Two-way tables and categorical relationships
Compare like response shares within clearly identified groups. Explain what the sample shows and what it cannot establish
- Within-group percentage Divide the response count by that group’s total, then multiply by 100. \(\displaystyle p=100c/n\)
- Fair comparison Keep the response category fixed. Different group sizes can reverse count and percentage comparisons
- Whole distributions With more than two responses, compare all categories. One matching share does not imply identical distributions
- Inference limits There is no universal percentage-point threshold proving a population association. Sample selection, uncertainty and context matter
Worked example
A has 18 Yes and 12 No responses. B has 24 Yes and 56 No responses. Compare the Yes shares
- \(\displaystyle 100\times18/30=60\%\)A’s denominator is 30.
- \(\displaystyle 100\times24/80=30\%\)B’s denominator is 80.
- \(\displaystyle 60-30=30\)A is 30 percentage points higher, although B has more Yes responses. This describes the sample and does not prove cause
Common mistakes
Difficulty levels
Questions about this topic
Does the explanatory variable have to be in rows?+
No. Read the labels and divide by the total for the named explanatory group, whether it is a row or a column
Why not just compare counts?+
Counts also reflect group size. Use counts for how many, and within-group percentages for comparing response shares
What if one group has no records?+
A within-group percentage is undefined when the group total is zero. Do not invent 0% or divide by zero
Does an equal response percentage mean no relationship?+
For two exhaustive responses, equal shares give matching displayed distributions. With three or more responses, the other shares may differ. Neither case alone proves population independence
Is a 10 percentage-point gap always significant?+
No. There is no universal cutoff here. Sample sizes, design, uncertainty and context matter for population inference
Can a two-way table prove cause?+
An observed sample association alone does not establish causation. Other variables and the collection method may explain it
How should missing responses be handled?+
State the rule and missing counts. A separate missing category may help; excluding records without explanation can distort the groups
What is the difference between percent and percentage points?+
From 30% to 60% is 30 percentage points. Relative to 30%, it is a 100% increase. Name the quantity being compared
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 5: Statistics