Normal Approximation to Binomial Calculator
Enter the number of trials, probability of success, and choose the probability type to get an instant normal approximation to the binomial distribution. The calculator applies the continuity correction automatically, shows the z-scores, and checks the np >= 5 and n(1-p) >= 5 conditions required for the approximation to be valid.
What is the normal approximation to the binomial?
The binomial distribution describes the number of successes in n independent Bernoulli trials, each with probability p of success. Computing exact binomial probabilities involves factorials and can be cumbersome when n is large. The normal approximation replaces the binomial with a normal distribution having the same mean (mu = np) and the same variance (sigma^2 = np(1-p)). Because a normal distribution is continuous and a binomial is discrete, a continuity correction of 0.5 is added or subtracted from the boundary before converting to a z-score. This single correction typically cuts the approximation error by an order of magnitude compared to the uncorrected version. The approximation is considered reliable when both np >= 5 and n(1-p) >= 5. When one of those conditions fails, the binomial distribution is too skewed for the bell curve to fit it well, and you should use the exact binomial CDF instead.
How to use this calculator
Enter the number of trials (n) and the probability of success on each trial (p, a value between 0 and 1). Then choose the probability type from the dropdown: exactly k successes, at most k, fewer than k, at least k, more than k, or between two values a and b. Fill in k (and b for the range option). The calculator instantly shows the probability, the mean and standard deviation of the approximating normal distribution, the z-scores after continuity correction, and a validity check. The "Show your work" panel walks through every algebraic step with your exact numbers. The chart below the result overlays the exact binomial PMF (bars) and the normal PDF (curve) so you can see how well the bell curve fits for your chosen n and p.
The continuity correction in detail
Because the binomial takes only integer values, the probability of exactly k is a bar of width 1 centered at k in a bar chart. To approximate the area under that bar using a continuous curve, the boundaries of that bar are 0.5 below and above k. That is why P(X = k) maps to P(k - 0.5 < X_normal < k + 0.5). The same principle extends to inequalities: P(X <= k) maps to P(X_normal < k + 0.5) because we include the upper half of the bar at k, and P(X < k) maps to P(X_normal < k - 0.5) because we exclude the lower half of the bar at k. Without the correction, the probability would be systematically off by half a standard probability mass unit.
Step-by-step calculation
Given n = 100 and p = 0.4, find P(X <= 35). First check the conditions: np = 40 >= 5 and n(1-p) = 60 >= 5, so the approximation is valid. The mean is mu = 100 x 0.4 = 40 and the standard deviation is sigma = sqrt(100 x 0.4 x 0.6) = sqrt(24) = 4.899. Applying the continuity correction for "at most 35": P(X_normal < 35.5). Convert to a z-score: z = (35.5 - 40) / 4.899 = -4.5 / 4.899 = -0.9185. Look up Phi(-0.9185) in a standard normal table or calculate it: approximately 0.1793 or 17.93%. The exact binomial probability using the CDF is about 17.76%, so the approximation is very close.
Continuity correction rules
| Probability type | Original boundary | After correction |
|---|---|---|
| P(X = k) | k | (k - 0.5, k + 0.5) |
| P(X <= k) | k | X < k + 0.5 |
| P(X < k) | k | X < k - 0.5 |
| P(X >= k) | k | X > k - 0.5 |
| P(X > k) | k | X > k + 0.5 |
| P(a <= X <= b) | a, b | (a - 0.5, b + 0.5) |
How to adjust the discrete binomial boundary before computing the z-score for each probability type.
Frequently asked questions
When can I use the normal approximation to the binomial?
The rule of thumb is that both np >= 5 and n(1-p) >= 5 must hold. Some textbooks use a stricter threshold of 10. When the distribution is very skewed (p close to 0 or 1) or n is small, the normal curve does not fit the binomial well and you should use the exact binomial CDF or a Poisson approximation instead. This calculator flags whether your inputs satisfy the condition.
Why do I need a continuity correction?
The binomial distribution is discrete (it takes integer values) while the normal distribution is continuous. When you replace a bar of probability at integer k with an area under a smooth curve, you need to account for the full width of that bar: 0.5 on each side. Without the correction, the probability estimate is systematically biased. Adding the correction is simple and substantially improves accuracy, so it should always be used when applying the normal approximation.
How is the z-score calculated in normal approximation?
After applying the continuity correction, the adjusted boundary x_c is converted to a z-score using z = (x_c - mu) / sigma, where mu = np and sigma = sqrt(np(1-p)). The standard normal CDF, Phi(z), then gives the probability to the left of that z-score. For two-sided events you compute Phi(z_high) - Phi(z_low).
What is the difference between P(X < k) and P(X <= k)?
Because the binomial is discrete, these are different events. P(X <= k) includes the probability mass at exactly k, while P(X < k) excludes it. In the continuity-corrected normal approximation, P(X <= k) maps to P(X_normal < k + 0.5) and P(X < k) maps to P(X_normal < k - 0.5). Mixing up which boundary to use is a common source of error when performing normal approximation by hand.
Is the normal approximation the same as a z-test?
They use the same mechanics but serve different purposes. A one-proportion z-test asks whether an observed number of successes is significantly different from a hypothesized p, while the normal approximation simply estimates a binomial probability. The z-score formula z = (x - mu) / sigma is identical; what changes is the question being answered and whether you compare the result to a critical value (z-test) or report it directly as a probability.
Can I use this calculator for proportions?
Yes. If you know the sample size n and the observed proportion p-hat and want to find probabilities about the count X = n * p-hat, enter those values directly. The binomial framework treats each trial independently with fixed probability p, so any situation that fits that model works here.