Inputs
Results appear above after submission.
Formula used
The central limit idea says that, for many independent samples, the distribution of an estimator becomes close to normal as n grows. This calculator uses the normal approximation for three common cases:
- X̄ (sample mean): mean μ, standard error σ/√n.
- Sₙ (sample sum): mean nμ, standard deviation √n·σ.
- p̂ (sample proportion): mean p, standard error √(p(1−p)/n).
Probabilities are computed from z scores: z = (x − mean)/sd, then P = Φ(z) using the standard normal CDF.
How to use this calculator
- Choose the estimator: mean, sum, or proportion.
- Enter the needed parameters and the sample size n.
- Select a probability type: between, left tail, or right tail.
- Provide one or two bounds that match your estimator units.
- Enable continuity correction when bounds come from counts.
- Press Calculate. Export results using CSV or PDF buttons.
Example data table
| Scenario | Inputs | Interpretation |
|---|---|---|
| Mean of noise samples | μ=0, σ=2, n=50, between −0.5 and 0.5 | Probability that the average noise stays near zero. |
| Sum of pulses | μ=1.2, σ=0.4, n=20, right tail ≥ 28 | Chance the total exceeds a threshold in one trial. |
| Detector hit rate | p=0.35, n=100, left tail ≤ 0.30 | Probability the observed hit fraction falls below 0.30. |