Normal Approximation Calculator

Normal approximation to the binomial distribution is a method where the normal (Gaussian) distribution is used to estimate probabilities for a binomial random variable. It works well when the number of trials is large (n ≥ 30), and both np ≥ 5 and n(1−p) ≥ 5 are satisfied. Under these conditions, the binomial distribution becomes approximately bell-shaped.

First, compute the mean μ = np and standard deviation σ = √(npq), where q = 1 − p. Then apply a continuity correction to x (e.g. use x + 0.5 or x − 0.5 depending on the probability type), and calculate the Z-score as Z = (x_corrected − μ) / σ. Finally, look up the corresponding probability from the standard normal distribution table.

Continuity correction is an adjustment made when approximating a discrete distribution (binomial) with a continuous one (normal). Since the binomial counts whole numbers, each value x is represented as an interval [x − 0.5, x + 0.5] in the normal distribution. Adding or subtracting 0.5 makes the approximation significantly more accurate.

It is generally not recommended. The rule of thumb requires that both np ≥ 5 and n(1−p) ≥ 5 for the normal approximation to be reliable. If these conditions are not met, the binomial distribution is too skewed, and the normal approximation will produce inaccurate results. In such cases, use the exact binomial probability instead.

The Z-value is calculated as Z = (x − μ) / σ = (60.5 − 50) / 5 = 2.1. This means the value 60.5 is 2.1 standard deviations above the mean. You would then look up P(Z ≤ 2.1) in a standard normal table to find the corresponding probability, which is approximately 0.9821.

The key steps are: (1) Verify the conditions — n ≥ 30, np ≥ 5, and n(1−p) ≥ 5. (2) Compute q = 1 − p, mean μ = np, and standard deviation σ = √(npq). (3) Apply the appropriate continuity correction to x. (4) Compute the Z-score. (5) Use the standard normal distribution to find the final probability.

The approximation is most accurate when p is close to 0.5 and n is large, because the binomial distribution is most symmetric under these conditions. As n increases, the binomial distribution converges to the normal distribution regardless of p, so even skewed cases become well-approximated with sufficiently large samples.

P(X = x) is the probability of exactly x successes. P(X ≤ x) is the cumulative probability of x or fewer successes (left-tail). P(X ≥ x) is the probability of x or more successes (right-tail). Each requires a different continuity correction: for P(X = x), use the interval [x − 0.5, x + 0.5]; for P(X ≤ x), use x + 0.5; for P(X ≥ x), use x − 0.5.