# Normal approximation to the binomial

A binomial with large n can be approximated by a normal with mean np and SD square root of np(1-p). It works well when np and n(1-p) are both at least 10. Add or subtract 0.5 (continuity correction) to bridge whole numbers and a smooth curve.

Statistics (university) · Part 04: Continuous distributions and the normal curve · Card 7 of 8 · Last updated 2026-10-19

## Analogy

A staircase with many tiny steps looks like a smooth ramp from far away.

## Example

**100 coin flips**

n = 100, p = 0.5
Mean = 50, SD = square root of 25 = 5
P(X <= 55) uses 55.5
z = (55.5 - 50) / 5 = 1.1
P = 0.8643 (exact: 0.8644)

## In practice

Used to estimate binomial tail chances with a z table when n is large and exact sums are tedious.

## Practice

### Flashcards

- **When is the normal approximation to the binomial fine?** When np and n(1-p) are both at least 10.

## Questions people ask

### What is a continuity correction?

Binomial counts are whole numbers, but the normal curve is continuous. The bar for 55 spans 54.5 to 55.5. So P(X at most 55) becomes P(Normal below 55.5). The half-unit adjustment improves accuracy, especially for modest n, where the gap between the staircase and the curve matters most.

### When is the normal approximation to the binomial reliable?

Check that both np and n(1-p) are at least 10. If p is near 0 or 1, the binomial is lopsided and the normal curve fits badly unless n is very large. In those cases an exact binomial or Poisson calculation is safer.

## Related concepts

- [Normal distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-4)
- [Standard normal and z-scores](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-5)
- [Normal probabilities and percentiles](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-6)
- [Sums of normal variables](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-4/statistics-university-part-4-concept-8)

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