Normal Distribution Calculator
Enter the mean and standard deviation of a normal distribution to evaluate z-scores, densities, cumulative probabilities, and interval probabilities.
Z-score: 1.5000
PDF f(x): 0.129518
CDF F(x): 0.933193
P(0 ≤ X ≤ 1.96): 0.475002
How to Use This Calculator
- Enter the normal distribution parameters μ (mean) and σ (standard deviation).
- Provide a value x to compute its z-score, PDF, and CDF.
- Set interval bounds to calculate the probability that X falls between them.
- Interpret the results to understand how observations relate to the distribution.
Formula
f(x) = (1 / (σ√(2π))) · exp(−(x − μ)² / (2σ²))
F(x) = Φ((x − μ) / σ)
Z = (x − μ) / σ
Φ denotes the standard normal cumulative distribution function and is computed via the error function approximation.
Frequently Asked Questions
How do I compute tail probabilities?
Set one bound to the point of interest and the other to a large magnitude (e.g., ±10σ) to approximate tail areas.
Can the standard deviation be zero?
No. σ must be strictly positive; otherwise the distribution degenerates to a single point at μ.
How do I work with the standard normal distribution?
Use μ = 0 and σ = 1 to operate directly on z-scores, which is typical when consulting z-tables.