Reading categorical displays
Read category labels, the scale and the sample total together. Count and percentage are different units
- Frequency A frequency is the number of observations in a category
- Percentage A percentage is a share per hundred. Use the relevant sample total to recover a count \(\displaystyle c=pn/100\)
- Mode and ties Report the category with the greatest frequency. If categories share the greatest frequency, report each tied mode
- Compare groups Use percentages to compare shares across different sample totals; use counts when the question asks how many
- Rounding A rounded percentage may not identify a unique original count. Keep precision and limits explicit
Worked example
Each of 120 members selects one preferred activity: swimming 25%, cycling 35%, walking 40%. Find the swimming count and the combined swimming-or-cycling count
- \(\displaystyle 25\times120/100=30\)Thirty members choose swimming
- \(\displaystyle 35\times120/100=42\)Forty-two choose cycling
- \(\displaystyle 30+42=72\)Seventy-two choose one of these two non-overlapping categories
Common mistakes
Difficulty levels
Questions about this topic
Why is the sample total needed?+
A percentage gives a share but not an absolute count. Multiply that share by the relevant total
Can a larger percentage represent fewer people?+
Yes. Different totals can reverse the ordering of counts. For example, 30% of 40 is 12 but 20% of 100 is 20
What is a percentage-point difference?+
It is the subtraction of two percentages. A change from 25% to 45% is 20 percentage points; the relative increase is 20 divided by 25, or 80%
Can there be more than one modal category?+
Yes. Every category tied at the greatest frequency is a modal category
Can a rounded percentage recover an exact count?+
Not always. Several integer counts can produce the same rounded percentage. State an estimate or use more precise information
When can I add category counts to find the sample total?+
When every observation belongs to exactly one listed category and the categories cover the whole sample. Multiple-response categories may overlap
Does the middle category in a table have special meaning?+
Not for nominal categories. Their order need not be numerical; the modal category is determined by frequency
Does a clear bar chart prove a population claim?+
No. Check how the sample was selected, who was excluded and whether the claim extends beyond the recorded evidence
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, General Mathematics, chapter 1: Data distributions