Interpreting a distribution in context
Choose appropriate summaries and explain what the data can and cannot support.
Before you begin
- Mean, median and spread
- Dot plots and box plots
Report more than one summary
A useful report names the attribute, units, number of observations, centre, variability and shape. For a roughly balanced distribution without strong unusual values, the mean and MAD can be informative. When a long tail or unusual extreme affects the mean, the median and IQR may better describe the central group. Neither choice replaces a plot or an explanation of the question being answered.
Limit a conclusion to its evidence
An unusual value might reflect a genuine event or a recording error; investigate before excluding it. Observational data can show an association but do not alone establish its cause. A small or unrepresentative sample also limits generalisation. State the group actually observed, acknowledge uncertainty and suggest what further data would make a claim more convincing.
Worked example
Nine journey times are 4,5,5,6,6,6,7,7,26 minutes. Summarise a typical journey and the middle spread.
Show the worked solution
- There are nine observations, with total 72 minutes and mean 8 minutes.
- The median is 6 minutes. Excluding it, the lower-half median is 5 and upper-half median is 7, so IQR = 2 minutes.
- Most journeys take 4–7 minutes. The 26-minute trip raises the mean, so median 6 and IQR 2 describe the central group well.
Median 6 minutes and IQR 2 minutes; one unusually long journey at 26 minutes.
Try it yourself
What can a survey of ten volunteers most directly describe?
Common mistakes
- Do not claim a cause solely from a descriptive comparison.
- Keep unusual valid observations unless there is a justified reason to exclude them.
What can you explain now?
Two groups both have median 12 minutes, but IQRs of 2 and 8 minutes. What can you conclude, and what else would you want to inspect?
Compare with the explanation
Their centres match by median, but the second group's middle half is more spread out.
Inspect the sample sizes, plots, extreme values and collection method. Matching medians do not imply identical distributions or explain why the spreads differ.
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Bring the ideas together
Investigating data
Try the unit reviewOriginal StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗