point of dots at $2$, $3$, $3$, $8$: ______

3. $6$, $9$, ______ has a mean of $8$.

4. Why must the mean lie between the least and greatest values?

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Part B · Adding, removing, changing

5. $4$, $6$, $8$, $10$, $12$ → add $20$. Before: mean ______ median ______ range ______ After: mean ______ median ______ range ______

6. $6$, $8$, $10$, $12$, $14$ → remove $14$. Before: mean ______ median ______ After: mean ______ median ______

7. $15$, $18$, $21$, $26$ → change $26$ to $30$. Before: mean ______ median ______ After: mean ______ median ______

8. $2$, $4$, $6$, $8$ has a mean of $5$. A value you could add without changing the mean: ______ Why?

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9. Why can the median stay the same when an end value changes?

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PAGE 15 — Chapter 17 review, part 2

Chapter 17 Review (continued)

Part C · Outliers

10. $12$, $14$, $15$, $15$, $16$, $60$ → outlier ______

11. $30$, $86$, $88$, $90$, $92$, $94$ → outlier ______ With: mean ______ median ______ range ______ Without: mean ______ median ______ range ______

12. Does $21$, $22$, $24$, $25$, $27$, $28$ have an outlier? ______ Mean: ______ Median: ______

13. Most affected by an outlier: ____________ and ____________ Least affected: ____________ and ____________

Part D · Application and reasoning

14. Homework minutes: $20$, $25$, $25$, $30$, $35$, $45$, $65$. Mean ______ median ______ mode ______ range ______ Outlier ______

    Effect on the mean: _______________________________________________

15. Mean score $80$, median score $88$. What do the data probably look like?

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16. "Removing the outlier always makes the mean smaller." Counterexample:

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17. A stopwatch recorded $600$ seconds when every other runner finished near $60$. What should you do, what should you report, and why is the median useful here?

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Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Blank number lines: pages 3 and 11 benefit from an unlabeled number line beneath each data set so students can plot before computing
• Figure widths: fig1-balance-point.png, fig3-dot-plot-siblings.png, and fig4-outlier-dot-plot.png full width; fig2-balance-deviations.png at half width beside its table
• Tables: before-and-after tables on pages 8 and 11 should keep both rows visible on the same page; do not split across pages
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow