Random Sample Margin of Error Calculator

Calculator Inputs

Estimate Type

Sample Size (n)

Confidence Level

Custom Z-Score

Estimated Proportion (%)

Sample Mean

Standard Deviation

Population Size (optional)

Target Margin of Error (%)

Target Margin of Error

Decimal Places

Apply finite population correction

Example Data Table

Use these example scenarios to understand common survey precision ranges.

Scenario	Mode	n	Confidence	Estimate	Population	Approx. MOE
Customer satisfaction pulse	Proportion	400	95%	50%	Large	±4.90%
Product launch feedback	Proportion	800	95%	50%	12,000	±3.42%
Average delivery time	Mean	100	95%	36.4	Large	±2.35 units
Call center quality score	Mean	225	99%	82.8	2,500	±1.63 units
Internal employee survey	Proportion	300	90%	62%	900	±4.21%

Formula Used

Proportion mode: MOE = z × √(p × (1 - p) / n) × FPC

Mean mode: MOE = z × (σ / √n) × FPC

Finite population correction: FPC = √((N - n) / (N - 1))

Required sample size: n = z² × variance / E², with finite adjustment when population size is known.

What each term means

z is the critical value linked to confidence level.
p is the estimated proportion as a decimal.
σ is the known or estimated standard deviation.
n is the sample size.
N is the full population size.
E is the target margin of error.

How to Use This Calculator

Select Proportion for percentages, or Mean for averages.
Enter your sample size and desired confidence level.
For proportion mode, enter the estimated proportion percentage.
For mean mode, enter the sample mean and standard deviation.
Add population size when sampling without replacement.
Keep finite population correction checked when population size matters.
Enter a target margin if you want a suggested sample size.
Submit the form to view results, interval bounds, and the Plotly graph.
Use the CSV or PDF buttons to export your results.

Frequently Asked Questions

1. What does margin of error represent?

Margin of error shows the expected sampling uncertainty around a survey estimate. A smaller value means your sample gives a more precise estimate, assuming random sampling and correct model assumptions.

2. Why does sample size reduce margin of error?

Larger samples reduce standard error because more observations stabilize the estimate. The gain is not linear, though. To cut the margin of error roughly in half, you usually need about four times the sample size.

3. When should I use proportion mode?

Use proportion mode when the result is a percentage or share, such as approval rate, conversion rate, defect rate, or survey response proportion. The calculator then uses the binomial-based standard error formula.

4. When should I use mean mode?

Use mean mode when you are estimating an average, like revenue, score, wait time, or delivery duration. You need a known or estimated standard deviation to compute the standard error and margin of error.

5. What is finite population correction?

Finite population correction adjusts uncertainty downward when the sample is a meaningful share of the full population. It matters most when sampling without replacement from a relatively small, known population.

6. Why is 50% often used for proportions?

A 50% proportion creates the largest binomial variance. Using it is conservative because it produces the largest margin of error and usually the largest required sample size for planning.

7. Does this replace a full survey design review?

No. This tool helps with statistical precision under random sampling assumptions. It does not correct for bias, bad questionnaires, nonresponse, weighting choices, clustering, or poor frame coverage.

8. Can I use this for experiments and A/B tests?

Yes, as a quick planning aid. For formal experiments, also review power, minimum detectable effect, baseline variance, and assignment design. Margin of error alone does not fully evaluate experimental adequacy.

Notes

This calculator is intended for random samples. Interpret results carefully when samples are biased, clustered, weighted, or collected with missing-data issues.