Normal Approximation to Binomial Calculator

Your details

Total number of independent Bernoulli trials (n). Must be a positive integer.
Probability of success on each trial (p). Must be between 0 and 1.
Choose the form of probability you want to compute.
The value of k for the chosen probability type.
ProbabilityApproximation valid
2.8716%

Normal approximation to the requested binomial probability.

Mean (mu)50
Std dev (sigma)5
Z-score (upper)-1.9
np50
n(1-p)50
Approximation valid?Yes
2.8716%99.8% below · Z-score

P(at most 40 successes) is approximately 2.8716% (0.028716).

  • With n = 100 trials and p = 0.5, the binomial mean is 50.000 and standard deviation is 5.000.
  • Both np >= 5 and n(1-p) >= 5, so the normal approximation with continuity correction is reliable here.
  • The continuity correction shifts the discrete boundary by 0.5 before computing the z-score, improving accuracy compared to using the raw value.

Next stepFor sample sizes where the condition is borderline, compare this result against the exact binomial CDF to see how close the approximation is.

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