← Evaluate statistically based reports
So if a report quotes 4% support and you apply the rule blindly, you will claim a far wider interval than is justified. Say the caveat when the percentage is extreme; it is exactly the kind of qualification that separates Merit from Excellence.
4 Margin of error
Rough margin of error (95%) \text{MoE}\approx\frac{1}{\sqrt{n}} n = sample size. For a proportion, gives roughly the ±% on the estimate. Confidence interval estimate ± MoE A 95% CI: we’re 95% confident the true value lies in this range. Worked example Interpreting a poll A poll of n = 1000 finds 52% support. Give the approximate margin of error and interpret. MoE ≈ 1/√1000 = 1/31.6 ≈ 0.032 = 3.2%. So the 95% CI is 52% ± 3.2% = 48.8% to 55.2%. Because the interval includes 50%, the poll does NOT provide strong evidence that support is actually above half. ✓ ≈ ±3.2%; the “majority support” claim is not justified: the interval spans 50%. ✓Bigger sample, smaller MoE
MoE ∝ 1/√n, so to halve the margin of error you need 4× the sample size. Diminishing returns. ⚠⭐ The rule of thumb only works between about 30% and 70%
MoE ≈ 1/√n is a rule of thumb for survey percentages roughly in the 30–70% range. Outside that band it overestimates the true margin of error — for a percentage near 5% or 95% the real margin is much smaller than 1/√n suggests.So if a report quotes 4% support and you apply the rule blindly, you will claim a far wider interval than is justified. Say the caveat when the percentage is extreme; it is exactly the kind of qualification that separates Merit from Excellence.
⭐ Confidence intervals — three possibilities, then a judgement
- Build the interval (estimate ± MoE), then look at what it contains. There are only three cases, and each licenses a different conclusion:
- 1. The interval lies entirely ABOVE the value of interest (e.g. all above 50%) → you have evidence the true value is greater. Make the claim.
- 2. The interval lies entirely BELOW it → evidence the true value is smaller.
- 3. The interval CONTAINS it → no evidence either way. The difference could easily be sampling variation. Do not claim a difference.
- For two groups, compare their intervals: if they overlap, you cannot claim a real difference between the groups.
- Always finish with the judgement in context — “so the claim that most voters support X is not justified, because the interval spans 50%.”