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Goal 3 Data Analysis And Probability

Expectation 3.1 The student will collect, organize, analyze, and present data.

Indicator 3.1.2 The student will use the measures of central tendency and/or variability to make informed conclusions.

Assessment Limits:

  • Measures of central tendency include mean, median, and mode.
  • Measures of variability include range, interquartile range, and quartiles.
  • Data may be displayed in a variety of representations which may include: frequency tables, box and whisker plots, and other displays.

Brief Constructed Response (BCR) Item - Released in 2005

A cereal company awards a prize to anyone who collects all 5 different game pieces. Each game piece has an equal chance of being placed in a box of cereal. Each box of cereal contains 1 game piece. A class of 20 students conducted a simulation to see how many boxes of cereal must be purchased to collect all 5 different game pieces. Their results are shown in the table below.

Complete the following in the Answer Book:

  • What are the mean, median, and mode of the data?
  • A box of cereal costs $2.80. Based on the simulation results, how much money must a consumer spend to collect all 5 different game pieces? Use measures of central tendency to justify your answer.
  • One student in the class suggested that the data point of 26 should be ignored. Will ignoring this data point have a greater influence on the mean or the median? Use mathematics to justify your answer.

The following 8 Anchor Papers represent a range of score points and are used in conjunction with the rubrics to assess student responses.

Anchor Paper #1

image of student response

Score for Anchor Paper #1: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. Although the answers for the mean, median, and mode are all correct, the student does not address the next part of the question concerning the amount of money a consumer must spend. The response addressing the data point of 26 shows a serious flaw in reasoning. The student indicates that both the mean and the median will be greatly changed by leaving out data point 26, when actually the median does not change at all. No valid justification is given for this answer. The response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #2

image of student response

Score for Anchor Paper #2: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The mode is correct; however, the mean and median are both incorrect. No response to the second part of the question is provided. The student correctly states that the mean will be more affected by ignoring the data point 26, but no justification is given. The response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #3

image of student response

Score for Anchor Paper #3: Rubric Score 1

Annotation: This response indicates little attempt to apply a reasonable strategy. The correct answers are given for the mean and median but the mode is not provided. No response to the second or third parts of the question is given. The response demonstrates a minimal understanding and analysis of the problem.


Anchor Paper #4

image of student response

Score for Anchor Paper #4: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The answers for the mean, median, and mode are all correct. The amount of money a consumer must spend ($14) is not based on a measure of central tendency and is the least amount of money it would take to receive all five game pieces. The student correctly states that removing data point 26 would have a greater influence on the mean and justifies by stating what the mean and median would be if 26 were ignored. The response shows a conceptual understanding and analysis of the problem.


Anchor Paper #5

image of student response

Score for Anchor Paper #5: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. Although both the mean and the median are correct, no mode is provided. A correct answer based on the mean is given for the amount of money a consumer must spend; however, no justification is supplied. The student correctly states that ignoring data point 26 will have a greater effect on the mean, and the justification (it will decreas it) supports the solution but does not address the median. The response shows a conceptual understanding and analysis of the problem.


Anchor Paper #6

image of student response

Score for Anchor Paper #6: Rubric Score 2

Annotation: This response indicates an incomplete application of a reasonable strategy. The answers for the mean, median, and mode are all correct. Although a correct answer ($25.20) is given for the amount of money a consumer would have to spend, no justification is provided. The student correctly states that removing data point 26 would have a greater influence on the mean. The justification is fully developed (the mean because it would be signaficantly lower, while the median would still be 9). The response shows a conceptual understanding and analysis of the problem.


Anchor Paper #7

image of student response

Score for Anchor Paper #7: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The answers for the mean, median, and mode are all correct. The response to the amount of money a consumer must spend ($28) is a correct answer. The justification uses a number that falls in between the three measures of central tendency ($2.80 × 10). The student correctly chooses the mean as the measure that will be most influenced if data point 26 is ignored. The justification is clearly presented and fully developed (the median stays the same even if the 26 is taken out. It is still 9. However the mean changes from 10.7 to 9.9). The response demonstrates a complete understanding and analysis of the problem.


Anchor Paper #8

image of student response

Score for Anchor Paper #8: Rubric Score 3

Annotation: This response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The answers for the mean, median, and mode are all correct. The response to the amount of money a consumer must spend contains two correct answers and a fully developed justification. The answer ($29.96) is justified by stating that this would be the cost using the mean, and the answer ($25.20) is justified by stating that this is the cost using the median or mode. The student correctly chooses the mean as the measure that will be most affected if data point 26 is ignored and provides a fully developed justification. (The median stayed the same, but the mean changed by a difference of .81.) The response demonstrates a complete understanding and analysis of the problem.


Brief Constructed Response (BCR) Rubric
Print: Scoring Rubric (pdf)
Score 3

The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are essentially correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.

Score 2

The response indicates application of a reasonable strategy that may be incomplete or undeveloped. It may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.

Score 1

The response indicates little or no attempt to apply a reasonable strategy or applies an inappropriate strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.

Score 0

The response is completely incorrect or irrelevant. There may be no response, or the response may state, “I don't know.”

Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.

Last Revised 8/16/00
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Resources for 3.1.2:
Skill Statements | PUBLIC RELEASE ITEMS | Lesson Plans |