Grade 7 Workbook — Chapter 17: The Data Cycle and Histograms

SOL 7.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/. Item numbers match the textbook exactly, and the problems are the same problems with the same numbers.

Blank histogram grids are printed as tables. Each column is one interval and each row is one frequency level; students shade one cell for each unit of frequency, filling upward from the bottom row, and the shaded columns touch because the cells touch.

PAGE 1 — Chapter opener

Chapter 17 · The Data Cycle and Histograms

Standard 7.PS.2

In this chapter you will:

• Run all four stages of the data cycle on a question of your own
• Write statistical questions whose answers are numbers, not categories
• Judge whether a sample is representative, using both randomness and sample size
• Group numerical data into equal-width intervals and build a frequency table
• Draw histograms whose bars touch, with every interval labeled
• Regroup the same data at a different interval width and explain what changed
• Compare a histogram with a dot plot, a stem-and-leaf plot, and a circle graph, and say which is best for a stated question
• Describe patterns a histogram reveals that a raw list of numbers hides

Words to know: data · data cycle · statistical question · numerical data · categorical data · observation · measurement · survey · experiment · population · sample · sample size · random sample · convenience sample · representative sample · biased sample · histogram · interval · frequency · frequency table · bar graph · dot plot (line plot) · stem-and-leaf plot · circle graph · peak · gap · symmetric · skewed

The interval convention for this whole chapter. Every interval is written $a \le x < b$, read "at least $a$ and less than $b$." The lower endpoint belongs to the interval; the upper endpoint belongs to the next interval. So 30 goes in $30 \le x < 40$, never in $20 \le x < 30$.

PAGE 2 — The four stages

17.1 The Data Cycle

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

Label the loop. Write the four stages in order.

1st ______________________________ 2nd ______________________________

3rd ______________________________ 4th ______________________________

Fill in the blanks.

Grade 6 focused on circle graphs, whose data are ______________ data. Grade 7 focuses on histograms, whose data are ______________ data.

A circle graph answers what share of the ______________. A histogram answers how the numbers are ______________.

The data cycle is drawn as a loop because answering one question usually raises the ______________ one.

Complete the table.

PAGE 3 — Guided practice 17.1

Naming the Stage

1. Name the four stages of the data cycle in order.

   ---

2. Which stage are you in when you decide to group measurements into intervals of width 5 and draw a histogram?

   ---

3. A class writes "How tall is each seventh grader in our school, in centimeters?" Which stage is this, and will the data be numerical or categorical?

   Stage: _______________________ Data: _______________________

4. A student says the data cycle ends as soon as the histogram is finished. Correct the statement.

   ---

5. In which stage would you write, "Half the class reads fewer than 20 minutes a night, so the reading block should be lengthened"?

   ---

PAGE 4 — Independent practice 17.1

Around the Cycle

6. Name the stage for each action.

7. Put these in the correct order: organize and represent data; analyze data and communicate results; formulate questions; collect or acquire data.

   1. ____________________ 2. ____________________ 3. ____________________ 4. ____________________

8. A class debates whether to measure every seventh grader or only a sample of 40. Which stage are they in? _______________________

9. Write one question about your class whose answers would be numerical, so a histogram could display them.

   ---

10. Label each numerical or categorical, then state which a histogram could display.

11. Application. The school nurse wants to know whether seventh graders get enough sleep on school nights. Write one sentence for each stage.

    Formulate: ____________________________________________________

    Collect: ______________________________________________________

    Organize: _____________________________________________________

    Analyze: ______________________________________________________

12. Reasoning. Explain why the data cycle is drawn as a loop rather than a straight line. Use an example in which a finished histogram raises a brand-new question.

    ---

    ---

PAGE 5 — Exit ticket 17.1

Exit Ticket 17.1

13. Name the four stages of the data cycle in order.

    ---

14. Which stage includes choosing the width of the intervals for a histogram? _______________________

15. Is "the number of pages each student read this week" numerical or categorical, and could a histogram display it?

    ---

16. Explain in your own words why analyzing data comes after representing it.

    ---

PAGE 6 — Sharpening a question

17.2 Asking a Question and Collecting Data

A sharp statistical question names three things. Fill them in.

• The ______________ — exactly whose answers you want
• The ______________ and its ______________ — exactly what number each response is
• The ______________ — when, or under what conditions

Sharpen each question.

Collection methods. Write one example of your own for each.

Data-collection plan. Fill this frame in for a question of your own.

My question: ____________________________________________________

Population: _____________________________________________________

Quantity and units: ______________________________________________

Time frame: _____________________________________________________

Method, and why: ________________________________________________

PAGE 7 — Guided practice 17.2

Statistical or Not

17. Is "How many minutes did Ellis practice piano on Monday?" a statistical question? Explain.

    ---

18. Is "How many minutes do seventh graders practice an instrument each day?" a statistical question? Explain.

    ---

19. Why must a question intended for a histogram ask for a number rather than a choice from a list?

    ---

20. Rewrite "Do students sleep enough?" as a question that produces numerical data suitable for a histogram.

    ---

21. Name the three things a sharp statistical question states.

    ---

PAGE 8 — Independent practice 17.2

Questions and Methods

22. Label each statistical or not statistical.

23. For each, say numerical or categorical, and whether a histogram could display it.

24. Sharpen "How far do students live from school?" so it names a population, units, and a precision.

    ---

25. Name the best collection method for each.

26. A class wants the heights of all seventh graders in Virginia. Explain why they must acquire data rather than collect it, and name two questions to ask about data they did not collect.

    ---

    Question 1: ___________________ Question 2: ___________________

27. Application. The PE teacher wants to know how many sit-ups seventh graders can do in one minute.

    Sharpened question: ___________________________________________

    Data needed: _________________________________________________

    Method, and why: _____________________________________________

28. Reasoning. A survey asks, "About how long do you read each night: a little, some, or a lot?" Explain why these responses cannot be put in a histogram, rewrite the question so they can, and explain what job you moved from the respondent to yourself.

    ---

    ---

PAGE 9 — Exit ticket 17.2

Exit Ticket 17.2

29. Is "How many minutes do seventh graders spend on homework each night?" a statistical question that produces numerical data? Explain in one sentence.

    ---

30. Name the four collection methods.

    ---

31. Rewrite "Are students tall?" so a histogram could display the answers.

    ---

32. Explain why a histogram cannot display the answers to "What is your favorite subject?"

    ---

PAGE 10 — Who gets chosen

17.3 Samples, Sample Size, and Randomness

FIGURE: fig2-convenience-versus-random.png (full width)

Fill in the blanks.

The ______________ is the entire group you want to describe. The ______________ is the part you actually collect data from.

In a ______________ sample, every member of the population has the same chance of being chosen.

A ______________ sample is chosen because it was easy to reach.

A sample chosen in a way that systematically favors certain values is a ______________ sample.

Size fixes ______________. Randomness fixes ______________. A large sample chosen badly is ______________ biased.

Sampling plan frame. Complete this for a question of your own.

Population, stated exactly: ______________________________________

Where the list of members comes from: ___________________________

Sample size, and why it is large enough: _________________________

Selection method, and what makes it random: _____________________

Coverage — who could not be selected under this plan: ____________

PAGE 11 — Guided practice 17.3

Population, Sample, Bias

33. A school has 360 seventh graders. A class measures 45 of them, chosen by drawing names from a container.

    Population: ___________________ Sample: ___________________

34. Explain in your own words what makes a sample representative.

    ---

35. Why is asking whoever happens to be standing near you not a random sample?

    ---

36. Which is more likely to represent a grade of 300: 8 students chosen at random, or 60 chosen at random? Explain.

    ---

37. A student surveys 100 students at a school basketball game about how many minutes they exercise each day. Is the large sample size enough to make it representative? Explain.

    ---

PAGE 12 — Independent practice 17.3

Judging a Sampling Plan

38. Name the population and the sample.

39. Give two reasons a sample of 5 students is a weak sample for a grade of 300.

    Reason 1: _____________________________________________________

    Reason 2: _____________________________________________________

40. Explain the difference between a random sample and a convenience sample, and give one example of each.

    ---

41. Describe three different ways to choose a random sample of 40 students from a numbered roster of 400.

    1. ---

    2. ---

    3. ---

42. Rewrite the plan "ask everyone in the library after school" so the sample is random and can include any student in the grade.

    ---

43. Application. A principal wants to know how many minutes seventh graders spend on homework each night. There are 320 seventh graders in eight homerooms. Describe a sampling plan.

    How many: ______ How chosen: __________________________________

    Why representative: ___________________________________________

44. Reasoning. Two students each survey 200 of the 600 students in their school. One draws 200 names at random from the full roster. The other stands outside the band room and asks the first 200 students who come out. Explain why the second sample is biased even though it is just as large, and name the direction its results about music practice would be pushed.

    ---

    ---

PAGE 13 — Exit ticket 17.3

Exit Ticket 17.3

45. A club measures 25 of the 400 students in a school.

    Population: ___________________ Sample: ___________________

46. Give one reason a larger sample is usually better than a smaller one.

    ---

47. Give one reason randomness still matters even when the sample is large.

    ---

48. A student surveys the chess club to find how many hours seventh graders spend playing sports. Explain why the sample is biased and describe a better plan.

    ---

PAGE 14 — Histogram or bar graph

17.4 Building a Histogram

FIGURE: fig3-histogram-versus-bar-graph.png (full width)

Fill in the blanks.

A histogram displays ______________ data grouped into ______________-width intervals.

The height of each bar is the ______________ of that interval.

Histogram bars ______________ because the horizontal axis is a ______________ ______________.

Bar graph bars have ______________ because the horizontal axis is a list of ______________.

A histogram bar of height 0 means ______________________________ and must not be deleted.

The one-sentence test. If the horizontal axis is a ______________ ______________, the bars touch and you have a histogram. If the horizontal axis is a list of ______________, the bars have gaps and you have a bar graph.

The reading data, tallied. Twenty-four students reported minutes read last night:

$$5,\; 8,\; 12,\; 15,\; 15,\; 18,\; 20,\; 22,\; 22,\; 25,\; 25,\; 27,\; 28,\; 30,\; 31,\; 33,\; 35,\; 38,\; 40,\; 42,\; 45,\; 48,\; 52,\; 55$$

FIGURE: fig4-reading-histogram.png (full width)

PAGE 15 — Guided practice 17.4

Tally, Total, Check

49. Group these values into intervals of width 10 starting at 0: $3,\, 7,\, 11,\, 14,\, 16,\, 19,\, 21,\, 24,\, 25,\, 29,\, 33,\, 38$.

50. The intervals are $20 \le x < 30$ and $30 \le x < 40$. In which interval does 30 belong? Explain.

    Interval: ______________ Because: ______________________________

51. A frequency table has frequencies 5, 8, 6, and 2. How many values are in the data set? ______

52. Explain why the bars of a histogram touch.

    ---

53. A frequency table for 25 values shows 6, 7, ______, and 4. Find the missing frequency.

PAGE 16 — Independent practice 17.4, part one

The Bus-Ride Data

Minutes each of 20 students spent riding the bus this morning:

$$4,\; 6,\; 9,\; 10,\; 12,\; 13,\; 15,\; 16,\; 17,\; 18,\; 20,\; 21,\; 22,\; 24,\; 26,\; 28,\; 31,\; 33,\; 37,\; 44$$

54. Make a frequency table using intervals of width 10 starting at 0, and show that the frequencies sum to 20.

55. Draw the histogram for your table. Shade one cell for each unit of frequency, filling upward from the bottom row.

Horizontal axis label: ___________________________________________

Vertical axis label: _____________________________________________

Title: __________________________________________________________

56. Using the reading data from page 14 (frequencies 2, 4, 7, 5, 4, 2), find how many students read fewer than 30 minutes and how many read 30 minutes or more. Show that your two answers sum to 24.

    Fewer than 30: ______ 30 or more: ______ Sum: ______

PAGE 17 — Independent practice 17.4, part two

Which Graph, and Why

57. Decide histogram or bar graph, and give a one-sentence reason.

58. A student draws a graph of numerical data grouped into intervals but leaves gaps between the bars. Explain what is wrong and what the gaps wrongly suggest to a reader.

    ---

59. Application. Twenty-four science-test scores: $52,\, 55,\, 58,\, 61,\, 63,\, 65,\, 67,\, 69,\, 70,\, 72,\, 73,\, 75,\, 76,\, 78,\, 79,\, 81,\, 82,\, 84,\, 86,\, 88,\, 90,\, 93,\, 95,\, 98$. Use intervals of width 10 starting at 50.

Describe what the histogram would look like: ____________________

_______________________________________________________________

60. Reasoning. Two students tally the same 24 values. One places every value equal to an interval's lower endpoint in that interval; the other places such values in the previous interval. Explain how their tables could differ even though neither miscounted, and why a stated convention prevents the problem.

    ---

    ---

PAGE 18 — Exit ticket 17.4

Exit Ticket 17.4

61. Group $2,\, 5,\, 9,\, 10,\, 14,\, 18,\, 21,\, 25$ into intervals of width 10 starting at 0.

62. A frequency table for 18 values shows 4, 7, ______, and 2. Find the missing frequency.

63. Name the two features that make a graph a histogram rather than a bar graph.

    1. ---

    2. ---

64. In which interval does 40 belong, $30 \le x < 40$ or $40 \le x < 50$? Explain.

    ---

PAGE 19 — Same data, different intervals

17.5 Choosing Intervals

Sit-ups completed in one minute by 24 students:

$$4,\; 7,\; 11,\; 12,\; 13,\; 15,\; 16,\; 17,\; 18,\; 19,\; 22,\; 24,\; 26,\; 28,\; 31,\; 32,\; 33,\; 34,\; 35,\; 36,\; 37,\; 38,\; 43,\; 46$$

Tally the same data twice.

FIGURE: fig5-same-data-two-interval-widths.png (full width)

FIGURE: fig6-too-many-too-few-intervals.png (full width)

Fill in the blanks.

Too many intervals: almost every frequency is ______ or ______, so the graph shows individual values instead of a ______________.

Too few intervals: the graph reports an ______________ spread and erases the clusters.

A workable rule of thumb for a class-sized data set is ______ to ______ intervals.

Whenever you present a histogram, state the interval ______________ and the ______________.

PAGE 20 — Guided practice 17.5

Regroup and Compare

65. Regroup the sit-up data into intervals of width 20 starting at 0. Give each frequency and verify the total.

    $0 \le x < 20$: ______ $20 \le x < 40$: ______ $40 \le x < 60$: ______ Total: ______

66. Two intervals of the width-10 histogram tie for the largest frequency. Name both and state the frequency.

    ______________________ ______________________ Frequency: ______

67. Explain in one sentence what the width-20 histogram hides that the width-10 histogram shows.

    ---

68. Which representation shows every individual value, a histogram or a stem-and-leaf plot? Explain.

    ---

69. Explain why a circle graph is a poor choice for showing how many minutes each student read.

    ---

PAGE 21 — Independent practice 17.5, part one

Four Views of Twenty Games

Points scored in each of 20 games:

$$8,\; 9,\; 12,\; 14,\; 15,\; 17,\; 18,\; 21,\; 21,\; 22,\; 23,\; 25,\; 26,\; 28,\; 29,\; 31,\; 33,\; 34,\; 36,\; 42$$

FIGURE: fig7-points-histogram.png (full width)

FIGURE: fig8-dot-plot-and-stem-and-leaf.png (full width)

FIGURE: fig9-points-circle-graph.png (full width)

70. Make a frequency table using intervals of width 10 starting at 0, and verify the total.

71. Write each frequency as a percent of 20, then find the central angle for a circle graph. Show both totals.

72. Use the stem-and-leaf plot.

    a) Highest score: ______

    b) Games with exactly 21 points: ______

    c) A value that occurs more than once: ______

73. Name the best representation for each question and justify in one sentence.

PAGE 22 — Independent practice 17.5, part two

Regrouping and Reasoning

74. Regroup the reading data from Lesson 17.4 (width-10 frequencies 2, 4, 7, 5, 4, 2) into intervals of width 20 starting at 0.

    $0 \le x < 20$: ______ $20 \le x < 40$: ______ $40 \le x < 60$: ______ Total: ______

    One thing the new grouping hides: ______________________________

75. Application. Use the reading histogram from Lesson 17.4.

    a) How many students read 30 minutes or more? ______

    b) What percent of the 24 students read fewer than 20 minutes? ______

    c) One conclusion the teacher could act on: _____________________

    The observation it rests on: ___________________________________

76. Reasoning. A student says, "The width-20 histogram of the sit-up data proves that most students do between 20 and 40 sit-ups, so the class is fairly consistent." Explain what the width-10 histogram shows that makes this misleading, then rewrite the claim so it is supported.

    Why it is misleading: __________________________________________

    ---

    Rewritten claim: ______________________________________________

PAGE 23 — Exit ticket 17.5

Exit Ticket 17.5

FIGURE: fig10-three-histogram-shapes.png (full width)

77. Name one thing a dot plot shows that a histogram hides.

    ---

78. Name one thing a histogram shows better than a dot plot.

    ---

79. A histogram of 24 values has width-10 frequencies 2, 8, 4, 8, and 2, starting at 0. Give the frequencies for intervals of width 20 starting at 0.

    ---

80. Explain in one or two sentences why changing the interval width changes the look of a histogram even though not one data value has changed.

    ---

    ---

PAGE 24 — Review, Parts A and B

Chapter 17 Review · Questions and Collection

Interval convention. Every interval is $a \le x < b$: the lower endpoint is included, the upper endpoint belongs to the next interval.

81. Which are statistical questions whose data are numerical?

82. Rewrite "Do students carry heavy backpacks?" as a sharp statistical question whose data would fit a histogram.

    ---

83. Write a statistical question about your grade whose data are numerical.

    Question: _____________________________________________________

    Population: ____________ Quantity and units: ____________ Time frame: ____________

84. Name the best collection method for each.

85. A class wants the heights of all seventh graders in their division. Explain why they should acquire existing data, and name two questions to ask about it.

    ---

    Question 1: ___________________ Question 2: ___________________

PAGE 25 — Review, Part C

Chapter 17 Review · Sample Size and Randomness

86. A school has 480 seventh graders. A class measures the arm span of 60, chosen with a random number generator applied to the numbered roster.

    Population: ___________________ Sample: ___________________

    Why this is likely representative: ______________________________

    ---

87. Give two reasons that timing 10 members of the track team is a biased way to learn how fast the whole grade runs 100 meters.

    Reason 1: _____________________________________________________

    Reason 2: _____________________________________________________

88. A student surveys 250 of a school's 600 students about minutes of daily music practice, asking everyone as they leave the winter concert. Explain why this large sample is still biased, and state the direction its results would be pushed.

    ---

    Direction: ____________________________________________________

PAGE 26 — Review, Part D

Chapter 17 Review · Building the Histogram

Minutes each of 20 students spent on math homework last night:

$$8,\; 12,\; 15,\; 17,\; 19,\; 20,\; 22,\; 25,\; 27,\; 28,\; 30,\; 31,\; 34,\; 36,\; 38,\; 41,\; 44,\; 47,\; 52,\; 58$$

89. Make a frequency table using intervals of width 10 starting at 0, and show the frequencies sum to 20.

Draw the histogram. Shade one cell for each unit of frequency, filling upward from the bottom row.

90. Describe the histogram: what goes on each axis, where the ticks go, and why the bars touch.

    ---

    ---

91. In which interval does 30 belong? State the convention you are using.

    Interval: ______________ Convention: ___________________________

92. A different frequency table, for 26 values, shows 3, 6, ______, 7, and 2. Find the missing frequency.

PAGE 27 — Review, Part E

Chapter 17 Review · Intervals and Representations

93. Regroup the homework data into intervals of width 20 starting at 0.

    $0 \le x < 20$: ______ $20 \le x < 40$: ______ $40 \le x < 60$: ______ Total: ______

94. Explain one thing the width-20 grouping hides that the width-10 grouping shows.

    ---

95. Explain why a histogram of the homework data cannot tell you whether any student spent exactly 30 minutes.

    ---

96. Name the best representation for each purpose and justify in one sentence.

97. Build a stem-and-leaf plot for the homework data. Use the tens digit as the stem.

Key: ______ | ______ means ______ minutes

98. State one advantage and one limitation of a circle graph built from the intervals of the homework data.

    Advantage: ____________________________________________________

    Limitation: ___________________________________________________

PAGE 28 — Review, Part F

Chapter 17 Review · Analyze and Conclude

99. Using your frequency table from item 89, find how many students spent fewer than 30 minutes and how many spent 40 minutes or more.

    Fewer than 30: ______ 40 or more: ______

100. Name one pattern the histogram of the homework data shows that the raw list of 20 numbers does not show easily.

     ---

101. Two intervals of the width-10 homework table tie for the largest frequency. Name both and state the frequency.

     ______________________ ______________________ Frequency: ______

102. Walk through all four stages of the data cycle for "How many hours of sleep do seventh graders at our school get on a school night?"

     Formulate: ___________________________________________________

     Collect: _____________________________________________________

     Organize: ____________________________________________________

     Analyze: _____________________________________________________

103. A student reports, "The histogram shows that 5 students slept exactly 8 hours." Explain what is wrong with this claim.

     ---

104. Reasoning. Two classes graph the same 24 sit-up values. One uses width 10 and reports two clusters with a dip between them. The other uses width 25 and reports a perfectly even spread. Neither made a counting mistake. Explain how both descriptions can come from the same data, and say which grouping you would trust and why.

     ---

     ---

     ---