📈 Normal Distribution Calculator
Compute normal and standard normal probabilities without spreadsheets. Enter mean, standard deviation, and the value or interval of interest to obtain the PDF, CDF, tail probabilities, and z-scores. Inverse CDF and a random sample generator are included for Monte Carlo checks.
Distribution Parameters
μ: mean • σ: standard deviation (> 0)
σ must be positive. σ = 1 gives the standard normal distribution.
Computes z-score, PDF( x ), and CDF(x).
Returns P(a < X < b) if both bounds are provided.
Find x such that P(X ≤ x) = p.
Instant Results
- Z-score ( (x − μ) / σ )
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- PDF at x
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- CDF at x
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- P(a < X < b)
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- Upper tail P(X > t)
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- Lower tail P(X < t)
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- Quantile x for P(X ≤ x) = p
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Random Sample Generator
Uses Box–Muller transform. Limited to 1000 draws for performance.
How to Use
- Set μ and σ; defaults μ = 0, σ = 1 recover the standard normal distribution.
- Enter the evaluation point x to obtain density, cumulative probability, and z-score values instantly.
- Provide optional interval bounds to compute P(a < X < b), or enter t in the tail box and choose upper/lower tail.
- Supply a probability between 0 and 1 to retrieve the corresponding quantile (inverse CDF). Use the sampler to generate synthetic data for simulations.
- Adjust parameters as needed; warnings will highlight invalid inputs such as non-positive σ or out-of-range probabilities.
Formula References
- Abramowitz, M., & Stegun, I. A. (1972). Handbook of Mathematical Functions. NIST.
- Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
Explore More Distributions
- Binomial Distribution Calculator
- Poisson Distribution Calculator (coming soon)
- Probability Simulators (coin, dice, Monty Hall)