NCEA Level 3Relative riskAS 91585

Relative risk: times as likely, and when you need a ratio

What this lesson covers

  • Relative risk compares the risk in one group with the risk in another.
  • 'Twice as likely' and 'twice more likely' do not mean the same thing.
Key idea

Risk is a conditional probability

  • Risk of late work, given a job.
  • Relative risk: one risk ÷ another.
  • Say it right; know when you need it.
Worked example

Part-time jobs and late work

  • 400 invented Year 13 students.
  • Job: 60 of 200 late.
  • No job: 40 of 200 late.
LateOn timeTotal
Job60140200
No job40160200
Total100300400
Worked example

The rest of the table

  • Each gap: total − known.
  • Check the grand total.
  1. job, on time: 200 − 60 = 140
  2. no job, on time: 200 − 40 = 160
  3. late: 60 + 40 = 100
  4. on time: 140 + 160 = 300
  5. 100 + 300 = 400
Worked example

Two risks, then divide

  • Each risk: its own group.
  • Then divide, job ÷ no job.
  1. P(late | job) = 60 ÷ 200job group only
  2. = 0.3
  3. P(late | no job) = 40 ÷ 200no-job group only
  4. = 0.2
  5. relative risk = 0.3 ÷ 0.2job v no job
  6. = 1.5
Key idea

Times as likely, not times more likely

  • Say "times as likely".
  • % more = (RR − 1) × 100
  • Avoid "times more likely".
  1. RR = 1.5: 1.5 times as likely
  2. (RR − 1) × 100percent more
  3. (1.5 − 1) × 100 = 50
  4. 50% more likely
  5. RR = 2: (2 − 1) × 100 = 100%
  6. RR = 3: (3 − 1) × 100 = 200%
  7. "1.5 times more likely"ambiguous
Worked example

Fifty percent more, drawn

  • No job: 0.2. Job: 0.3.
  • The extra 0.1 is half of 0.2.
  • The whole risk, plus half again.
Fifty percent more, drawn
Key idea

Percentage points or percent?

  • Subtract: % points.
  • Divide: percent more.
  1. 30% − 20% = 10 percentage points
  2. 0.3 ÷ 0.2 = 1.5relative risk
  3. (1.5 − 1) × 100 = 50% morepercent
  4. "10% more likely"mixes the two
Worked example

The other way round

  • No job ÷ job.
  • Below 1: less likely.
  1. P(late | no job) ÷ P(late | job)
  2. = 0.2 ÷ 0.3
  3. = 0.66674 decimal places
  4. about two thirds as likely
  5. (0.6667 − 1) × 100 = −33.33percent change
  6. about 33% less likely
Key idea

Does the claim need a ratio?

  • A ratio by habit: a named 2025 error.
  • Read the claim first.
  • Only a multiplier needs relative risk.
The claim saysYou need
"more likely"compare two risks
"most", "over half"one probability
"twice as likely"relative risk
"50% more likely"relative risk
Worked example

Claim 1: "more likely to be late"

  • Compare the two risks.
  • No ratio needed.
  1. P(late | job) = 0.3
  2. P(late | no job) = 0.2
  3. 0.3 > 0.2supported, this sample
Worked example

Claim 2: "mostly from job holders"

  • Given late: a new group.
  • Over half means > 0.5.
  1. late: 60 + 40 = 100
  2. P(job | late) = 60 ÷ 100late students only
  3. = 0.6
  4. 0.6 > 0.5over half: supported
Worked example

Claim 3: "twice as likely"

  • Multiplier: use the ratio.
  • No cause from a survey.
  1. relative risk = 0.3 ÷ 0.2 = 1.5
  2. 1.5 < 2claim: twice as likely
  3. claim not supported
  4. 1.5 times as likely
Key idea

Where the marks are

  • Achieved: risks from a table.
  • Merit: a claim justified with numbers.
  • Excellence: relative risk explained, claim judged.
Summary

What to take away

  • Read the claim, then choose the comparison it needs.
RiskP(outcome | group)
Relative riskrisk ÷ risk, named
Saytimes as likely
Percent more(RR − 1) × 100
"More likely"compare two risks
Multiplierneeds the ratio
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Wet days: 54 of 180 trips late.
  • Dry days: 36 of 360 trips late.
GivenFind
1wet 54/180, dry 36/360both risks
2the same tripswet v dry RR
3"200% more likely"true?
4"most late trips wet"check it
Show answer

1. P(late | wet) = 54 ÷ 180 = 0.3; P(late | dry) = 36 ÷ 360 = 0.1. 2. RR = 0.3 ÷ 0.1 = 3: wet-day trips were 3 times as likely to run late. 3. True: (3 − 1) × 100 = 200% more likely. 4. P(wet | late) = 54 ÷ 90 = 0.6 > 0.5, so supported; no ratio needed.

Worked example

The answers: questions 1 and 2

  1. 1. P(late | wet) = 54 ÷ 180wet days only
  2. = 0.3
  3. P(late | dry) = 36 ÷ 360dry days only
  4. = 0.1
  5. 2. relative risk = 0.3 ÷ 0.1wet v dry
  6. = 3
  7. wet: 3 times as likely to run late
Worked example

The answers: questions 3 and 4

  1. 3. (3 − 1) × 100 = 200percent more
  2. 200% more likely: true
  3. 4. late: 54 + 36 = 90
  4. P(wet | late) = 54 ÷ 90late trips only
  5. = 0.6
  6. 0.6 > 0.5: supportedno ratio needed

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