NCEA Level 1AS 91944Comparing groupsNCEA Level 1 Maths

Comparing groups: 6 questions solved step by step

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

6 questions, one method

  • 1. Sort each group.
  • 2. Median, quartiles, IQR.
  • 3. Shift, overlap, spread.
  • 4. Make a call only if you can.
Your turn

Question 1 · Achieved

  • Minutes to get to school (10 students):
  • 12, 25, 8, 15, 30, 18, 10, 22, 14, 20
  • Find the median and the IQR.
Worked example

Q1: median and IQR

  • Sort before anything.
  • Quartiles: middle of halves.
  1. 8, 10, 12, 14, 15lower half, sorted
  2. 18, 20, 22, 25, 30upper half
  3. n = 10: median between 5th, 6th
  4. median = (15 + 18) ÷ 2 = 16.5
  5. Q1 = 12middle of lower half
  6. Q3 = 22middle of upper half
  7. IQR = 22 − 12 = 10 minutesQ3 − Q1
  8. median 16.5 min, IQR 10 min
Your turn

Question 2 · Achieved

  • Hours of part-time work (8 students):
  • 6, 15, 9, 12, 4, 10, 18, 8
  • Find the five-number summary and IQR.
Worked example

Q2: five-number summary

  • n = 8: medians of pairs.
  • A quartile can fall between.
  1. 4, 6, 8, 9, 10, 12, 15, 18sorted
  2. min = 4, max = 18
  3. median = (9 + 10) ÷ 2 = 9.5n = 8: 4th and 5th
  4. lower half: 4, 6, 8, 9
  5. Q1 = (6 + 8) ÷ 2 = 7
  6. upper half: 10, 12, 15, 18
  7. Q3 = (12 + 15) ÷ 2 = 13.5
  8. IQR = 13.5 − 7 = 6.5 hoursQ3 − Q1
  9. 4, 7, 9.5, 13.5, 18 (hours)
Your turn

Question 3 · Merit

  • Screen time on a school day (hours)
  • Year 9: 2, 3, 3, 4, 5, 6, 6, 9
  • Year 13: 3, 5, 6, 6, 7, 7, 8, 12
  • Medians, IQRs, then shift and spread.
Worked example

Q3: both groups

  • Lists already sorted.
  • Same steps for each group.
  1. Year 9: 2, 3, 3, 4, 5, 6, 6, 9
  2. median = (4 + 5) ÷ 2 = 4.5
  3. Q1 = 3, Q3 = 6(3 + 3) ÷ 2, (6 + 6) ÷ 2
  4. IQR = 6 − 3 = 3 hours
  5. Year 13: 3, 5, 6, 6, 7, 7, 8, 12
  6. median = (6 + 7) ÷ 2 = 6.5
  7. Q1 = 5.5, Q3 = 7.5(5 + 6) ÷ 2, (7 + 8) ÷ 2
  8. IQR = 7.5 − 5.5 = 2 hours
Worked example

Q3: shift and spread

  • Shift, then spread.
  • Too few to make a call.
  1. Year 9: median 4.5, IQR 3
  2. Year 13: median 6.5, IQR 2
  3. 6.5 − 4.5 = 2 hoursshift
  4. Year 13 sits about 2 h higher
  5. IQR 2 h < IQR 3 hspread
  6. Year 13 middle half less spread
  7. Year 13: typically 2 h a day more
  8. 8 per group: too few for a call
Your turn

Question 4 · Merit

  • Resting pulse, 30 Year 11s per group.
  • Breakfast: median 72, box 67 to 78
  • Skipped breakfast: median 75, box 70 to 81
  • Can you make a call?
Question 4 · Merit
Worked example

Q4: is there a call?

  • Each median vs other box.
  • Inside both: no call.
  1. Rule: median outside other box
  2. 72 vs no-breakfast box 70 to 81
  3. 72 is inside
  4. 75 vs breakfast box 67 to 78
  5. 75 is inside
  6. medians 3 bpm apart
  7. boxes overlap from 70 to 78
  8. No call can be made
Your turn

Question 5 · Merit, in context

  • Sleep on a school night, 30 per group.
  • Phone in the bedroom at night, or not.
  • Make a call if you can, then write it.
Sleep (h)PhoneNo phone
Minimum55.5
Q16.57
Median78
Q37.58.5
Maximum9.510
Worked example

Q5: can we make a call?

  • Median outside the box?
  • Then state the overlap.
  1. Medians: phone 7 h, no phone 8 h
  2. Phone box: 6.5 h to 7.5 h
  3. 8 > 7.5: outside the phone box
  4. So I can make a call
  5. 8 − 7 = 1 hourdifference of medians
  6. Boxes overlap 7 h to 7.5 h
Worked example

Q5: write the comparison

  • Claim, population, evidence.
  • Say "tend to".
  1. I think Year 11s at this school
  2. with no phone in the bedroom
  3. tend to sleep longer on a school
  4. night than those with one.
  5. Medians 8 h vs 7 h: 1 h more
  6. Boxes overlap: so "tend to"
  7. Observed only: not proof of cause
Your turn

Question 6 · Excellence

  • Kahikatea seedlings, 10 in each group.
  • Compare the two groups fully.
  • Make a call if you can.
GroupHeight after 6 weeks (cm)
Sun22, 15, 19, 26, 17, 21, 14, 24, 18, 22
Shade16, 11, 23, 13, 18, 10, 14, 19, 12, 15
Worked example

Q6: sun and shade

  • Sort, then split in halves.
  • Quartile: middle of a half.
  1. Sun: 14, 15, 17, 18, 19,
  2. 21, 22, 22, 24, 26
  3. median = (19 + 21) ÷ 2 = 20
  4. Q1 17, Q3 22, IQR 22 − 17 = 5
  5. Shade: 10, 11, 12, 13, 14,
  6. 15, 16, 18, 19, 23
  7. median = (14 + 15) ÷ 2 = 14.5
  8. Q1 12, Q3 18, IQR 18 − 12 = 6
Worked example

Q6: compare and call

  • Shift, overlap, spread.
  • Call, but say how sure.
  1. Sun: median 20, IQR 5
  2. Shade: median 14.5, IQR 6
  3. 20 − 14.5 = 5.5 cmshift
  4. boxes overlap 17 to 18 cmoverlap
  5. IQR 5 and 6: similarspread
  6. 20 > 18: outside shade boxso make a call
  7. I think kahikatea grown in sun
  8. tend to be taller than in shade
  9. Only 10 each: a tentative call
Summary

The method that solved all six

  • Sort, summarise, describe, then call.
StepHow
Sortsmallest to largest first
Medianmiddle, or mean of the two
Quartilesmedians of the two halves
IQRQ3 minus Q1
Describeshift, overlap, spread
Callmedian outside other box
Writetend to, units, population

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