Your survey says 40% of customers would buy the new product, and someone in the meeting asks how sure you are. The margin of error is the honest answer to that question. It tells you how far a survey result could be from the true value just because you asked a sample rather than everyone. It is one part of the bigger picture of survey data quality, and it is the part you can put an exact number on. Below are the margin formula, a worked example, a table by sample size and the rules for crosstabs, comparisons and weighted data.
Key takeaways
- The margin of error is the range around a survey estimate within which the true value probably falls, usually stated at 95% confidence.
- For a proportion, the formula is z × √(p(1 − p) ÷ n), with z = 1.96 at 95% confidence.
- With 1,000 respondents the maximum margin is about ±3.1 points; a subgroup of 150 has about ±8.0.
- The margin for a difference between two results is not the sum of their margins, so overlapping ranges can still be significantly different.
- The margin covers sampling error only. It says nothing about bias, and weighting makes it wider.
What is the margin of error in a survey?
A survey measures a sample and uses it to estimate a value for the whole population. Draw a different random sample and the estimate moves a little, purely by chance. That chance variation is sampling error, and the margin of error puts a size on it.
The confidence level says how often the range would capture the true value if you repeated the survey many times. At 95%, about 19 out of 20 such ranges would contain it. Higher confidence means a wider margin.
What the margin of error does not cover matters as much. It assumes a random sample and questions that measured what you intended. It says nothing about survey bias from leading questions, or about nonresponse bias when the people who answered differ from those who did not. A survey of 10,000 can have a tiny margin and still be wrong.
What is the margin of error formula?
For a percentage (a proportion), the margin of error at a given confidence level is:
Margin of error = z × √(p × (1 − p) ÷ n)
- p is the survey result as a decimal, for example 0.40 for 40%.
- 1 − p is the share who did not give that answer, here 0.60.
- n is the number of respondents who answered the question.
- z is the z-score for the confidence level: 1.645 at 90%, 1.96 at 95% and 2.576 at 99%.
The square root term is the standard error, the typical chance variation in a sample result; z turns it into a range. Because p × (1 − p) is largest when p is 0.50, a 50% result has the widest margin. That is why surveys often quote the margin “at 50%” as the worst case.
How do you calculate margin of error?
Here is a worked example with illustrative numbers. A survey of 1,000 respondents finds that 40% say yes.
- Multiply p by 1 − p: 0.40 × 0.60 = 0.24.
- Divide by n: 0.24 ÷ 1,000 = 0.00024.
- Take the square root: √0.00024 = 0.0155. This is the standard error.
- Multiply by z: 1.96 × 0.0155 = 0.0304.
- Convert to points: 0.0304 × 100 = ±3.0 percentage points.
The true value is likely between 37.0% and 43.0%. At 50%, the same sample gives 0.50 × 0.50 = 0.25, then √(0.25 ÷ 1,000) = 0.0158, and 1.96 × 0.0158 = ±3.1 points. That is the maximum margin for 1,000 respondents.
How does sample size change the margin of error?
The table shows margins at 95% confidence. The figures follow from the formula; they are not from any particular survey.
| Sample size | Margin at 50% | Margin at 20% or 80% |
|---|---|---|
| 100 | ±9.8 | ±7.8 |
| 200 | ±6.9 | ±5.5 |
| 400 | ±4.9 | ±3.9 |
| 600 | ±4.0 | ±3.2 |
| 1,000 | ±3.1 | ±2.5 |
| 1,500 | ±2.5 | ±2.0 |
| 2,000 | ±2.2 | ±1.8 |
The returns diminish. Going from 100 to 400 respondents cuts the margin in half. Halving it again takes 1,600.
Why do subgroups have a wider margin of error?
The n in the formula is the base for the number you are reading, not the total sample. In a crosstab of purchase intent by age, the total of 1,000 has a maximum margin of ±3.1 points, but an age group with a base of 150 has ±8.0 points. A 46% in that column could plausibly be anywhere from 38% to 54%.
So every table should show its bases. When you build crosstabs and banner tables with weighting and significance testing, read the base before the percentage, and treat small columns as directional.
How do you compare two survey results?
The margin for the difference between two results is smaller than their two margins added together, because both are unlikely to miss by their full margin in opposite directions at once.
Take an illustrative brand tracker with 600 respondents per wave. Consideration rises from 38% to 44%.
| Result | Margin (95%) | Range | |
|---|---|---|---|
| Wave 1 | 38% | ±3.9 | 34.1% to 41.9% |
| Wave 2 | 44% | ±4.0 | 40.0% to 48.0% |
The ranges overlap, so it is tempting to call the change noise. Yet the margin for the difference is 1.96 × √(0.38 × 0.62 ÷ 600 + 0.44 × 0.56 ÷ 600) = ±5.6 points. The change of 6 points is larger than that, so it is significant at 95%. Our guide to statistical significance explains the test in more detail.
What else changes the margin of error?
Finite population correction
The standard formula assumes the population is much larger than the sample. When you survey a large share of a small population, such as 500 of a company’s 2,000 business customers, multiply the margin by √((N − n) ÷ (N − 1)). Here that is √(1,500 ÷ 1,999) = 0.866, which takes the maximum margin from ±4.4 to ±3.8 points.
Weighting and design effect
Weighting corrects the sample’s profile but reduces precision. The effective sample size measures the loss: the sum of the weights squared, divided by the sum of the squared weights.
A simple illustration: 1,000 respondents, half with a weight of 0.5 and half with 1.5. The sum of the weights is 1,000, and the sum of the squared weights is 500 × 0.25 + 500 × 2.25 = 1,250. The effective sample size is 1,000,000 ÷ 1,250 = 800, a design effect of 1.25. The maximum margin grows from ±3.1 to ±3.5 points, the margin you would get from 800 unweighted respondents.
How do margin of error, sampling error and confidence interval differ?
| Term | What it is | Example |
|---|---|---|
| Sampling error | The chance gap between a sample result and the true population value | The true value is 41%; your survey found 40% |
| Margin of error | The half-width of the range that sampling error probably stays within | ±3.0 points at 95% confidence |
| Confidence interval | The full range: estimate minus and plus the margin | 37.0% to 43.0% |
How should you report the margin of error?
State the base, the confidence level and the margin together, such as “n = 1,000; margin of error ±3.1 points at 95% confidence”. Give the margin for the largest subgroup you discuss, not only the total. If the data are weighted, base the margin on the effective sample size. Add a line that the margin covers sampling error only.
What are the common mistakes?
- Applying the total margin to subgroups. A ±3.1 for the total says nothing about a column with a base of 150.
- Treating the margin as covering bias. A small margin does not fix a skewed sample or a leading question.
- Ignoring weighting. Quoting the unweighted margin on weighted data makes results look more precise than they are.
- Adding two margins to judge a difference. It overstates the uncertainty and hides real changes.
- Reading the margin as a hard boundary. Values near the edge of the range are less likely than values near the estimate, but not impossible.
How do you put this into practice?
Before you report a number, check its base, its weighting and what you are comparing it with, and test differences rather than eyeballing ranges. On large trackers, keep bases, weights and definitions in one governed platform for wide, complex studies and global brand trackers, so the margin you quote this wave means the same as last wave.