Margin of Error Calculator
Calculate the sampling margin of error, an approximate confidence interval, or the sample size needed for a survey. You can also apply a finite population correction.
Your Results
Your calculation details will appear here.
Enter values and click Calculate.
Estimates assume simple random sampling and an appropriate normal approximation. They do not account for survey bias, nonresponse, weighting, or complex sampling designs.
How to Calculate Margin of Error
Choose a calculation mode, enter the relevant information, and click Calculate.
The result and formula appear directly below the buttons.
Try an Example loads a survey of 400 respondents at 95% confidence. Reset restores the starting values.
Margin of Error for a Percentage
MOE = z × √[p(1 − p) / n]
Here, z is the confidence-level critical value, p is the sample proportion expressed as a decimal, and n is the sample size.
Multiply by 100 to express the result in percentage points.
Margin of Error for a Mean
MOE = z × SD / √n
The mean’s confidence interval is the sample mean plus or minus the margin of error.
This calculator uses the normal approximation and accepts sample sizes of at least 30.
Finite Population Correction
For simple random sampling without replacement from a finite population, multiply the margin of error by:
√[(N − n) / (N − 1)]
This correction is not intended for a convenience sample.
Required Sample Size
For a proportion:
n₀ = z² × p(1 − p) / E²
E is the desired error as a decimal.
If population size N is supplied, use:
n = N × n₀ / (N + n₀ − 1)
Round up to the next whole respondent.
Example: 400 Survey Respondents
With 400 respondents, a 50% observed proportion, 95% confidence, and no finite population correction, the approximate margin of error is 4.90 percentage points.
The corresponding interval runs from roughly 45.10% to 54.90%.
Frequently Asked Questions
What is margin of error?
It estimates the sampling uncertainty around a statistic under a specified sampling model and confidence level.
Is a 5% margin of error always acceptable?
No. The precision needed depends on the decision, research question, and consequences of estimation error.
Why is 50% the default proportion?
It maximizes p(1 − p) and therefore gives the largest conventional margin of error for a fixed sample size.
How does a larger sample change the result?
Without finite population correction, increasing sample size fourfold approximately halves the margin of error.
What does a 95% confidence level mean?
Over repeated samples, approximately 95% of intervals constructed using a valid 95% confidence procedure would contain the true parameter.
Does margin of error measure survey bias?
No. It does not capture nonresponse, coverage errors, misleading questions, or other nonsampling errors.
When should I enter a population size?
Use it when sampling without replacement from a known finite population, such as a list of eligible employees.
Can the observed percentage be 0% or 100%?
The conventional normal (Wald) method used here is unreliable at these boundaries. Use a Wilson score or exact binomial interval instead.
Why does the mean mode require 30 observations?
It uses a large-sample normal approximation. Small samples with an estimated standard deviation typically require a Student’s t method and appropriate distribution assumptions.
How many responses are needed for ±5 percentage points?
For 95% confidence, 50% expected proportion, and a large population, the standard formula rounds up to 385 responses.