Quartiles, box plots and distribution comparisons · Higher
| English | 中文 | Pinyin |
|---|---|---|
| interquartile range/ˌɪntəˈkwɔːtaɪl reɪndʒ/ | 四分位距 | sì fēn wèi jù |
Two classes can have the same median but different consistency. A box plot shows the middle half separately from the extreme observations.
- Two classes can have the same median but different consistency. A box plot shows the middle half separately from the extreme observations.
- This lesson studies interquartile range 四分位距: The upper quartile minus the lower quartile.
Choose the mathematical structure
- A box plot marks minimum, lower quartile Q1, median, upper quartile Q3 and maximum, or identifies separately shown outliers according to the stated convention. IQR=Q3-Q1 describes the middle half. Compare median and IQR in context; a smaller IQR suggests less middle-half variation, not necessarily a smaller full range. Quartile conventions vary for a finite raw list, so follow the task or supplied summaries.
- 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 upper quartile minus the lower quartile.
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.
Group A has summary 2,5,8,11,18 and group B has 1,6,8,10,20. Both medians are 8. A has IQR=6 and range=16; B has IQR=4 and range=19. B’s middle half is more consistent although its full range is larger. Under a stated 1.5×IQR outlier rule, A’s fences are 5-9=-4 and 11+9=20; a new value 23 lies above the upper fence. This rule is a specified convention, not a compulsory rule for every box plot. When using cumulative frequency, read Q1 at N/4 and Q3 at 3N/4, then subtract; small reading errors affect the estimated IQR.
Quartiles, box plots and distribution comparisons
A box plot marks minimum, lower quartile Q1, median, upper quartile Q3 and maximum, or identifies separately shown outliers according to the stated convention
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find IQR for Q1=5,Q3=11.
11-5=6.
Test a tempting shortcut
- The median line need not be halfway along the box. Equal medians do not mean identical distributions. Whisker meanings depend on whether outliers are drawn separately; read the stated convention.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Equal medians prove that two groups have the same spread. This claim is false. Explain which definition or assumption it violates.
Find range for minimum 2,maximum 18.
18-2=16.
Equal medians prove that two groups have the same spread.
The median line need not be halfway along the box. Equal medians do not mean identical distributions. Whisker meanings depend on whether outliers are drawn separately; read the stated convention.
Interpret a new situation
- AQA S4 Higher adds box plots, quartiles and IQR to distribution comparisons. Name both centre and spread, with units and context, and avoid claims about all observations from just the box.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Under the stated 1.5 IQR rule, find the upper fence for those quartiles.
11+1.5×6=20.
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
- 8300 · Higher · 3.6. 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 upper quartile minus the lower quartile. Choose the relationship, show the method, check its assumptions and interpret the result.