Margin of error: the three rules of thumb
What this lesson covers
- The margin of error measures sampling variation only, not bias or other errors.
- The 1/√n rules of thumb are not in the booklet, so they must be learned.
One sample, one answer
- A different sample gives a slightly different result.
- The margin of error measures how much.
- It turns a sample result into a range.
What a margin of error covers
- Sampling variation, and nothing else.
- Not bias, non-response or wording.
- Keep the two ideas apart.
Rule 1: one proportion
- Learn it: not in the booklet
- Then form the interval.
- MOE ≈ 1/√none proportion
- n = 1000: MOE ≈ 1/√1000
- = 0.0316 ≈ 3.2%4 decimal places
- 42% ± 3.2%
- 42 − 3.2 = 38.8
- 42 + 3.2 = 45.2
- 38.8% to 45.2%
Form the interval, then say what it means
- A margin of error on its own is not an answer.
- Name the population and the variable.
- Between 38.8% and 45.2% of all voters.
Four times the sample, half the margin
- n = 100: 10%.
- n = 400: 5%.
- n = 1600: 2.5%.
Rule 2: a difference within one group
- Two results, one sample.
- One up, the other down.
- So the margin doubles.
- A: 42%, B: 35%same 1000 voters
- MOE ≈ 2 × 1/√nsame sample
- = 2 × 1/√1000 = 2 × 0.0316
- = 0.0632 ≈ 6.3%
- 42 − 35 = 7 pointsthe difference
- 7 − 6.3 = 0.7
- 7 + 6.3 = 13.3
- 0.7 to 13.3 points
Is zero in the interval?
- Difference: 7 points ± 6.3.
- 0.7 to 13.3 points.
- All above zero: A leads B.
Rule 3: a difference between two groups
- Two independent samples.
- Each has its own 1/√n.
- A: n = 400, 58% yes
- B: n = 625, 47% yes
- A: 1/√400 = 1/20 = 5%
- B: 1/√625 = 1/25 = 4%
1.5 × the average margin of error
- Average, then times 1.5.
- Difference ± that margin.
- A: 5%, B: 4%
- average = (5 + 4)/2 = 4.5%
- MOE ≈ 1.5 × 4.5 = 6.75%two samples
- 58 − 47 = 11 pointsthe difference
- 11 − 6.75 = 4.25
- 11 + 6.75 = 17.75
- 4.25 to 17.75 points
Pick the rule first
- The wrong rule: a named 2025 error.
- One proportion, same sample, or two samples?
- The answer picks the rule.
| Situation | MOE |
|---|---|
| One proportion | 1/√n |
| Two, same sample | 2 × 1/√n |
| Two samples | 1.5 × average MOE |
When the rules do not apply
- Proportions roughly 30% to 70%.
- A random sample.
- Never a cover for bias.
| Condition | Needs |
|---|---|
| Proportion | about 30% to 70% |
| Sample | random |
| Covers | sampling error only |
Where the marks are
- Achieved: a margin of error and an interval.
- Merit: interpret it in context.
- Excellence: right rule, its limits, a claim.
What to take away
- Pick the rule, form the interval, then interpret it in context.
| One proportion | 1/√n |
| Same sample | 2 × 1/√n |
| Two samples | 1.5 × average MOE |
| Then | form the interval |
| Only if | 30% to 70%, random |
| Not covered | bias, non-response |
Your turn: four questions
- Pause here and answer each one.
- Decide which rule fits first.
- Write each interval in full.
| Given | Find | |
|---|---|---|
| 1 | n = 900, 45% say yes | interval |
| 2 | same poll: 45% v 38% | A − B |
| 3 | 60% of 100, 45% of 400 | difference |
| 4 | 4% of 1000 | rule OK? |
Show answer
1. MOE 1/√900 = 3.3%; 41.7% to 48.3%. 2. Same sample: 2/√900 = 6.7%; difference 7: 0.3 to 13.7 points. 3. Two samples: 1.5 × (10% + 5%)/2 = 11.25%; difference 15: 3.75 to 26.25 points. 4. No: 4% is outside 30% to 70%, so the rule overstates the margin.
The answers: questions 1 and 2
- 1. MOE ≈ 1/√900 = 1/30one proportion
- = 0.0333 ≈ 3.3%
- 45 − 3.3 = 41.7, 45 + 3.3 = 48.3
- 41.7% to 48.3%
- 2. MOE ≈ 2/√900 = 2/30same sample
- = 0.0667 ≈ 6.7%
- 45 − 38 = 7 pointsthe difference
- 7 − 6.7 = 0.3, 7 + 6.7 = 13.7
- 0.3 to 13.7 points
The answers: questions 3 and 4
- 3. 1/√100 = 10%, 1/√400 = 5%two samples
- average = (10 + 5)/2 = 7.5%
- MOE ≈ 1.5 × 7.5 = 11.25%
- 60 − 45 = 15 pointsthe difference
- 15 − 11.25 = 3.75
- 15 + 11.25 = 26.25
- 3.75 to 26.25 points
- 4. p = 4%: outside 30% to 70%
- rule does not applyit overstates the margin
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