Grade 6

End-of-Unit Assessment

End-of-Unit Assessment
1.

Select all the true statements.

A.

Given a dot plot, it is always possible to construct a corresponding histogram.

B.

Given a dot plot, it is always possible to calculate the mean of the data.

C.

Given a box plot, it is always possible to calculate the IQR of the data.

D.

Given a histogram, it is always possible to calculate the mean of the data.

E.

Given a histogram, it is always possible to calculate the median of the data.

Answer: A, B, C

Teaching Notes

An important fact is that only a dot plot can be used to reconstruct the entire data set. Given a histogram or box plot, only certain information is readily known. Students who select Choices D and E may have a significant misconception about what a histogram represents, notably, that it indicates only ranges of information, not specific values.

2.

Here is a dot plot of a data set.

<p>A dot plot</p>

Which statement is true about the mean of the data set?

A.

The mean is less than 8.

B.

The mean is equal to 8.

C.

The mean is greater than 8.

D.

There is not enough information to determine the mean.

Answer:

A

Teaching Notes

The “balancing” interpretation of mean is very useful here. There are three data points that are not 8: two that are 7 less than 8, and one data point that is 2 more than 8. Students who select Choice B may be confused about the difference between the mean and median. Students who select Choice C may think the mean is larger because 10 is significantly larger than 1. Students who select Choice D may think that the dot plot does not contain full information about the data, but it does.

3.

The ages of people dining in two restaurants are shown in the box plots.

<p>A box plot. Ages of people in Restaurant A.</p>

<p>A box plot. Ages of people in Restaurant B.</p>

Select all the statements that must be true.

A.

The median age of people dining in Restaurant B is greater than the median age of people dining in Restaurant A.

B.

The MAD (mean absolute deviation) for Restaurant B is greater than the MAD (mean absolute deviation) for Restaurant A.

C.

The youngest person was dining in Restaurant A.

D.

The IQR (interquartile range) for Restaurant A is equal to the IQR (interquartile range) for Restaurant B.

E.

Every person dining in Restaurant B is older than everyone dining in Restaurant A.

Answer: A, C

Teaching Notes

Students who select Choice B are attempting to judge the mean absolute deviation by visually assessing the box plots, but no such judgment can be made. Students who select Choice D may have calculated the IQR incorrectly or made an incorrect visual assessment of the IQR. Students who select Choice E did not realize that the oldest person dining in Restaurant A is older than the youngest person dining in Restaurant B.

4.

This box plot displays information about the distance in miles teachers drive to school each day.

<p>A box plot. Distance in miles.</p>

  1. What is the IQR (interquartile range)?

  2. What is the median distance in miles driven by teachers?

  3. Is this data set symmetric? Explain how you know.

Answer:

  1. 10, because 45−35=1045-35=10.
  2. 40
  3. No. If a vertical line is drawn at the median, the left and right sides are not mirror images.

Tier 1 response:

  • Accurate, correct work.
  • Correct answers to all three questions, including a correct explanation for why the data set is not symmetric.
  • Acceptable errors: Arithmetic error computing the IQR using correct values for quartiles 1 and 3; different wording for the last question that conveys the same idea.

Tier 2 response:

  • Work shows general conceptual understanding and mastery, with some errors.
  • Sample errors: Incorrect median; incorrect IQR; incorrect answer or explanation on data symmetry question, including a general statement that the box plot is not symmetric with no explanation.

Tier 3 response:

  • Significant errors in work demonstrate lack of conceptual understanding or mastery.
  • Sample errors: Two or more error types from Tier 2 response.

Teaching Notes

Students use a box plot to make conclusions about a data set, including the median, the IQR, and the shape of the distribution.

5.

Two groups of students had a car wash fundraiser every day for a week in the summer. Here are the funds raised from each group.

Group P

  • 60
  • 65
  • 70
  • 70
  • 80
  • 80
  • 85

Group Q

  • 40
  • 50
  • 50
  • 60
  • 80
  • 90
  • 140

  1. Draw two box plots, one for the data in each group.

    <p>A number line</p>

  2. Which group shows greater variability? Explain your reasoning.

Answer:

Solution

  1.  

  2. Group Q shows greater variability. It has a wider range (100 to Group Q’s 25), and a wider IQR (40, compared to Group P’s 15).

Tier 1 response:

  • Accurate, correct work.
  • Both box plots are drawn correctly, correctly stating that Group Q shows greater variability and using the range or IQR as evidence.

Tier 2 response:

  • Work shows general conceptual understanding and mastery, with some errors.
  • Sample errors: 1 or 2 types of minor errors in creating box plots (incorrect placement of median, quartiles, max or min, badly drawn box); incorrectly stating Group P shows greater variability or omitting the second question.

Tier 3 response:

  • Significant errors in work demonstrate lack of conceptual understanding or mastery.
  • Sample errors: More than 2 types of minor errors in creating box plots; major errors in creating box plots, such as not using 7 numbers to generate box plot; creating only one box plot.

Teaching Notes

Because box plots are being constructed, students are very likely to use the IQR or range as measures of variability, but Group A also has the higher MAD if students decide to compute it.

6.

Ten students took a history quiz. This list shows how many questions each student answered correctly.

9

6

8

10

5

8

10

7

9

8

  1. What is the IQR (interquartile range)?
  2. What is the median?

Answer:

  1. 2 questions. (The ordered list is 5, 6, 7, 8, 8, 8, 9, 9, 10, 10. The first half of the data is 5, 6, 7, 8, 8; its median is 7; The second half of the data is 8, 9, 9, 10, 10; its median is 9. The IQR is 2, since 9−7=29-7=2.)
  2. 8 questions.

Teaching Notes

Watch for students attempting to answer the question without first sorting the data. Also watch for students excluding the center values from the quartile calculation, which leads to an incorrect (larger) IQR.

7.

Priya asked some student athletes at her school how many hours each month they spent practicing their sport. Here are a box plot and histogram for the data she collected.

<p>A histogram. Time in hours.</p>

<p>A box plot. Time in hours.</p>

  1. How many students did Priya ask?
  2. Is the mean or the median a more appropriate measure of center for this data set? Explain your reasoning.
  3. Can Priya use these data displays to find the exact median? Explain how you know.
  4. Can Priya use these data displays to find the exact mean?
  5. What would be an appropriate measure of variability for this data set? Find or estimate its value.

Answer:

Solution

  1. Priya asked 100 students.
  2. The median is more appropriate because the data is not symmetric.
  3. Yes, the box plot gives the median, which is 57 hours.
  4. No.
  5. Sample response: The IQR (interquartile range) is appropriate, because the median is being used as a measure of center. The box plot gives the IQR of 19 hours because 64−45=1964-45=19.

Tier 1 response:

  • Accurate, correct work.
  • Correct answer to each question, description of why IQR is an appropriate measure of spread, correct IQR.
  • Acceptable errors: Mistake in determining median or IQR caused by a misreading of the box plot.

Tier 2 response:

  • Work shows good conceptual understanding and mastery, with minor errors.
  • Sample errors: Incorrect response for total number of students; stating that the data is symmetric; attempt to calculate precise mean; incorrect or missing IQR calculation.
  • Acceptable errors: Incorrect MAD estimation, given (incorrect) statement that data is symmetric.

Tier 3 response:

  • Work shows a developing but incomplete conceptual understanding, with significant errors.
  • Sample errors: Two or more error types from Tier 2 response; incorrect response for histogram total, 6 or fewer; incorrect median; invalid use of box plot to determine mean.

Tier 4 response:

  • Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
  • Sample errors: Three or more error types from Tier 2 response; two or more error types from Tier 3 response; multiple omitted parts.

Teaching Notes

This question is about the limitations of the histogram and box plot, which provide only partial information about a distribution. Notably, one display may be more useful than another, depending on the question asked about the data.