Calculator Inputs
The page stays in a single content flow, while the input grid expands to three columns on large screens, two on medium screens, and one on phones.
Example data table
| Scenario | Inputs | Requested event | Approximate probability |
|---|---|---|---|
| Binomial | n = 100, p = 0.40 | P(35 ≤ X ≤ 50) | 0.8532 |
| Poisson | λ = 36 | P(X ≥ 40) | 0.2798 |
| Sampling proportion | n = 250, p = 0.52 | P(p̂ ≥ 0.58) | 0.0288 |
| Sampling mean | μ = 80, σ = 12, n = 64 | P(78 ≤ X̄ ≤ 83) | 0.8860 |
Formula used
Binomial approximation: If X ~ Bin(n, p), then X is approximated by a normal model with μ = np and σ = √(np(1-p)).
Poisson approximation: If X ~ Pois(λ), then use μ = λ and σ = √λ.
Sampling proportion: For p̂, use μ = p and σ = √(p(1-p)/n).
Sampling mean: For X̄, use μ = μ and σ = σ/√n.
Standardization: Convert a cutoff to a z score with z = (x - μ) / σ, then read the probability from the standard normal curve.
Continuity correction: For discrete counts, replace an integer cutoff with a half-step boundary. For example, P(X ≤ k) becomes P(Y ≤ k + 0.5).
How to use this calculator
- Select the approximation mode that matches your problem.
- Choose whether you want a lower tail, upper tail, interval, or exact count result.
- Enter the model parameters such as
n,p,λ,μ, orσ. - Turn continuity correction on for discrete distributions unless you want to compare the effect without it.
- Set the decimal precision and a reference confidence level for the central interval summary.
- Press the calculate button to show the result above the form.
- Review the chart, approximation checks, and optional exact comparison.
- Use the CSV or PDF buttons to save the result for reporting.
Frequently asked questions
1. When should I trust a normal approximation?
Trust it when the expected counts are not too small. For binomial work, many instructors use np and n(1-p) greater than 5 or 10. For Poisson work, a larger λ gives a smoother shape and a better fit.
2. Why does continuity correction matter?
A normal curve is continuous, but binomial and Poisson variables jump by whole counts. Continuity correction shifts cutoffs by 0.5 so the continuous area better matches the discrete event you really want.
3. Does this tool give exact probabilities too?
Yes, for many discrete inputs. The calculator compares the approximation with an exact binomial or Poisson value when the problem size remains practical. Very large inputs may skip exact comparison to keep performance reasonable.
4. Can I use it for sample proportions?
Yes. Choose the sampling proportion mode, enter n and the population proportion p, then evaluate a threshold or interval for p̂. The tool also checks whether the expected success and failure counts are large enough.
5. Can I use it for sample means?
Yes. Enter the population mean, population standard deviation, and sample size. The calculator uses the standard error σ/√n and reminds you that larger sample sizes or a normal population improve the approximation.
6. What does the central interval mean?
It is a quick reference band centered on the approximate mean. For example, a 95% central interval shows the middle 95% of the approximating normal model using the selected standard deviation or standard error.
7. Why is an exact probability sometimes missing?
Some exact calculations become expensive when counts get very large. In that case the tool still returns the approximation, displays the assumptions, and explains that the exact comparison was skipped for performance.
8. What does the graph show?
The graph shows the approximating normal density. The shaded region marks the probability event you requested, such as a left tail, right tail, or interval. This makes the z-score boundaries easier to interpret.