Comparing recorded samples
Read the two sides independently, using their shared key and units
- Centre Compare medians after ordering each sample separately
- Spread Compare ranges and state that they describe full spans. An extreme value can strongly affect a range
- Shape and overlap Inspect clusters, gaps, tails and overlap as well as the numerical summaries
- Limits Do not generalise to every individual or infer causation from the plot alone
Worked example
Compare the two recorded waiting-time samples shown below
AB
5 5 214 8
4 020 2 5
Key: A (left): 2 | 1 means 1.2 min
B (right): 1 | 4 means 1.4 min
B (right): 1 | 4 means 1.4 min
\[n_A=n_B=5\]
- \(\displaystyle m_A=1.5,\quad m_B=2.0\)B has the higher sample median, by 0.5 min.
- \(\displaystyle R_A=2.4-1.2=1.2\)A’s range is 1.2 min.
- \(\displaystyle R_B=2.5-1.4=1.1\)B has a slightly smaller range, 1.1 min.
- \(\displaystyle 1.4<2.4\)The samples overlap: a B time is below an A time. Describe these samples; the plot does not establish the cause of a difference
Common mistakes
Left leaf 5 beside stem 1
✕\(\displaystyle \text{value}=5.1\)✓\(\displaystyle \text{value}=1.5\)
With a tenths key, the central stem 1 and leaf 5 give 1.5, even when the leaf is shown on the leftCentre does not determine spread
✕\(\displaystyle m_B>m_A\Rightarrow R_B>R_A\)✓\(\displaystyle m_B>m_A,\quad R_B<R_A\)
The worked example has the larger median in the group with the smaller rangeRange remains a valid full span
✕\(\displaystyle R=\text{middle half}\)✓\(\displaystyle R=\max-\min\)
Range uses both extremes. IQR, taught later, measures the middle halfDifficulty levels
introRead counts and sample medians
standardCalculate and compare medians and ranges
challengeCompare unequal samples and qualify a conclusion
examCheck numerical claims and identify an unsupported inference
Questions about this topic
Why do left-hand leaves run in the opposite order?+
On both sides the smaller leaves are nearer the stem. Left leaves therefore appear descending across the page
Can the plot use decimal measurements?+
Yes. The explicit key and units determine the value of each stem-and-leaf pair
Should I combine both sides to find a median?+
No. Compare group medians by ordering each group separately
Does a larger median mean every value is larger?+
No. The two samples can overlap substantially
Does a larger sample have a larger median?+
Not necessarily. The median depends on ordered values, not just their number
Is range invalid when an extreme value is present?+
No. It correctly describes the full span, but may not describe where most observations lie
Does an unusual point have to be deleted?+
No. Investigate its accuracy and context. It may be a genuine observation
Can the comparison prove causation?+
No. Group labels and observed differences alone do not identify a cause
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 5: Statistics
In this chapter
Classify data types5A Data collection and types: Collect data and evaluate biasBuild a frequency and percentage frequency tableBuild a bar chart from a frequency tableGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramDescribe the shape of a distributionChoose an appropriate display for a variableRead values from a frequency table or bar chart5B Frequency tables and histograms: Critique statistical displays and claimsConstruct a dot plot from raw dataConstruct an ordered stem plot from raw data5C Dot plots and stem-and-leaf plots: Compare a numerical variable across groupsMean, mode, median and range of a data listFind the mean and the median, and choose between themFind measures of centre from a frequency displayFind the range, the quartiles and the IQRFind the fences and identify outliers5E Quartiles, interquartile range and outliers: Produce a five-number summary and construct a boxplot5F Box plots: Produce a five-number summary and construct a boxplotRead values and percentages from a boxplot5F Box plots: Compare a numerical variable across groupsPlot, read and classify a basic time-series graphConstruct and read a time series plotIdentify qualitative features in a time seriesIdentify the explanatory and response variablesConstruct a scatterplot from a data tableRead a scatterplotDescribe an association from a scatterplotAssociation is not causationFit a line of best fit by eye and make estimatesCompare categorical groups using two-way tables