Statistics & Probability

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Normal Approximation Calculator.

Approximate a binomial cumulative probability with continuity correction.

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Set your values

Results update as you type.

mean: 50

mean

50.000000
standardDeviation: 5

standardDeviation

5.000000
zScore: 1.1

zScore

1.100000
probabilityAtMost: 0.864334

probabilityAtMost

0.864334
continuityCorrection: 0.5

continuityCorrection

0.500000

Results update automatically as you type.

Use Cases

Quality Control Sampling

Estimate the probability that the number of defective items in a sample is within a limit, using normal approximation for large sample sizes.

Example: If 5% of products are defective, what is the probability that at most 10 out of 200 are defective?

Survey Response Analysis

Approximate the likelihood of observing a certain number of positive responses in a large survey, given a known response rate.

Example: With a 30% response rate, what is the chance that at most 50 out of 300 respond?

Frequently Asked Questions

What is the normal approximation to the binomial distribution?
The normal approximation uses the normal distribution to approximate binomial probabilities when the number of trials is large. It relies on the central limit theorem and is often used when n is large and p is not too close to 0 or 1.
What is continuity correction and why is it used?
Continuity correction adjusts for the fact that a discrete distribution (binomial) is approximated by a continuous one (normal). It involves adding or subtracting 0.5 to the discrete value to improve the approximation, especially for small n.
When is it appropriate to use this calculator?
It is appropriate when you have a binomial scenario with a fixed number of independent trials, each with the same success probability, and you want the probability of at most a certain number of successes. It works best when n is large (e.g., n > 20) and p is not extreme.

Tips & Common Mistakes

Tips

  • Ensure the number of trials (n) is large enough; a common rule is n*p > 5 and n*(1-p) > 5 for a good approximation.
  • Use the continuity correction by entering the exact 'at most' value; the calculator automatically applies the correction.
  • Remember that the result is an approximation; for exact probabilities, use the binomial distribution directly if n is small.
  • Double-check that the success probability is between 0 and 1, and that k is between 0 and n.

Common Mistakes to Avoid

  • Using the calculator when n is too small, leading to inaccurate approximations.
  • Forgetting that the result is for 'at most' k successes, not 'exactly' k.
  • Entering a success probability outside the 0-1 range or a k value greater than n.

Last updated: August 13, 2026