How do we detect and describe association between two categorical variables?
Construct and interpret two-way frequency tables to investigate association between categorical variables.
Constructing two-way frequency tables, calculating row and column percentages, and judging association between categorical variables in TCE Mathematics Applications.
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What this dot point is asking
Bivariate data is not always numerical. When both variables are categorical, such as gender and preferred transport, you cannot draw a scatterplot or compute a correlation coefficient. Instead you organise the data in a two-way frequency table and look for patterns in the percentages.
Reading the table
The cells inside the table are the joint frequencies. The totals down the right edge and along the bottom are the marginal frequencies, and the single number in the bottom-right corner is the grand total. Always check that the row totals and the column totals each add up to the grand total before going further.
Judging association
To decide whether two categorical variables are associated, you compare the percentage breakdown of the response variable across each category of the explanatory variable. If those percentage breakdowns are noticeably different, the variables are associated. If they are about the same, there is no association.
Choosing the right percentages
The percentage direction must match the question. If sex is the explanatory variable and you want to know how drink preference depends on sex, calculate the percentages within each sex (so each sex column adds to 100 percent). Calculating the percentages the other way around answers a different question and usually hides the pattern you were asked about.
Describing the strength
You are not expected to run a formal significance test in this course. Instead, describe the association in words and quantify the gap. Phrases like a higher proportion of, compared with, and a difference of so many percentage points show the marker you have read the table correctly. A larger gap between the comparison groups suggests a stronger association.
When you report, state which variable you treated as explanatory, quote the comparison percentages, and finish with a plain-language conclusion about whether an association exists.
A complete answer converts the relevant cells to percentages on a common base, compares them across the explanatory categories, and concludes clearly about association rather than restating the original counts.
Exam-style practice questions
Practice questions written in the style of TASC exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
TCE 20245 marksA table shows results from two territories (ACT and NT) in a national vote (ACT: Yes , No ; NT: Yes , No ). a) Complete a percentage table. b) Complete a bar chart of your percentages. c) What is meant by the statement: there appears to be some association between these two variables?Show worked answer →
a) (1 mark) Convert each count to a percentage of its own column (territory) total. ACT: Yes , No . NT: Yes , No . Each column totals .
b) (2 marks) Draw a segmented or side-by-side bar chart with one bar per territory, heights showing the Yes and No percentages, the vertical axis clearly labelled as a percentage and scaled correctly.
c) (2 marks) Association means the distribution of one variable (the vote) changes depending on the value of the other (the territory). Here the Yes percentage is much higher in the ACT () than in the NT (), so knowing the territory changes the likely vote. Because the column percentages differ noticeably, the two categorical variables appear associated rather than independent.
TCE 20214 marksA study of 250 students cross-classifies year level (Junior, Senior) against whether they walk to school. Of 150 juniors, 90 walk; of 100 seniors, 40 walk. a) Construct the two-way frequency table with totals. b) Using year level as the explanatory variable, decide whether walking to school is associated with year level. Justify with percentages.Show worked answer →
a) (2 marks) Juniors: walk , do not walk , total . Seniors: walk , do not walk , total . Column totals: walk , do not walk , grand total .
b) (2 marks) Percentage who walk within each year level (the explanatory variable): juniors ; seniors . The gap of percentage points means the proportion walking changes with year level, so walking to school is associated with year level in this sample. Markers want percentages on a common base and a clear conclusion.