STEM Calculators

The Normal Approximation to the Binomial Distribution

Statistics • Continuous Random Variables and the Normal Distribution

Written by STEM Calculators Team Published Updated
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The Normal Approximation to the Binomial Distribution

Compute exact binomial probabilities, then approximate them using a normal distribution with the continuity correction. The chart overlays the binomial histogram with the normal curve and shades the selected probability.

Binomial model: fixed n trials, each trial is success/failure, success probability p is constant, and trials are independent.
Normal approximation rule of thumb: use it when n·p > 5 and n·(1 − p) > 5. Then μ = n·p and σ = √(n·p·(1 − p)).
Continuity correction: adjust integer boundaries by ±0.5 when converting the discrete binomial event into a continuous normal interval.

Inputs

Enter n and p, choose an event, then click Calculate.

CSV data (copy/paste or import)

Generate a binomial distribution table as CSV, copy/paste it anywhere, or import a CSV file. Format supported: x,px or x,px,cdf (header optional).

Tip: Generate CSV from n and p, or import your own file.

Visualization

Binomial bars Selected event (exact) Normal curve Selected area (normal)

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Frequently Asked Questions

When can I use the normal approximation to the binomial distribution?

A common rule of thumb is to use it when n*p > 5 and n*(1 - p) > 5. When these conditions hold, the binomial distribution is usually close enough to normal for many probability calculations.

What is the continuity correction and why does it use 0.5?

The continuity correction adjusts discrete integer bounds to continuous bounds by adding or subtracting 0.5. It improves accuracy because the normal distribution is continuous while the binomial distribution is discrete.

How does the calculator compute mu and sigma for the approximation?

It uses the binomial mean and standard deviation: mu = n*p and sigma = sqrt(n*p*(1 - p)). These parameters define the normal model used to approximate the binomial probabilities.

Why is the exact binomial probability different from the normal approximation?

The exact probability sums binomial terms at integer values, while the approximation uses the area under a continuous normal curve. The difference depends on n, p, and the event bounds, which is why the calculator also reports the absolute error |exact - approx|.

What does the z-table rounding option change?

It rounds z-scores to 2 decimals to match how standard normal tables are typically used. This can make your approximation align with table-based homework or textbook methods.

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