The idea, step by step
Plan, collect and organise
Begin with a question. Use: “How many books were borrowed in one fortnight by these twelve Year 7 students?” Define a loan and use the same fortnight for every record. The fictional teaching data are 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9 and 13 books.
Check the records, keep them anonymous and organise them in a frequency table. There are twelve observations. This variable measures borrowing, not books read or reading ability.
| Books borrowed | Frequency |
|---|---|
| 4 | 1 |
| 5 | 2 |
| 6 | 3 |
| 7 | 2 |
| 8 | 2 |
| 9 | 1 |
| 13 | 1 |
Analyse, conclude and question
The total is 84 books and the mean is 7. The median is 6.5, the mode is 6 and the range is 9. A dot plot shows eleven observations from 4 to 9, with one at 13. These summaries describe this particular set of records.
A sensible conclusion is that borrowing in this teaching group was usually around 6–8 books in the fortnight, with one higher count of 13. Do not claim this proves a reading program works or describes every Year 7 student. State the small group, short period and difference between loans and reading.
A common trap
The conclusion can be stronger than the question and data if the average looks convincing.
Instead: Only make claims supported by the defined group, variable, method and period.
See the thinking
Build a supported conclusion
Write a conclusion using two pieces of numerical evidence from the loan data.
| Books borrowed | Frequency |
|---|---|
| 4 | 1 |
| 5 | 2 |
| 6 | 3 |
| 7 | 2 |
| 8 | 2 |
| 9 | 1 |
| 13 | 1 |
- Median=(sixth value 6 + seventh value 7)÷2=6.5 books.
- The most frequent count is 6, occurring three times.
- Tie both summaries back to borrowing during the chosen fortnight.
Reject an unsupported causal claim
A report says, “The new library posters made all Year 7 students read seven books.” Explain problems with this claim.
- There is no comparison or experimental evidence showing posters caused borrowing.
- The records concern twelve students’ loans, not all Year 7 students or books actually read.
- Seven is the mean, not a statement that everyone borrowed seven.
Check your understanding
The investigation records books borrowed. Can it conclude how many were finished?
Show answer
No.
A borrowing record does not show whether a book was read or completed.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Check the total frequency in the loans table.2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 - Use the loans table to calculate the total number of loans.2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 - Find the mean loans per student.2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 - Why can the median be 6.5 books when counts are whole numbers?2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 - The record of 13 loans is confirmed correct. Should it be removed because it is higher than the others? Explain.2 marks
- Suggest an improvement if the question becomes “How much do students across the whole year borrow during a term?”2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Check the total frequency in the loans table.2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 1+2+3+2+2+1+1=12.
12 students. - Use the loans table to calculate the total number of loans.2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 4×1+5×2+6×3+7×2+8×2+9×1+13×1=84.
84 books. - Find the mean loans per student.2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 84 loans divided by 12 students gives 7.
7 books. - Why can the median be 6.5 books when counts are whole numbers?2 marks
Fictional teaching data: books borrowed in one fortnight Books borrowed Frequency 4 1 5 2 6 3 7 2 8 2 9 1 13 1 (6+7)÷2=6.5.
A summary need not be an observed count.
For 12 observations, the median is the average of the sixth and seventh values, 6 and 7. - The record of 13 loans is confirmed correct. Should it be removed because it is higher than the others? Explain.2 marks
It is a genuine observation belonging to the question.
Keep it and describe its influence.
No. - Suggest an improvement if the question becomes “How much do students across the whole year borrow during a term?”2 marks
Match the collection period and group to the new question.
A fortnight and twelve selected students do not establish the broader pattern.
Use the whole term and a suitably selected larger group across the year, with the same loan definition.
Curriculum reference & lesson scope
Australian Curriculum v9 / South Australian Mathematics Curriculum
Year 7 statistical variables, representations, summary statistics, outliers and an original small investigation; deeper sampling methods develop in Year 8.
Official curriculum source ↗