NCEA Level 1AS 91944Comparing groupsNCEA Level 1 Maths

Comparing groups: box plots, shift and overlap

What this lesson covers

  • Comparing two groups means comparing their whole distributions, not one number from each.
  • Shift, overlap, spread and shape together decide how strong a comparison is.
Key idea

A comparison question

  • One numerical variable, two groups.
  • Thirty students in each, chosen at random.
  • Does one group tend to sleep longer?
The investigation
QuestionDo sporty Year 11s tend to sleep longer?
Variablehours of sleep on a school night
Groupsplays team sport: yes or no
Sample30 randomly chosen from each group
Worked example

The five-number summary

  • Sort the 30 values from smallest to largest.
  • The median splits them into two halves of 15.
  • Each quartile is the median of one half.
Sport groupWhereValue (h)
Minimumsmallest value6.5
Lower quartile8th value of 308
Medianmean of 15th and 16th8.75
Upper quartile23rd value of 309.25
Maximumlargest value10.5
Worked example

Both groups side by side

  • The interquartile range is Q3 minus Q1.
  • It measures the spread of the middle 50%.
  • Unlike the range, one extreme value cannot stretch it.
SportNo sport
Median8.75 h7.75 h
Middle 50%8 to 9.257.25 to 8.5
IQR1.25 h1.25 h
Range4 h4.5 h
Worked example

Two box plots, one scale

  • The box runs from Q1 to Q3, with a line at the median.
  • The whiskers reach the minimum and maximum.
  • One shared scale, or no comparison is possible.
Two box plots, one scale
Key idea

Shift: where the groups sit

  • The sport median is 8.75 hours.
  • The no-sport median is 7.75 hours.
  • The whole sport box sits about an hour higher.
Shift: where the groups sit
Key idea

Five features to describe

  • Shift and overlap compare the groups.
  • Spread and shape describe each group.
  • Unusual values get named and explained.
FeatureIn this data
Shiftsport box sits about 1 hour higher
Overlapmiddle 50% overlap from 8 to 8.5
SpreadIQRs equal at 1.25 hours
Shapeboth roughly symmetric
Unusualno values far from the rest
Key idea

Overlap: how separate they are

  • The middle halves overlap from 8 to 8.5 hours.
  • So plenty of non-sporty students outsleep sporty ones.
  • That is why the claim says "tend to".
Worked example

Can you make a call?

  • Rule of thumb for samples of about 30:
  • is one median outside the other group's box?
  • 8.75 is above 8.5, so yes.
CheckResult
Sport median8.75 hours
No-sport box7.25 to 8.5 hours
Outside the box?yes, above it
Somake a call: "tend to"
Key idea

When you cannot make a call

  • If each median sits inside the other box...
  • ...the difference could be sampling variation.
  • Say so. That is a valid conclusion too.
Worked example

The most common mistake

  • Comparing one value from each group.
  • A maximum is one student, not a tendency.
  • Use the whole distribution.
Weak"The sport group has the highest value, 10.5"
Why weakone person, not the group
Strongmedians, boxes and overlap together
Worked example

Writing the comparison

  • A claim about the population, with "tend to".
  • Evidence: statistics with units.
  • An honest line about the overlap.
PartSentence
Claimsporty Year 11s tend to sleep longer
Evidencemedian 8.75 h vs 7.75 h
Evidencesport median above no-sport box
Honestboxes overlap: not every sporty student
Key idea

Bring in the context

  • Why might sport go with more sleep?
  • Tiredness from training, earlier routines.
  • A reason is a suggestion, not a proof.
Summary

What to take away

  • Compare whole distributions, and say "tend to".
Five numbersmin, Q1, median, Q3, max
IQRQ3 minus Q1, the middle 50%
Same scalealways, or shift cannot be seen
Describeshift, overlap, spread, shape, unusual
Callone median outside the other box
Words"tend to", in context, with numbers
Nevercompare just one value from each
Your turn

Your turn: four questions

  • Pause here. Sketch both box plots on one scale.
  • Then answer questions 1 to 4.
  • Use units and the context in every answer.
Resting pulse (beats per minute)
Apulse: min 52, Q1 60, median 66, Q3 72, max 88
Bpulse: min 58, Q1 66, median 74, Q3 80, max 96
1Find the IQR of group A.
2Can you make a call? Explain.
3Where do the middle 50% overlap?
4"B is higher because its max is 96." Why is this weak?
Show answer

1. 72 − 60 = 12 beats per minute. 2. Yes. B's median, 74, is above A's upper quartile, 72, so B's median is outside A's box, and group B tends to have higher resting pulse rates. 3. From 66 to 72 beats per minute. 4. It compares one value from each group, not the distributions. Compare medians, boxes and overlap instead.

Worked example

The answers

  • 72 − 60 = 12 beats per minute.
  • Yes: B's median 74 is above A's box, which ends at 72.
  • From 66 to 72 beats per minute.
  • One value each, not the distributions.

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