Normal Approximation Calculator

What This Normal Approximation Calculator Does

This tool approximates binomial probabilities with a normal curve. Enter the number of trials $N$, the success probability $p$, the target number of successes $n$, and the probability statement you want, such as $P(X \ge n)$ or $P(X = n)$. The calculator then applies continuity correction, converts the boundary to a z-score, and returns the approximate probability.

Normal Approximation to Binomial Distribution

A binomial random variable has mean $ \mu = Np $ and variance $ \sigma^2 = Np(1-p) $. When both expected successes and failures are large enough, the binomial distribution can be approximated by a normal distribution:

$$X \approx N(\mu, \sigma^2)$$

In practice, the usual rule of thumb is to check whether $Np \ge 5$ and $N(1-p) \ge 5$. If both conditions are met, the approximation is usually acceptable for quick probability estimates.

Continuity Correction Rules

Because binomial values are discrete and the normal model is continuous, you should shift the integer boundary by $0.5$. This small correction improves the result:

• $P(X = n) \approx P(n - 0.5 < Y < n + 0.5)$
• $P(X \le n) \approx P(Y < n + 0.5)$
• $P(X < n) \approx P(Y < n - 0.5)$
• $P(X \ge n) \approx P(Y > n - 0.5)$
• $P(X > n) \approx P(Y > n + 0.5)$

How to Compute It by Hand

1. Calculate $q = 1 - p$.
2. Find the mean with $ \mu = Np $.
3. Find the standard deviation with $ \sigma = \sqrt{Np(1-p)} $.
4. Rewrite the event using continuity correction.
5. Convert the corrected boundary to a z-score with $ z = \frac{x-\mu}{\sigma} $.
6. Use the standard normal table or this calculator to read the probability.

When to Use This Tool

This calculator is useful when you need fast binomial approximations for quality control, survey sampling, exam-score modeling, or repeated success/failure experiments. If you need an exact discrete answer, try the Binomial Probability Distribution Calculator. For raw z-score work, the Normal Distribution Calculator is also helpful.