Margin of Error Examples in Statistics

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 Statistics.

Concept Recap

The margin of error is the maximum expected difference between a sample statistic and the true population parameter, typically expressed as a plus-or-minus value. It equals half the width of a confidence interval and decreases as sample size increases.

When a poll says '52% \pm 3%,' that 3% is the margin of error. It means the true value is probably within 3 percentage points of 52%, so between 49% and 55%.

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: The margin of error is half the width of a confidence interval. It quantifies the maximum expected sampling error for the stated confidence level.

Common stuck point: Students think a larger margin of error means the survey was poorly done. It simply reflects a smaller sample size or higher desired confidence level.

Sense of Study hint: First, find the standard error of your statistic. Then multiply by the appropriate critical value (e.g., 1.96 for 95% confidence): margin of error = critical value * SE. Finally, report the result as 'statistic +/- margin of error' to give a range of plausible values.

Worked Examples

Example 1

hard
A poll of 400 voters found 55% support a policy. Calculate the margin of error for a 95% confidence interval.

Solution

  1. 1
    Step 1: For proportions, \text{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
    Step 2: Margin of error = z^* \times \text{SE} = 1.96 \times 0.0249 \approx 0.049.
  3. 3
    Step 3: The margin of error is approximately ±4.9 percentage points.

Answer

Margin of error ≈ ±4.9 percentage points.
The margin of error quantifies the precision of an estimate. A smaller margin means a more precise estimate. Increasing sample size reduces the margin of error.

Example 2

hard
How does quadrupling the sample size affect the margin of error?

Practice Problems

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

Example 1

hard
A 95% CI for a mean is 50 \pm 3. What is the margin of error, and what would it be if the sample size were quadrupled?

Example 2

hard
If the confidence level is increased from 90% to 99% while the sample size and variability stay the same, what happens to the margin of error?
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Background Knowledge

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

confidence intervalstandard error