Margin of Error Calculator
Enter your sample size and desired confidence level to calculate the margin of error — how many percentage points a survey result could realistically be off from the true population value.
How the Margin of Error formula works
The margin of error formula, using the most conservative (maximum variability) assumption:
Margin of error = z × √( p(1−p) / n ) × 100
Where z is the z-score for the chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), p is the assumed population proportion (0.5 gives the most conservative, widest margin), and n is sample size.
Step-by-step calculation
- Choose a confidence level and its corresponding z-score (95% and z=1.96 is the most common default).
- Multiply p(1−p) — commonly 0.5 × 0.5 = 0.25 for the most conservative estimate — divided by sample size, then take the square root.
- Multiply by the z-score and by 100 to express margin of error as a percentage.
Worked example
A survey of 1,000 people at 95% confidence: Margin of error = 1.96 × √(0.25/1000) × 100 ≈ 1.96 × 0.01581 × 100 ≈ 3.1%.
Frequently asked questions
Why does a larger sample size reduce margin of error?
Larger samples give a more precise estimate of the true population value, which directly shrinks the margin of error — but the relationship isn't linear: quadrupling sample size only halves the margin of error, since it depends on the square root of sample size.
Why use p = 0.5 by default?
p(1−p) is maximized at p = 0.5, producing the widest (most conservative) possible margin of error — a safe default when the true population proportion isn't known in advance, which is the most common situation before a survey is run.