Back-to-back stem plots
Use the group labels and shared key before calculating or comparing
- Left side Leaves descend as displayed, so read from right to left within each row for increasing values
- Right side Leaves increase from left to right
- Key Read the stem first on either side. A left leaf does not become the leading digit
- Comparison Compare centres, ranges, shapes and unusual observations in the data’s context
- Limits A difference between observed groups does not establish a causal explanation
Worked example
Group A has 21, 25, 27, 32, 36, 38, 43. Group B has 23, 24, 30, 32, 35, 39, 41, 46. Compare their centres and ranges
Group AGroup B
7 5 123 4
8 6 230 2 5 9
341 6
Key: Group A (left): 1 | 2 means 21 marks
Group B (right): 2 | 3 means 23 marks
Group B (right): 2 | 3 means 23 marks
\[n_A=7,\quad n_B=8\]
- \(\displaystyle \operatorname{median}_A=32\)Use the fourth of seven values
- \(\displaystyle (32+35)/2=33.5\)Average the fourth and fifth of eight values
- \(\displaystyle R_A=43-21=22\)Use largest minus smallest
- \(\displaystyle R_B=46-23=23\)Group B has median 1.5 marks higher and range 1 mark larger. The groups still overlap
Common mistakes
Reversing place value
✕\(\displaystyle 1\mid2\text{ means }12\)✓\(\displaystyle 1\mid2\text{ means }21\)
Here 1 is the left leaf and 2 is the shared stem, under a tens keyTaking only one middle value
✕\(\displaystyle \operatorname{median}_B=32\)✓\(\displaystyle (32+35)/2=33.5\)
There are eight values, requiring both central valuesComparing unlike summaries
✕\(\displaystyle \operatorname{median}_A>R_B\)✓\(\displaystyle 32<33.5\)
For a centre comparison, compare the two medians, not a median with a rangeDifficulty levels
introRead counts, minimum, maximum, median or range on either side
standardComplete missing leaves and calculate a requested statistic
challengeComplete a plot and compare centre, range and shape
examRecognise a correct completed plot or an evidence-based comparison
Questions about this topic
Why does the left row look reversed?+
The smallest leaf is nearest the shared stem on each side
Can the groups have different sizes?+
Yes. Count each side separately and account for the sizes when comparing frequencies
Can the key use decimals?+
Yes. The same decimal place value applies to both groups
What makes a useful comparison?+
Say which centre/range is higher or larger and support that direction with the actual values and units
Does a higher median mean every value is higher?+
No. The groups can overlap substantially
Does the display explain why the groups differ?+
No. A descriptive comparison alone does not establish a cause
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics
In this chapter
Finite events, probability assignments and complementsDescribe and order probability values9B Venn diagrams and tables: Venn diagrams, two-way tables and set notation for two eventsComplete and use a two-way probability tableCompound events, the addition rule and mutual exclusivity9C Set notation: Event notation and shaded regionsFinite sets and set operationsBuild two-step sample spaces with and without replacementUse two-step probability treesExperimental probability from observed frequenciesDesign and interpret a probability simulationFind the mean and the median, and choose between themWork backwards from a mean or medianMode, bimodal and no-mode data, and the effect of an outlierConstruct an ordered stem plot from raw dataConstruct and read back-to-back stem-and-leaf plotsDescribe the shape of a distributionGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramFind the range, the quartiles and the IQRProduce a five-number summary and construct a boxplotRead values and percentages from a boxplotCompare a numerical variable across groupsSurveys and statistical investigations