Margin of Error Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Margin of Error.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The maximum expected difference between the sample statistic and the true population parameter; it is half the width of a confidence interval.

When a poll says 'the approval rating is 52\% with a margin of error of \pm 3\%,' it means the true value is likely between 49\% and 55\%. The margin of error is the '\pm' part—it tells you how much wiggle room to give the estimate. Larger samples and less variability shrink the margin of error.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Margin of error depends on three things: confidence level (z^*), variability (s), and sample size (n). You can shrink it by increasing n.

Common stuck point: The margin of error only accounts for random sampling error—it doesn't cover bias from bad survey design or non-response.

Worked Examples

Example 1

easy
A poll of n=400 voters finds \hat{p}=0.55 supporting a candidate. Calculate the margin of error at 95% confidence and interpret the result.

Solution

  1. 1
    SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.55 \times 0.45}{400}} = \sqrt{\frac{0.2475}{400}} = \sqrt{0.000619} \approx 0.0249
  2. 2
    E = z^* \times SE = 1.96 \times 0.0249 \approx 0.049 \approx \pm 5\%
  3. 3
    Confidence interval: 0.55 \pm 0.049 = (0.501, 0.599)
  4. 4
    Interpretation: we are 95% confident between 50.1% and 59.9% of voters support the candidate

Answer

E \approx \pm 5\%; 95% CI: (50.1\%, 59.9\%). Majority support is likely.
Margin of error quantifies polling uncertainty. 'Candidate leads 55% ± 5%' means the true support could be anywhere from 50% to 60%. A margin of error that doesn't include 50% still suggests the candidate leads, but we can't be certain.

Example 2

medium
A news report claims: 'The margin of error for this poll is ±3%.' Explain three things a careful reader should know about this margin of error.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A 95% CI for average commute time is (28, 36) minutes. What is the margin of error, and what does it tell us about the survey's precision?

Example 2

hard
Two polls: Poll A (n=100): margin of error ±10%. Poll B (n=1600): margin of error ±2.5%. By what factor was n increased, and by what factor did E decrease? Verify using E = z^* \sqrt{\frac{0.5 \times 0.5}{n}}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

confidence intervalstandard deviation