Comparing categorical groups
Keep the groups, sample totals and units visible
- Counts Counts answer how many. A larger sample may have a larger category count without a larger share
- Percentages Use each group's own total to compare proportions
- Percentage points Subtract percentages to describe their gap in percentage points
- Whole distributions A comparison of one category does not establish that all category shares are the same
Worked example
A has 12 Bus responses out of 40 and B has 16 out of 80. Compare the counts and shares
- \(\displaystyle 16>12\)B has the larger Bus count
- \(\displaystyle \frac{12}{40}\times100=30\)Use A's total: 30% chose Bus
- \(\displaystyle \frac{16}{80}\times100=20\)Use B's total: 20% chose Bus
- \(\displaystyle 30-20=10\)A has the larger share by 10 percentage points, although B has the larger count
Common mistakes
Using the combined total
✕\(\displaystyle \frac{12}{120}\times100\)✓\(\displaystyle \frac{12}{40}\times100=30\)
The question asks for A's percentage, so use A's total. The result is 30%Assuming more means a larger share
✕\(\displaystyle 16>12\Rightarrow p_B>p_A\)✓\(\displaystyle 20\%<30\%\)
Different group totals can reverse the ranking of counts and percentagesDifficulty levels
introCompare category counts in two samples
standardCalculate both within-group percentages and compare shares
challengeResolve conflicting count/share rankings and equal-share cases
examSelect a supported comparative report with correct group totals
Questions about this topic
Why compare percentages when sample sizes differ?+
Percentages put both groups on a per-hundred basis. Counts alone can reflect the difference in sample sizes
Can two groups have equal shares but different counts?+
Yes. 12 of 40 and 24 of 80 are both 30%
What is a percentage-point gap?+
It is the difference between percentages. 30% minus 20% is 10 percentage points
Does a sample comparison prove a population difference?+
No. You also need to consider how each sample was selected and the uncertainty in the results
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 2: Data distributions
In this chapter
Classify a variable as nominal, ordinal, discrete or continuousDistinguish interval and ratio measurement scalesBuild a frequency and percentage frequency tableBuild a bar chart from a frequency table2A Data types and categorical displays: Read values from a frequency table or bar chart2B Reading categorical displays: Read values from a frequency table or bar chartReport on the distribution of a categorical variableCompare categorical distributions across two groupsGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramDescribe the shape of a distributionConstruct a dot plot from raw dataConstruct an ordered stem plot from raw dataChoose an appropriate display for a variableFind the mean and the median, and choose between themFind measures of centre from a frequency displayWork backwards from a mean or medianFind the range, the quartiles and the IQRCalculate the standard deviation from the formulaApply the 68–95–99.7% ruleProduce a five-number summary and construct a boxplotFind the fences and identify outliersRead values and percentages from a boxplotCompare a numerical variable across groups