Compare two raw datasets with back-to-back stems
Use shared stems and two directional keys to compare raw distributions without reversing their place value or erasing repeats.
Section 5.1 · Representation of data
Exam scope and assessment notes
Probability & Statistics 1 (Paper 5, AS & A Level) · Mathematics 9709, 2026–2027, section 5.1. Label variables, units, class boundaries and diagram keys. In these raw-data quartile exercises, use the median of each ordered half, excluding the overall median when the number of observations is odd; follow any convention supplied in an examination question. Cumulative-frequency quartiles use ranks n/4 and 3n/4, with a median rank n/2. Grouped-data interpolation and midpoint moments are estimates, not recovered raw measurements. Histogram area represents frequency, with height proportional to frequency density. Use descriptive variance with divisor n, not the sample-estimator divisor n − 1. Coded and combined totals must refer to the same units and compatible coding before addition. Retain unrounded working and distinguish variance units from standard-deviation units. Written practice is self-checked; human subject-teacher review remains pending.
Before you begin
- Build a single ordered stem-and-leaf diagram.
- Find a median from an ordered list.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For tens stems on the left, what is leaf 4 beside stem 2?
What you will learn
- Construct ordered back-to-back stem-and-leaf diagrams.
- Recover values and make numerical comparisons on a shared scale.
Share the stem and order leaves away from it
Group A times are 12, 14, 14, 18, 21, 23, 26, 29; Group B times are 11, 16, 18, 19, 22, 24, 24, 28 minutes. Put A on the left and B on the right of shared stems 1 and 2. The left rows, read from left to right, are 8 4 4 2 and 9 6 3 1; the right rows are 1 6 8 9 and 2 4 4 8. On either side, the smallest leaf is nearest the stem.
Label both groups and use keys 2 | 1 on the left means 12 minutes, while 1 | 1 on the right means 11 minutes. The first part of the left key is the leaf and the second the stem. Both datasets must use the same units and compatible precision. Do not read the left row as a separate decimal number or discard its repeated leaves.
Compare values, not the visual row length alone
Recovering A in order gives median (18 + 21)/2 = 19.5 minutes. B has median (19 + 22)/2 = 20.5 minutes. A has a lower median by 1 minute in these observations. Their ranges are 29 − 12 = 17 and 28 − 11 = 17 minutes, so the overall spreads by range are equal.
Equal ranges do not force equal IQRs, means or distribution shapes. Each group here has eight observations; a diagram with different counts needs separate medians from its own count. The comparison describes these measurements, without establishing why the times differ. A shared stem helps comparison but does not replace numerical evidence.
Pause and explain
Which left row is correctly ordered for 12, 14, 14, 18 beside stem 1?
Worked example
Construct a back-to-back diagram for the two eight-time datasets above and compare their medians.
Show the worked solution
- Use A-left rows 8 4 4 2 | 1 and 9 6 3 1 | 2; B-right rows 1 | 1 6 8 9 and 2 | 2 4 4 8.
- Provide a direction-aware key on each side and retain all eight leaves per group.
- A median is 19.5 minutes and B median 20.5 minutes; A is lower by 1 minute.
Answer A median 19.5 minutes; B median 20.5 minutes; A is lower by 1 minute.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Write Group A = 12, 14, 14, 18 and B = 11, 16, 18, 19 on shared stem 1, A-left and B-right, with keys.
Give me a hint
Left leaves decrease from left to right; right leaves increase.
Compare my reasoning
- The shared row is A: 8 4 4 2 | 1 | 1 6 8 9 :B.
- Left key 2 | 1 = 12 minutes; right key 1 | 1 = 11 minutes.
- Each side has four leaves, including A’s two values at 14.
8 4 4 2 | 1 | 1 6 8 9; four leaves per group, with directional time keys.
Look for these in your work
- Order both sides consistently.
- Retain the duplicate and explain place value on both sides.
A times are 12, 14, 14, 18, 21, 23, 26, 29; B times are 11, 16, 18, 19, 22, 24, 24, 28 minutes. Compare medians and ranges.
Give me a hint
Both groups have eight values, so each median is the average of its fourth and fifth values.
Compare my reasoning
- A median = (18 + 21)/2 = 19.5; B median = (19 + 22)/2 = 20.5 minutes.
- A range = 29 − 12 = 17; B range = 28 − 11 = 17 minutes.
- A has a lower median by 1 minute, while the ranges are equal.
Medians A 19.5, B 20.5 minutes; both ranges 17 minutes.
Look for these in your work
- Use each group’s ordered middle pair.
- Compare centre and spread separately.
A diagram uses stem 2 with leaves 0 5 9 on the right under key 2 | 0 = 2.0 seconds. Another group records 20, 25, 29 seconds. Can those entries share the same key as if they were comparable? Explain.
Give me a hint
A shared key fixes measurement scale, not just matching digits.
Compare my reasoning
- The first leaves encode 2.0, 2.5 and 2.9 seconds.
- The second group’s values are ten times as large; the same leaves under that key would misstate them.
- Use compatible place value and units for both groups, or a different representation with a shared numerical scale.
No: 2.0, 2.5, 2.9 seconds differ from 20, 25, 29 seconds; align the scale and key first.
Look for these in your work
- Decode the supplied key before comparing.
- Do not infer equivalence from matching leaf digits.
Common mistakes
- On the left, the smallest leaf is nearest the stem.
- Give a key for each direction.
- Equal ranges do not imply identical distributions.
What can you explain now?
What is the left entry leaf 8 beside stem 1 under a tens-and-units key?
Compare with the explanation
18, not 81.
The shared stem still represents tens regardless of which side holds the leaf.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Recover values with the directional keys, count each group separately, then compare numerical centre and spread.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.