Cumulative frequency and box plots · Extension
| English | 中文 | Pinyin |
|---|---|---|
| interquartile range/ˌɪntəˈkwɔːtaɪl reɪndʒ/ | 四分位距 | sì fēn wèi jù |
Which route is more consistent?
- Two bus routes have similar typical times but different reliability. Quartiles show the middle half of the journeys.
- This lesson studies interquartile range 四分位距: The difference between the upper and lower quartiles.
Choose the mathematical structure
- A cumulative frequency counts observations below successive class boundaries. Read quartiles at one quarter, one half and three quarters of the total frequency. A box plot represents minimum, lower quartile, median, upper quartile and maximum.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines interquartile range?
The difference between the upper and lower quartiles.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For 80 observations, read Q1 at cumulative frequency 20, median at 40 and Q3 at 60. If Q1=12,Q3=21, then IQR=9. Compare the medians for typical journey time and the IQRs for consistency.
Cumulative frequency and box plots
A cumulative frequency counts observations below successive class boundaries
Compare the model with the worked case and explain one change.
Find the IQR when Q1=12,Q3=21.
IQR=Q3-Q1=21-12=9.
Test a tempting shortcut
- Plot against class boundaries rather than midpoints. Grouped quartiles are estimates. The range is sensitive to extremes; the IQR describes only the middle half.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The interquartile range always equals the maximum minus the minimum. This claim is false. Explain which definition or assumption it violates.
For 80 observations, find the cumulative frequency used for the median.
The median position is half the total cumulative frequency: 80/2=40.
The interquartile range always equals the maximum minus the minimum.
Plot against class boundaries rather than midpoints. Grouped quartiles are estimates. The range is sensitive to extremes; the IQR describes only the middle half.
Interpret a new situation
- Explain a comparison in the context of the measured quantity. An outlier rule may use Q1-1.5IQR and Q3+1.5IQR; use the rule specified in the task rather than assuming every graph follows it.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For Q1=12,IQR=9, find Q1-1.5IQR.
The lower outlier fence is 12-1.5×9=-1.5.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 9260 · Extension · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The difference between the upper and lower quartiles. Choose the relationship, show the method, check its assumptions and interpret the result.