# Margin of Error Calculator

URL: https://speedycalculator.com/calculators/statistics/margin-of-error/
Last updated: 2026-10-25

> The margin of error calculator finds the ± range around a survey percentage for a given sample size and confidence level, with an optional finite population correction.

## Inputs

- Observed percentage (%)
- Sample size
- Population size (0 = very large)
- Confidence level

## Results

- z value
- Margin of error
- Range from
- Range to

## Formula

```
MOE = z × √(p(1 − p) ÷ n) × √((N − n) ÷ (N − 1))
```

## Explanation

Polls report a margin of error so readers know how much the result could differ from the true population value.

### How to use the Margin of Error Calculator

- **Observed percentage**
- **Sample size**
- **Population size (0 = very large)**
- **Confidence level**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Multiply the z value for the confidence level by the standard error of the proportion, then correct for finite populations.

### Inputs explained

FieldWhat to enterObserved percentageIn %Sample sizeA numberPopulation size (0 = very large)A numberConfidence levelChoose one: 90%, 95%, 99%

### What each result means

- **z value** = ninv(1 - (1 - [Confidence level])÷2)
- **Margin of error** = [z value]×sqrt([Observed percentage]÷100×(1 - [Observed percentage]÷100)÷[Sample size])×[Finite population correction]×100
- **Range from** = [Observed percentage] - [Margin of error]
- **Range to** = [Observed percentage] + [Margin of error]

### Example values

How the main result changes when **Observed percentage** changes and everything else stays at the example values above:

Observed percentage (%)Margin of error (± percentage points)252.6837.53503.162.53752.681000

## Worked example

Inputs — Observed percentage: 50 %; Sample size: 1,000; Population size (0 = very large): 0; Confidence level: 95%.

Results — z value = 1.96; **Margin of error = 3.1 ± percentage points**; Range from = 46.9 %; Range to = 53.1 %.

## FAQ

**Why do polls of 1,000 people have about ±3%?**
At 95% confidence and p = 50%, MOE ≈ 1.96 × √(0.25/1000) ≈ 3.1%.

**Does population size matter?**
Only when the sample is a sizeable share of the population.

**What formula does the Margin of Error Calculator use?**
MOE = z × √(p(1 − p) ÷ n) × √((N − n) ÷ (N − 1)).


## Sources

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)