Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Scatter plots work for numbers, but many interesting questions involve categories: do students who play a team sport prefer cats or dogs? Does owning a phone go with walking to school? A two-way table organizes this kind of bivariate categorical data, with one variable across the top, the other down the side, and counts in each cell plus row and column totals.
Raw counts can mislead when the groups are different sizes, so students compute relative frequencies. Dividing each cell by its row total answers questions like 'of the students who play a sport, what fraction prefer dogs?', and comparing that with the non-sport row reveals whether the two variables are associated. If 70% of sport players prefer dogs but only 40% of non-players do, there is evidence of an association; if the percentages are close, there is little. Choosing whether to divide by row or column totals depends on the question being asked, and students must be able to explain that choice.
If 30 of 100 seventh graders and 20 of 40 eighth graders like a club, the eighth graders are more keen (50% versus 30%), even though 20 is less than 30.
To compare groups, divide by the group's row or column total. The grand total answers a different question about the whole sample.
Students compute 'of the dog lovers, what fraction play sport' when the question asked the reverse. Underline the group named after 'of the'.
Percentages of 52% and 49% are very close, which suggests little or no association. A clear gap is needed before claiming a link.
A survey of 80 students: of 50 who have a curfew, 35 do chores at home; of 30 with no curfew, 9 do chores. Is there evidence that students with a curfew are more likely to do chores?
Answer: Yes. 70% of students with a curfew do chores compared with 30% of those without, which suggests an association.
Survey the class on two yes-or-no questions that students choose, then build the two-way table together on the board. Seeing their own data in the table, and then converting rows to percentages, shows why relative frequencies are fairer than counts when groups differ in size.
Assessment items often give a partially completed table to fill in, then ask for a relative frequency and whether the data suggest an association. Train students to write the fraction first, with the right total in the denominator, before converting to a percentage.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 70 (also accepted: 70%)
28 ÷ 40 = 0.7, which is 70%.
Answer: 18
The cells in a row add to the row total: 60 - 25 - 17 = 18.
Answer: C) Bus riders are more likely to eat breakfast at school (40% versus 20%)
20 ÷ 50 = 40% of bus riders and 6 ÷ 30 = 20% of car riders eat breakfast at school, so bus riders are more likely.
Answer: B) The total number of 8th graders
The question is about 8th graders, so divide the number of 8th graders who prefer soccer by the total number of 8th graders.
Answer: A) Little or no association
51% and 49% are almost the same, so the data show little or no association between group and opinion.
A full lesson with slides, activities and an exit ticket on two-way tables and relative frequency, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.SP.A.4 with an answer key, ready in about a minute.
Make a worksheet →Turn two-way tables and relative frequency into a quiz students answer online that marks itself, with a class summary for you.
Build a test →A count divided by a total, such as a row total or column total, written as a fraction, decimal or percentage. It lets you compare groups of different sizes fairly.
Compare the relative frequencies across groups. A clear difference, such as 70% versus 30%, suggests an association; similar values suggest none.