Margin of error calculator

How much sample results may differ from the true value

Margin of error — half-width of the confidence interval. How much the sample proportion or mean may differ from the population value.

FAQ about margin of error

Parameters

Sample size (n)

Number of respondents who completed the survey

Population size (N)

Total size of target audience. Leave empty if unknown or very large

Confidence level

Probability that the true value falls within the confidence interval

90% 95% 99%

Expected proportion (p)

If unknown, use 50% — gives maximum (conservative) margin of error

50%

Assumptions and limitations

Formula is for proportion, not mean
p = 50% gives maximum (conservative) margin of error
For small samples (n < 30) use the result with caution
FPC (finite population correction) applied when N is given

Margin of error

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How margin of error is calculated

Margin of error formula

MOE = z × √(p(1−p)/n) × FPC, where z is the z-value for confidence level (1.645/1.96/2.576), p = proportion, n = sample size, FPC = finite population correction.

Z-values

90% confidence → z = 1.645. 95% confidence → z = 1.96 (standard). 99% confidence → z = 2.576. Higher confidence means wider interval and larger MOE.

FPC correction

FPC = √((N−n)/(N−1)). Used when the sample is a significant part of the population (n/N > 5%). Reduces MOE. For infinite population FPC = 1.

Why p = 50%

At p = 50% the product p(1−p) is maximum (0.25), giving the largest MOE. If the true proportion differs from 50%, the actual MOE will be smaller.

Typical margin of error values

Sample size

MOE (90%)

MOE (95%)

MOE (99%)

100

±8,2%

±9,8%

±12,9%

200

±5,8%

±6,9%

±9,1%

400

±4,1%

±4,9%

±6,4%

1000

±2,6%

±3,1%

±4,1%

2000

±1,8%

±2,2%

±2,9%

* At p = 50% and infinite population

Frequently asked questions about margin of error

What is margin of error?

Margin of error is the range within which the true value falls with a given probability. With MOE ±5% and a result of 60%, the true value lies between 55% and 65%.

What margin of error is acceptable?

Standard for market research is ±5% at 95% confidence. For critical decisions use ±3% or ±2%. For pilot studies ±10% is acceptable.

How to reduce margin of error?

Increase sample size. MOE is inversely proportional to √n: to halve MOE you need to quadruple the sample.

What is FPC (finite population correction)?

FPC reduces MOE when the sample is a significant part of the population. If n/N > 5%, FPC has a substantial effect. For infinite population FPC = 1.

Why is MOE maximum at p = 50%?

Mathematically: p(1−p) is maximum at p = 0.5 (equals 0.25). If the true proportion differs from 50% (e.g. 80% or 20%), the actual MOE is smaller than the calculated one.

How are margin of error and sample size related?

Inverse relationship: larger sample, smaller MOE. Use the sample size calculator to find n for a desired MOE.

What formula is used for margin of error?

MOE = Z × √(p(1−p)/n). For finite population N multiply by √((N−n)/(N−1)) (FPC). Z from confidence level (95% → 1.96).

Why use a margin of error calculator after a survey?

To understand how precise the results are. If MOE is too large — avoid strong conclusions; consider a larger sample or cautious interpretation.

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