Margin of Error Calculator
Enter a survey's sample size, a confidence level and the observed percentage, and get the margin of error — plus the range the true population value probably falls in. Add the population size to apply the finite population correction. The formula is worked through with your numbers. Free, no sign-up.
Enter a sample size of at least 2 and a proportion between 0 and 100. If you give a population size, it must be larger than the sample.
How to Use the Margin of Error Calculator
- 1. Enter the sample size. This is the number of completed responses, n, and it must be at least 2.
- 2. Choose the confidence level. 90%, 95% or 99% (z = 1.645, 1.96 or 2.576). Polls usually report 95%.
- 3. Enter the observed percentage. Use the share that gave the answer you are reporting, between 0 and 100. If you want the most cautious figure, or the survey has many questions, use 50.
- 4. Add the population size, if known. It must be larger than the sample. Leave it blank for a large population.
- 5. Read the margin. You get the margin of error as ± percentage points, the z-score used, the finite correction if any, and a sentence giving the likely range for the true value.
The Formula
For a proportion, margin of error = z × √(p × (1 − p) ÷ n). With a known population N the result is multiplied by the finite population correction √((N − n) ÷ (N − 1)). The margin is largest at p = 0.5 and shrinks as the observed share moves toward 0% or 100%.
1,000 respondents, 95% confidence, 54% said yes
Margin = 1.96 × √(0.54 × 0.46 ÷ 1,000) = 1.96 × 0.015761 = ±3.09 points
Likely range for the true share: 54% ± 3.09 = about 50.9% to 57.1%
At p = 50% the same sample gives ±3.10 points, which is the figure usually quoted for a poll of 1,000
If those 1,000 were drawn from a population of 5,000: correction = √(4,000 ÷ 4,999) = 0.8945, so the margin falls to ±2.77 points at p = 50%
How Sample Size Changes the Margin
| Sample size | Margin of error (95%, p = 50%) |
|---|---|
| 100 | ±9.80% |
| 500 | ±4.38% |
| 1,000 | ±3.10% |
| 2,000 | ±2.19% |
Margin shrinks with the square root of the sample, so halving it takes four times as many respondents. The calculator draws this table for your chosen confidence level.
What the Margin Does and Does Not Cover
- It describes random sampling error only. It assumes a simple random sample. Biased question wording, a non-representative panel, and people who do not respond are not in it, and they can matter more.
- It is not a guarantee. At 95% confidence, about one interval in twenty built this way misses the true value.
- Subgroups are noisier. The margin for women aged 18 to 24 uses the number of respondents in that group, not the whole sample.
- Comparing two numbers needs more room. The margin on a difference between two independent results is about 1.4 times the single margin, so a 3-point lead with ±3-point margins is not clear evidence of a lead.
- It is for proportions. For a mean, such as average spend, use the Confidence Interval Calculator.
Which Tool Do You Need?
| Your question | Use |
|---|---|
| The margin of error for a sample you already have | This Margin of Error Calculator |
| How many people to survey for a target margin | Sample Size Calculator |
| A confidence interval for a proportion or a mean | Confidence Interval Calculator |
| The sample for a study that must detect an effect | Statistical Power & Sample Size Calculator |
| The standard error behind the margin | Standard Error Calculator |
Frequently Asked Questions
- What is the margin of error?
- It is the plus-or-minus that goes with a survey percentage to reflect sampling error. A result of 54% with a margin of ±3 points at 95% confidence means the true population value is likely between 51% and 57%.
- How is the margin of error calculated?
- Multiply the z-score for your confidence level by the standard error of the proportion: z × √(p(1 − p) ÷ n). With a known population you also multiply by the finite population correction √((N − n) ÷ (N − 1)).
- How can I reduce my margin of error?
- Increase the sample size, accept a lower confidence level, or, for a small population, sample a larger share of it. Because of the square root, a smaller margin gets expensive quickly: ±3 points needs about four times as many people as ±6.
- Why is 50% used as the default proportion?
- p × (1 − p) is largest at 50%, so it gives the largest, most cautious margin. Poll results at 50% always have a margin at least as big as the same poll at any other percentage.
- What is the finite population correction?
- When the sample is a large fraction of a small population, each additional respondent removes uncertainty faster, so the margin shrinks by the factor √((N − n) ÷ (N − 1)). It matters when the sample is more than about 5% of the population; below that it is negligible.
- What is a good margin of error for a survey?
- Most polls and market surveys aim for ±3 to ±5 points at 95% confidence, which takes roughly 400 to 1,100 respondents. Smaller margins may be worth the cost for close decisions, and larger ones are acceptable for exploratory work.