Two-Way Frequency Tables - TCE Mathematics Applications (Tasmania)
“Construct and interpret two-way frequency tables to investigate association between categorical variables.”
You must build a two-way frequency table from categorical data, convert the counts into row or column percentages, and decide whether the two variables are associated by comparing those percentages across categories.
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.
A table that cross-classifies each individual by two categorical variables at once. One variable labels the rows, the other labels the columns, and each cell holds the count of individuals in that combination.
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.
A row percentage divides each cell by its row total. A column percentage divides each cell by its column total. Choose the one that lets you compare the response variable across the categories of the explanatory variable.
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.
A survey of 200 people records sex and whether they prefer tea or coffee. Of 120 women, 78 prefer tea; of 80 men, 32 prefer tea.
Construct the table
Women: tea 78, coffee 42, total 120. Men: tea 32, coffee 48, total 80. Column totals: tea 110, coffee 90, grand total 200.
Convert to column percentages within each sex
Treat sex as the explanatory variable, so use the sex totals as the base.
Women preferring tea: .
Men preferring tea: .
Interpret
of women prefer tea against only of men, a gap of 25 percentage points. Because the breakdown differs markedly between the sexes, sex and drink preference are associated in this sample.
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.
Decide which variable is explanatory, then make each of its categories total 100 percent. Comparing those equal-base percentages is what reveals association. Comparing raw counts can mislead when the group sizes differ.
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.
In another survey, 50 percent of younger respondents and 52 percent of older respondents owned a pet.
Compare the percentages
The two percentages, 50 percent and 52 percent, are almost identical.
Conclude
Since the proportion owning a pet barely changes between age groups, there is little or no association between age group and pet ownership in this sample.
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.
Students often say one group is bigger because its cell count is larger, but raw counts are misleading when the groups are different sizes. With 120 women and 80 men, more women prefer tea partly because there are more women overall. Convert to percentages within each group before comparing, every time.
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 questions
Questions in the style of TASC exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.
Practice5 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.
Practice4 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.