Understand · explore · practise
Compare location and spread with units and context, choose robust summaries, account for sample sizes and avoid unsupported claims about individuals, thresholds or causation.
Before you startMean, median, standard deviation, IQR, box plots and grouped diagrams.
01 / Make two supported comparisons
State the statistic, compare its values, then interpret it in the variable’s units.
A summary describes the dataset; it does not guarantee every individual outcome.
For journey times, a lower median means a shorter typical journey in the median sense. A smaller IQR means the middle half is less spread out. Neither statement proves the service is always quicker or explains why the difference occurred.
A: median journey time 24 minutes, IQR 8 minutes. B: median 30 minutes, IQR 14 minutes.
A has a shorter median journey and a less spread-out middle half.
Supported. The medians and IQRs support these sample comparisons, not an ordering of every individual journey.
02 / Interpret the centre
A’s median is 6 minutes lower
Name the statistic and size of the difference.
Its typical journey time is shorter, using the median
Use the meaning of the variable.
We cannot say every A journey is shorter
The distributions may overlap.
Class A has mean score 62 and class B has mean 68 on the same test. Give a supported comparison.
Compare means in marks, not every student.
B’s mean score is 6 marks higher. This does not imply every B student outscored every A student.
Two machines have median absolute errors 0.2 mm and 0.5 mm. Is the larger median preferable for accuracy?
Absolute error measures distance from the target.
No. The 0.2 mm median indicates a smaller typical absolute error. Context determines what a higher value means.
03 / Interpret variability precisely
Route A has IQR 8 minutes; B has IQR 14 minutes. State a comparison.
IQR describes the middle half.
A’s middle half of journey times is less spread out by the IQR measure; its IQR is 6 minutes smaller. This does not describe the full range.
Two delivery samples have SDs 3 hours and 5 hours. Compare variability.
Use SD in the original units.
The sample with SD 3 hours has smaller variability by the SD measure. Variance would have units hours squared.
04 / Choose useful summaries
Median and IQR focus on order and the middle half.
Mean and SD use all numerical observations and can be strongly affected by extreme values.
Most incomes are moderate but a few are extremely high. Why might median and IQR help describe a typical income and central spread?
They are less sensitive to the sizes of extreme observations.
They describe the centre and middle half without the high values pulling the mean upward as strongly. The extreme incomes still matter and should not be deleted merely to simplify the summary.
Must the mean always be reported only with SD, never with a range or quartiles?
Useful standard pairings are not universal bans.
No. Mean/SD and median/IQR are common informative pairings, but additional summaries can answer different questions. Explain the purpose and limitations of the chosen statistics.
05 / Normalise unequal samples
A sample of 80 has 20 journeys over 40 minutes. Another of 200 has 40. Which sample has the higher proportion?
20/80 compared with 40/200.
First sample: 25%; second: 20%. The second has more long journeys in absolute count but a lower sample proportion.
Why can raw-count histogram areas mislead when comparing samples of very different sizes?
A larger sample can make all counts larger.
Relative-frequency areas, using each sample’s own total, permit proportional comparisons. Use consistent classes and units, or explicitly account for differences.
06 / Separate summaries from shape
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Does mean 20 and SD 4 determine the percentage above 24?
No distributional shape has been stated.
No. Many distributions have that mean and SD but different threshold counts. Do not silently assume a normal distribution.
Two samples share median 10 and IQR 4. Must their ranges, means and sample sizes be equal?
The two summaries contain limited information.
No. They can differ in tail values, overall range, mean, size and detailed shape.
07 / Keep conclusions within the study
One self-selected group has a higher mean score than another. Does this alone establish that a teaching method caused the difference?
Consider prior attainment, selection and other factors.
No. The comparison describes these samples. Without an appropriate design, confounding and selection offer alternative explanations.
A convenience sample of commuters has a low median journey time. Can it establish the median for all commuters?
Check representativeness before generalising.
No. Sampling method, coverage, timing and sample size affect what can be inferred. A precise sample calculation does not remove sampling bias.
08 / Write a defensible conclusion
Compare an appropriate centre and spread, attach units and interpret their meaning. Use percentages for unequal sample sizes. State what is observed and what requires assumptions or stronger evidence.
Section 1 of 8 · Make two supported comparisons