Case 1
- Sample proportion p (0 to 1)
- 0.5
- Sample size n
- 1000
- z-value (confidence)
- 1.96
3.0990%
Open with these valuesMargin of Error Calculator
3.0990%
Result: 3.0990 %The margin of error is the plus-or-minus band around a poll result. Enter the share as a decimal, the number of people asked, and the z-value for your confidence level — 1.96 is the usual 95 percent. The band shrinks with the square root of the sample, so four times the people buys only half the error.
The numbers at a glance
Held fixed: Sample proportion p (0 to 1) 0.500000, Sample size n 1,000.
| z-value (confidence) | Result (%) |
|---|---|
| 0.5000 | 0.7906 |
| 1.0000 | 1.5811 |
| 1.5000 | 2.3717 |
| 1.9600Your value | 3.0990 |
| 2.0000 | 3.1623 |
| 2.5000 | 3.9528 |
| 3.0000 | 4.7434 |
| 3.5000 | 5.5340 |
Worked examples
Case 2
- Sample proportion p (0 to 1)
- 0.3
- Sample size n
- 500
- z-value (confidence)
- 1.96
4.0168%
Open with these valuesCase 3
- Sample proportion p (0 to 1)
- 0.1
- Sample size n
- 200
- z-value (confidence)
- 1.645
3.4896%
Open with these valuesHow it's calculated
MoE = z × √(p × (1 − p) ÷ n) × 100
- StepEnter the share of the sample with the outcome, as a decimal.
- StepEnter how many people or items were sampled.
- StepPick the z-value: 1.645 for 90 %, 1.96 for 95 %, 2.576 for 99 %.
- ResultRead the margin as a plus-or-minus percentage.
What this number means
Covers sampling error only
The margin captures the random variation that comes from measuring a sample instead of everyone. It says nothing about bias from a badly drawn sample, and no sample size fixes that.
It shrinks with the square root, not linearly
The margin follows 1 ÷ √n, so quadrupling the sample only halves it. At p = 0.5 and z = 1.96, a hundred people give ±9.8 % and a thousand give ±3.099 %.
Enter a proportion, not a percentage
The field runs from 0 to 1, so 47 percent goes in as 0.47. The margin itself comes back as a percentage.
p = 0.5 is the widest case
The term p(1 − p) is largest at 0.5, so that share gives the largest margin for a given sample. Use it when planning a sample before the true share is known.
Commonly misread
±3 % means the true value is certainly within 3 points.
It holds at the stated confidence level only — 95 % for z = 1.96. Read at 99 %, the same thousand-person poll gives ±4.073 %.
Twice as many people asked halves the margin.
Four times as many halves it, because the margin follows 1 ÷ √n. Going from 100 to 1000 people takes ±9.8 % down to ±3.099 %, not to a tenth.
A large enough sample removes the error from a poll.
It only narrows the sampling error, the part this margin measures. A sample drawn badly stays wrong at any size.
Reference table
| p, n, z | Confidence and sample | Margin in % |
|---|---|---|
| 0.5, 1000, 1.96 | The classic thousand-person poll | 3.099 |
| 0.1, 200, 1.645 | 90 % confidence, a rare outcome | 3.4896 |
| 0.3, 500, 1.96 | 95 % confidence, a mid-sized sample | 4.0168 |
| 0.5, 1000, 2.576 | Same poll read at 99 % confidence | 4.073 |
| 0.5, 100, 1.96 | 95 % confidence, only a hundred asked | 9.8 |
Questions
How do I calculate the margin of error?
Use MoE = z × √(p(1 − p) ÷ n) × 100, where p is the sample proportion as a decimal, n is the sample size, and z is the confidence multiplier. For p = 0.5, n = 1000 and z = 1.96 that is 1.96 × √(0.25 ÷ 1000) × 100, about ±3.1 percent.
What does the margin of error actually mean?
It is the plus-or-minus band around a sample estimate. A poll reporting 47 percent with a margin of ±3 means the true value plausibly lies between 44 and 50 percent at the stated confidence level. It captures sampling error only — the random variation from measuring a sample instead of everyone — not bias from a flawed sample.
Which z-value should I use?
The z-value sets the confidence level. The common choices are 1.645 for 90 percent, 1.96 for 95 percent, which is by far the most reported, and 2.576 for 99 percent. A higher confidence level uses a larger z-value, which widens the margin — more certainty costs precision.
Why is p = 0.5 the most conservative choice?
The term p(1 − p) is largest at p = 0.5, so that share produces the widest margin. When the true proportion is not yet known, for instance while planning a sample size, using 0.5 gives the worst-case margin, so the estimate cannot turn out more uncertain than planned.
How does sample size change the margin?
The margin shrinks in proportion to 1 ÷ √n, so it falls off slowly. Quadrupling the sample size only halves the margin of error. Moving from a thousand-person poll to a four-thousand-person poll roughly halves the error but costs four times the fieldwork.
Sources and last check
Information, not professional advice.