Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Bivariate data
- Categorical data
- Joint frequency
- Joint relative frequency
Purpose and Instructional Strategies
In grade 8, students began their exploration of bivariate data. Students focused on informally finding the line of fit. In Algebra I, students studied bivariate categorical data and displayed it in tables showing joint frequencies and marginal frequencies. In Mathematics for College Statistics, students continue this exploration with the use of real-world data.
- Instruction includes the connection to MA.912.DP.1.1 where students work with categorical bivariate data and display it in tables. A two-way frequency table is just a way to display frequencies jointly for two categories.
- In later courses, students may hear the term for two-way tables as contingency tables. Instruction includes the use of both terms interchangeably so that students are familiar with them.
- Students are expected to interpret the joint and marginal frequencies and therefore must know the differences between these two.
- Marginal frequencies
- Entries in the “Total” row and “Total” columns are called marginal frequencies. The areas highlighted in yellow in the table below represents the marginal frequencies.
- Joint frequencies
- Entries in the body of the table represent joint frequencies. The area highlighted in yellow in the table represents the joint frequencies.
- Instruction includes students creating two-way (contingency) tables from data that has either been collected or researched. Students can then analyze the data by either examining joint relative frequencies or marginal relative frequencies.
- Instruction includes determining possible association. Two categorical variables are associated if the row of conditional relative frequencies (or column of relative
frequencies) are different for the rows (or columns) of the table. Evidence of an association is strongest when the conditional
relative frequencies are quite different. If the conditional relative frequencies are nearly equal for all categories,
then there is probably not an association between variables.
- Instruction includes determining if variables are independent. If the conditional relative frequencies are nearly equal for all categories, there may be no association between the variables and they are said to be independent.
- Instruction includes experience with the Simpson’s Paradox. Simpson’s Paradox is an
association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group.
- For example, a university was concerned with the number of females who were being accepted into their graduate programs.
Of the 384 men who applied to the school, 285 were admitted which is an admission rate of 74.2%. Of the 435 women who applied to the school, 126 were admitted which is an admission rate of 29.0%. This may suggest there is a bias where men are favored in graduate
admissions. However, upon closer inspection of the graduate programs they
offered, the following was discovered:
- When looking at the individual programs and comparing the acceptance rate of women versus that of men, they noticed that more women were being accepted into these programs then men were. This is an example of a Simpson’s Paradox. When the data is combined into a single group, we get one result. When the data is divided into several groups, the results are reversed.
Common Misconceptions or Errors
- Students may have difficulty when completing the table based on the given data.
- Students may not distinguish the differences between marginal and joint frequencies.
- Students may make errors when identifying the relationship and possible associations in the data in terms of the given context.
- Students may find the relative joint frequencies of the column only and not realize that to
find the relative joint frequency depends on what is being analyzed and could be based on
Strategies to Support Tiered Instruction
Instructional Task 1
The U.S. Department of Transportation
Federal Highway Administration
National Household Travel Survey. Within that survey, they
looked at trends in the annual
number (millions) of person trip by mode of transportation. The
following is the data from
- Private Vehicle: to and from work: 56,981; work-related business: 4,844; shopping
and errands: 126,268; school or church: 28,427; social and recreational: 78,890;
- Public Transit: to and from work: 3,537; work-related business: 208; shopping and
errands: 2,586; school or church: 1,009; social and recreational: 1,618; other: 487
- Walk: to and from work: 2,523; work-related business: 510; shopping and errands:
11,496; school or church: 4,146; social and recreational: 18,483; other: 1,790
- Other: to and from work: 1,540; work-related business: 486; shopping and errands:
2,404; school and church: 6,721; social and recreational: 3,330; other: 1,873
- Part A. Create a contingency table using the data provided.
- Part B. What is the joint relative frequency of people who used public transit traveled for
work related business?
- Part C. What is the marginal relative frequency of those who traveled for shopping and
- Part D. What is the joint relative frequency of people who traveled to and from work by
using a private vehicle?
- Part E. Which mode of transportation was most popular as it relates to social and
- Part F. Does the data show an association between using a personal vehicle and going to
school or church? Justify your reasoning.
Instructional Item 1
The results from a survey about whether students (male or female) at a university were from England, Wales or Scotland is summarized in the following two-way table:
- Part A. Complete the table with the totals.
- Part B. What proportion of the students are women?
- Part C. What proportion of students are from Wales?
- Part D. Is it fair to say that more students from Wales are women? Explain your reasoning.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.