Getting Started 
Misconception/Error The student is unable to determine if data in a twoway table suggest a relationship between categories. 
Examples of Student Work at this Level The student:
 Establishes a decision based on personal opinion rather than the data in the table.
 Writes about the structure of the table rather than data presented in the table.
 Writes about vegetarianism and nonvegetarianism rather than gender and vegetarianism.
 Writes a yes or no answer using the wording of the question without justification. When asked, the student states that the response is a â€śguess.â€ť

Questions Eliciting Thinking How might you examine the data in the table to draw a conclusion? What is the same/different about the two tables?
What is the question asking you to do? Did you answer the question asked? What does it mean to â€śsuggest a relationshipâ€ť between gender and vegetarianism? What would you look for in the table? Which rows/columns?
Your response correctly describes the structure of the table, but is the question asking about the structure or the data in the cells of the table? 
Instructional Implications Provide instruction on the structure of twoway frequency tables. Assist the student in identifying the two variables described in the context (e.g., gender and vegetarianism) and the number of levels of each variable [e.g., two levels of gender (male and female) and two levels of vegetarianism (vegetarian and nonvegetarian)]. Since the row (i.e., gender) totals are equal, guide the student to calculate and compare the relative frequencies of males and females who are vegetarian for each table. Assist the student in interpreting similarities or differences in the relative frequencies as evidence of a relationship between the two variables, gender and vegetarianism. Emphasize that for a relationship to exist between two variables, there should be a (wide) gap or difference between the relative frequencies within a category. A significant difference suggests that there may be a relationship between the two variables. Explain to the student that the only valid conclusion is to say that this discrepancy â€śsuggests a relationship.â€ť Any statement explaining or describing the reasoning for the relationship is a hypothesis or conjecture.
Provide additional opportunities to examine twoway frequency tables for evidence of a relationship between two variables. Include examples in which both column totals are unequal and row totals are unequal. 
Making Progress 
Misconception/Error The student provides a partially correct interpretation that either contains errors or lacks specificity. 
Examples of Student Work at this Level The student has some understanding of how to examine a twoway table for evidence of a relationship between two variables. However, the student completes only a partial examination or is unable to correctly communicate using appropriate language. The student:

Questions Eliciting Thinking What is the question asking you to do? Did you answer the question asked?
What does it mean to â€śsuggest a relationshipâ€ť between gender and vegetarianism? Is it equally likely for both males and females to be vegetarians or does the table show that one gender is more likely to be vegetarian than the other?
In the first table, the number of male and female vegetarians differs by one person. Is this enough of a difference to assume a relationship or a trend?
What did you base your decision on? Can you explain your rationale in more detail? 
Instructional Implications Assist the student in interpreting similarities or differences in the relative frequencies as evidence of a relationship between the two variables, gender and vegetarianism. Emphasize the importance of comparing relative frequencies of one variable at both levels of the other variable when computing relative frequencies with unequal totals. For example, since the total number of vegetarians is not equal to the total number of nonvegetarians, explain that it is not enough to calculate only the percent of males and females who are vegetarians; the percent of males and females who are not vegetarian must also be calculated and compared. Both sets of relative frequencies must be considered when looking for patterns and relationships between the two variables.
Since the row (i.e., gender) totals are equal, guide the student to calculate and compare the relative frequencies of males and females who are vegetarian for each table. Emphasize that for a relationship to exist between two variables, there should be a (wide) gap or difference between the relative frequencies within a category. A significant difference suggests that there may be a relationship between the two variables. Explain to the student that the only valid conclusion is to say that this discrepancy â€śsuggests a relationship.â€ť Any statement explaining or describing the reasoning for the relationship is a hypothesis or conjecture.
Remind the student that writing a clear and concise response includes using the context of the data to describe frequencies and relative frequencies. Guide the student to describe 48% (14/29) as the percent of vegetarians who are male and 52% (15/29) as the percent of vegetarians who are female. Provide the student with a model response to the task and have the student compare his or her answer to this response. Ask the student to identify what was missing, incorrect, or unnecessary in his or her answer and explain why the model response is more precise and complete.
Provide additional opportunities to examine twoway frequency tables for evidence of a relationship between two variables. Include examples in which both column totals are unequal and row totals are unequal. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student examines the data in the table to investigate the relationship between the two variables. The student references relative frequencies rather than raw data and explains:
 The data in table one do not suggest that there is a relationship between gender and vegetarianism. The percent of males who are vegetarians is nearly equal to the percent of females who are vegetarian, 28% and 30%, respectively. Consequently, gender does not seem to influence or be related to vegetarianism.
 The data in table two suggests a relationship between gender and vegetarianism. The percent of males who are vegetarians is much greater than the percent of females who are vegetarian, 76% and 34%, respectively. The data suggest that if you are male, you are more likely to be a vegetarian.
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Questions Eliciting Thinking It looks like you compared the percent of males who are vegetarian to the percent of females who are vegetarian. Is it necessary to also compare the percent of males who are not vegetarian to the percent of females who are not vegetarian? Why or why not? 
Instructional Implications Ask the student to explain why organizing data in a twoway frequency table is helpful for identifying relationships and trends in data. Challenge the student to explain how examining raw data (as opposed to relative frequencies) might lead to an inaccurate conclusion when analyzing a twoway frequency table.
Consider implementing MFAS task Conditional Relative Frequency (912.SID.2.5). 