Missing values, modes and display conversions
Recover a missing observation from a mean and express the same frequency data as counts, percentages and pie sectors.
What you will tackle
- Recover a missing value using total = count × mean
- Find modes including ties
- Translate frequency tables and charts
- Check whether rounded summaries support an exact conclusion
Recover the total first
For n observations with mean m, the total is nm. Subtract known observations to find a missing value. A mode is a most frequent value; a dataset can have several tied modes. Frequencies, percentages and pie-sector angles describe the same categories when they use the same total.
Change the representation, preserve the data
A category with frequency f out of total n has percentage 100f/n and pie-sector angle 360f/n degrees. A bar chart's bar height can be a count or a proportion, so read its axis. Rounded reported means or percentages can represent intervals, so they may not determine one exact missing observation.
| Category | Frequency |
|---|---|
| A | 12 |
| B | 18 |
| C | 30 |
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Five observations have mean 6; four are 3,5,6,7. Find the fifth. Find the modes of 1,1,2,2,3. Convert the displayed category frequencies A=12, B=18, C=30 into percentages and pie-sector angles.
Show a starting hint
All pie sectors together must total 360°.
Compare with the explained answer
- The total is 5·6 = 30; the known sum is 21, so the missing value is 9.
- Both 1 and 2 occur twice, so both are modes.
- The total frequency is 60: percentages 20%,30%,50% correspond to angles 72°,108°,180°.
Answer / response criteria: 9; modes 1 and 2; 20%,30%,50% and 72°,108°,180°.
Independent practice
Independent 1
Six values have mean 10. Five are 4,8,10,12,14. Find the sixth and the resulting mode. A pie chart of 80 people has sectors of 90°,135°,135°; recover the three counts.
Compare with the explained answer
- The total is 60 and known sum 48, so the sixth value is 12.
- 12 now occurs twice, more often than any other value, so it is the mode.
- The sector fractions are 1/4,3/8,3/8; multiplying by 80 gives 20,30,30.
Answer / response criteria: 12; mode 12; counts 20,30,30.
Application practice
Application 1
A class of 20 has mean score 7.5, while another 30 students have mean 8. Compute their combined mean. If the first mean was merely reported as 7.5 after rounding to one decimal place, explain why that combined mean would no longer be uniquely exact. Convert 30 of the 50 students into a pie-sector angle.
Compare with the explained answer
- Use weighted totals: (20·7.5 + 30·8)/50 = 390/50 = 7.8.
- A rounded first mean can lie from 7.45 up to 7.55, so the total and combined mean can vary; 7.8 would be based on the reported representative value.
- 30/50 of a full circle is 216°.
Answer / response criteria: 7.8 if the supplied means are exact; otherwise the combined mean is an estimate; 216°.
What can you explain now?
A class of 20 has mean score 7.5, while another 30 students have mean 8. Compute their combined mean. If the first mean was merely reported as 7.5 after rounding to one decimal place, explain why that combined mean would no longer be uniquely exact. Convert 30 of the 50 students into a pie-sector angle.
Compare with the explanation
7.8 if the supplied means are exact; otherwise the combined mean is an estimate; 216°.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide