Why does margin of error shrink as sample size grows?

The formula divides by √n, so quadrupling the sample size only halves the margin of error — this is why large surveys need proportionally huge sample sizes to make small further gains in precision.

Why does the sample proportion (p) affect the margin of error?

The formula includes p(1−p), which is largest at p = 50% and smaller as p moves toward 0% or 100%. A proportion far from 50% (e.g. a rare outcome) produces a smaller margin of error for the same sample size.

Margin of Error Calculator

Calculate the margin of error for a survey sample.

How this is calculated

MOE = z × √(p(1 − p) ÷ n)

Assumptions used in this calculation

Simple random sampling: Assumes the sample was drawn via simple random sampling from a large population; it does not account for design effects from stratified or clustered sampling.

About this calculator

The margin of error calculator finds the margin of error for a sample proportion at a given confidence level and sample size.

Worked example

95% confidence, n=1000, p=50%

Result: ±3.1%

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Frequently asked questions

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Margin of Error Calculator

About this calculator

Frequently asked questions

Why does margin of error shrink as sample size grows?

Why does the sample proportion (p) affect the margin of error?

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