Getting Started 
Misconception/Error The student is not able to interpret marginal and joint relative frequencies of data in a twoway table. 
Examples of Student Work at this Level The studentâ€™s explanation of marginal and joint relative frequencies is unclear or incorrect. For example, the student:
 Is unable to write a coherent explanation of either relative frequency.
Â
 Correctly converts each relative frequency to a percent without further interpretation.
Â
 Incorrectly describes each relative frequency.
Â Â Â
Â

Questions Eliciting Thinking Can you explain the meaning of the terms marginal relative frequency and joint relative frequency?
The relative frequencies are given as fractions. What does the numerator of each fraction represent? The denominator?
Where is the value of the denominator of each fraction located in the table? How can this location help you understand its meaning?
In the table, the value of 17 combines which two categories? What does the value of 100 represent in the table? Can you use the structure of a fraction to interpret how these two categories are related?
What does the value of 29 represent in the table? What does the value of 100 represent in the table? Can you use the structure of a fraction to interpret how 29 and 100 are related? 
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)]. Provide the student direct instruction onÂ interpreting marginal, joint, and conditional relative frequencies for data in twoway tables. Remind the student that a marginal frequency represents a row or column total out of the entire sample size. Ask the student to identify the marginal cell containing 29 and use the appropriate column title and the context of the data to explain the meaning of . Likewise, remind the student that a joint frequency represents a number in a cell from the body of the table out of the sample size. Ask the student to identify the cell containing 17 and use the appropriate column and row titles and the context of the data to explain the meaning ofÂ .
Explain the difference between frequencies and relative frequencies. Indicate to the student that since the sample size of this set of data is 100, the values in the table are easily converted to percents and essentially represent relative frequencies.
Provide the student practice writing marginal, joint, and conditional relative frequencies for data in twoway tables. Assist the student with interpreting marginal, joint, and conditional relative frequencies when given in fraction form. Tell the student to use the relationship between the numerator and denominator of a fraction to guide the word arrangement for a clear and concise response. Give feedback as necessary. 
Making Progress 
Misconception/Error The studentâ€™s interpretation lacks specificity. 
Examples of Student Work at this Level The student indicates an understanding of the marginal relative frequencies in the context of the data but omits an important piece of information in his or her interpretation. For example, the student:
 Does not reference the actual values in the marginal relative frequencies when describing them.
Â Â Â
Â
Â
 Explains the numerator and denominator independently (separated by a fraction bar) rather than describing the relationship between them.
Â

Questions Eliciting Thinking How did you determine your interpretation? Which part of the table did you use?
Do you remember the meaning of the terms, marginal relative frequency and joint relative frequency? How can the key terms, marginal and joint, help you remember their meaning?
How does understanding the structure (numerator and denominator) of a fraction help you answer these questions? What does the numerator of a fraction represent? The denominator? How might you edit your response to reflect a more clear and concise interpretation of each relative frequency?
Your response is somewhat correct, but can you refine it or describe it more clearly and concisely? 
Instructional Implications Ask the student to identify the two categories related by the cell with an entry value of 29. Next, ask the student to identify what the value of 100 represents in the table. Remind the student to use wording that supports the meaning of these values when interpreting the marginal and joint relative frequency.
Remind the student that writing a clear and concise response starts with using the wording of the question. For example, guide the student to describe the meaning of the relative frequencies as follows:
 The marginal relative frequency of Â is the number of (the category related to the numerator) out of (the category related to the denominator).
 The joint relative frequency ofÂ Â is the number of (the two categories related to the numerator) out of (the category related to the denominator).
Provide the student practice writing the marginal, joint, and conditional relative frequencies for data in twoway tables. Assist the student with interpreting marginal, joint, and conditional relative frequencies when given in fraction form. Tell the student to use the relationship between the numerator and denominator of a fraction to guide the word arrangement for a clear and concise response. Give feedback as necessary. 
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 interprets the relative frequencies as follows:
 The marginal relative frequency ofÂ Â means that 29 of the 100 people surveyed are vegetarians.
 The joint relative frequency ofÂ means that of the 100 people surveyed, 17 of them are both female and vegetarian.

Questions Eliciting Thinking Can you interpret the meaning of ? ? What if the sample size had not been 100? Would that change how you determine the relative frequencies? 
Instructional Implications Provide the student with data presented in a twoway table with a sample size that is not easily converted to a percent and with unequal row totals and unequal column totals. Ask the student to consider how to calculate relative marginal and joint frequencies (i.e., which values should serve as denominators for each relative frequency). 