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Compare a numerical variable across groups

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Upper plot A: 2,5,9,14,18. Lower plot B: 4,8,12,16,30 minutes.AB010203040Time (minutes)

A has median 9 and IQR 9 minutes. B has median 12 and IQR 8 minutes. Which comparison is supported?

Year 11, VCE Units 1 & 2, Chapter 2, 4 difficulty levels, Calculator allowedBuild a worksheet

Comparing distributions

A useful comparison identifies both the measure and the context

\[\mathrm{range}=\max-\min\]
  1. Centre Compare the same measure in both groups. A median is a middle value, not an IQR
  2. Spread Range and IQR describe different spans. Their rankings can disagree
  3. Outliers Include separate plotted observations in the full range. State the rule if formally identifying outliers
  4. Inference Summaries alone do not establish cause, representativeness or a population difference

Worked example

Waiting-time summaries in minutes: A = (2,5,9,14,18), B = (4,8,12,16,30)

\[\begin{gathered}A:(2,5,9,14,18)\\B:(4,8,12,16,30)\end{gathered}\]
  1. \(\displaystyle 12-9=3\)B has the higher median by 3 minutes
  2. \(\displaystyle \mathrm{IQR}_A=14-5=9\)A’s middle-half spread is 9 minutes
  3. \(\displaystyle \mathrm{IQR}_B=16-8=8\)B’s middle-half spread is 8 minutes, slightly less than A’s.
  4. \(\displaystyle \mathrm{range}_A=18-2=16\)A’s full span is 16 minutes
  5. \(\displaystyle \mathrm{range}_B=30-4=26\)B’s full span is larger. Explain both aspects rather than declaring range an invalid measure

Common mistakes

Comparing different measures
✕\(\displaystyle \mathrm{median}_B>\mathrm{IQR}_A\)✓\(\displaystyle \mathrm{median}_B>\mathrm{median}_A\)
Compare centre with centre and spread with the same spread measure
Treating a larger range as a larger IQR
✕\(\displaystyle 26>16\Rightarrow\mathrm{IQR}_B>\mathrm{IQR}_A\)✓\(\displaystyle \mathrm{IQR}_A=9>8=\mathrm{IQR}_B\)
The range and middle-half spread can give different rankings

Difficulty levels

introCompare two groups’ medians and IQRs
standardInclude plotted outliers and evaluate a report
challengeCompare three groups or use back-to-back stems
examIdentify an unsupported comparison statement

Questions about this topic

Is range a wrong measure of spread?+

No. It measures the full span. Use IQR when discussing the middle-half spread and state which measure you use

Does a wider box mean a bigger sample?+

No. A wider box means a larger IQR on a common scale

Can I compare plots drawn on different scales?+

Compare numerical values carefully; visual widths alone are not comparable unless the scales match

Are more outliers proof of greater IQR?+

No. Read or calculate each IQR independently

Can these plots explain the cause of a difference?+

No. You need information about the design, sampling and relevant variables

Can two matching boxplots hide different modes?+

Yes. A five-number summary does not preserve individual frequencies. Inspect raw dots or an appropriate histogram for shape and modality

What does cumulative frequency mean here?+

At each marked class boundary, it counts all observations strictly below that boundary. Connecting segments guide the eye; unmarked thresholds require raw data or an explicitly qualified estimate

How should I compare unequal sample sizes?+

Report both sizes and compare appropriate summaries. Use common histogram classes and consider percentage frequencies rather than comparing raw column heights alone

How do I compare three groups?+

Calculate and compare each group’s median and spread separately. Name the groups, values and units. A group with the highest median need not have the largest IQR; do not combine the observations into a single summary

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 2: Data distributions

In this chapter

Classify a variable as nominal, ordinal, discrete or continuousDistinguish interval and ratio measurement scalesBuild a frequency and percentage frequency tableBuild a bar chart from a frequency table2A Data types and categorical displays: Read values from a frequency table or bar chart2B Reading categorical displays: Read values from a frequency table or bar chartReport on the distribution of a categorical variableCompare categorical distributions across two groupsGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramDescribe the shape of a distributionConstruct a dot plot from raw dataConstruct an ordered stem plot from raw dataChoose an appropriate display for a variableFind the mean and the median, and choose between themFind measures of centre from a frequency displayWork backwards from a mean or medianFind the range, the quartiles and the IQRCalculate the standard deviation from the formulaApply the 68–95–99.7% ruleProduce a five-number summary and construct a boxplotFind the fences and identify outliersRead values and percentages from a boxplotCompare a numerical variable across groups