Normal Probability Calculator (Sampling)
Evaluate the probability that the sample mean falls within a specified range when sampling from a normal or large population.
Standard error: 2.0000
Z(lower): -1.0000
Z(upper): 2.5000
P(lower ≤ x̄ ≤ upper): 0.835135
How to Use This Calculator
- Enter the population mean and standard deviation.
- Specify the sample size n (ensure independence or n large enough for CLT).
- Provide the interval for the sample mean.
- Review resulting z-scores, standard error, and interval probability.
Formula
Standard error = σ / √n
Z = (x̄ − μ) / (σ / √n)
P(L ≤ x̄ ≤ U) = Φ((U − μ) / (σ/√n)) − Φ((L − μ) / (σ/√n))
Φ denotes the standard normal CDF. For small samples with unknown σ, use the t-distribution instead.
Frequently Asked Questions
When is the normal approximation valid?
It is exact for normally distributed populations and approximate for other populations when n is sufficiently large (typically n ≥ 30).
What if σ is unknown?
Use the sample standard deviation and apply the t-distribution with n − 1 degrees of freedom.
Can I compute tail probabilities?
Yes. Set either the lower or upper bound to ±∞ (or a large magnitude) to approximate single-tail probabilities.