What Is Find the Margin of Error?
The margin of error is the half-width of a confidence interval. It says how far the sample estimate is likely to be from the true value because of random sampling alone.
Choose a proportion, a mean or an existing interval. Enter your values and confidence level to get the margin of error, standard error, critical value, the formula used and the sample size that would halve it, with a chart of how the margin shrinks as n grows.
Below are the formulas, a worked poll example and answers on reducing the margin of error.
How to Use the Find the Margin of Error Calculator
- Choose proportion, mean or interval.
- Enter the sample values.
- Set the confidence level.
- Read the margin of error and the chart.
Find the Margin of Error Formula
Proportion
Mean
From an interval
Find the Margin of Error Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| E | Margin of error | |
| z*, t* | Critical value | |
| p̂ | Sample proportion | |
| n | Sample size |
Find the Margin of Error Worked Example
A poll of 1000 people finds 52 % support. The standard error is √(0.52 × 0.48 ÷ 1000) = 0.0158, and z* for 95 % is 1.96, so the margin of error is 0.0310, about ±3.1 percentage points.
With the example values loaded in the calculator above, the steps are:
- Standard error = √(0.52 × 0.48 ÷ 1000) = 0.015798734.
- z* (95 %) = 1.95996.
- Margin of error = 1.95996 × 0.015798734 = 0.03096495.
Result: Margin of error = 0.03096495; As percentage points = 3.0965 points.
How to Read a Margin of Error
The margin of error is the half-width of a confidence interval. A poll of 1,000 people with 52% support has a standard error of 1.58 points, and at 95% confidence the margin is 1.96 × 1.58 ≈ 3.1 points. The reported range is 52% ± 3.1, or about 48.9% to 55.1%.
The margin covers sampling error only. It does not cover biased questions, people who did not respond, or a sample that does not represent the population. Because the margin shrinks with √n, four times the sample gives half the margin.
Where Find the Margin of Error Is Used
Polls and surveys
Find the margin of error for a poll from its sample size and result.
Means
Use the sample standard deviation and a critical value to get the margin for an average.
Existing intervals
Recover the margin from a published interval by taking half its width.
Planning
See how much a larger sample would shrink the margin.
Common Find the Margin of Error Mistakes to Avoid
- The margin of error is largest at a proportion of 50%, which is why polls quote a single worst-case figure.
- Higher confidence widens the margin. A 99% interval is wider than a 95% interval for the same data.
- The margin only describes random sampling error.
- Quote the confidence level with the margin. A margin without one is incomplete.
Find the Margin of Error Terms and Definitions
- Margin of error
- The maximum expected difference between the sample estimate and the true value at a stated confidence.
Find the Margin of Error FAQs
How do you find the margin of error?
Multiply the critical value by the standard error. For a proportion it is z × √(p̂(1 − p̂) ÷ n), and for a mean it is t × s ÷ √n.
What is the margin of error for a poll of 1000 people?
At 95% confidence and a result near 50%, it is about ±3.1 percentage points. A result further from 50% has a slightly smaller margin.
How do I get the margin of error from a confidence interval?
Subtract the lower limit from the upper limit and divide by 2. An interval from 48.9% to 55.1% has a margin of 3.1 points.
How can I reduce the margin of error?
Increase the sample size, accept a lower confidence level, or reduce the variability of the data. Halving the margin needs four times as many respondents.