This normal distribution calculator converts a z-score into cumulative probability, upper-tail probability, and density.
Statistics
Normal Distribution Calculator
Compute Z-score, density and tail probabilities for a normal variable.
Calculator
About this calculator
Formula notes
Φ(z) is the area under the standard normal curve to the left of z; the upper tail is 1 − Φ(z); the density is φ(z) = e^(−z²/2)/√(2π).
Worked examples
- Example: z = 1.96 gives Φ ≈ 0.975 and an upper tail of 0.025 — the classic 95% two-sided split.
- For mean 100 and standard deviation 15, a value of 130 has z = 2 and an upper-tail probability of about 2.28% under a normal model.
How to use it
- Enter a z-score (or convert your value to z first).
- Read the cumulative probability Φ(z).
- Read the upper tail for one-sided questions.
Frequently asked questions
Is Φ(1.96) really 0.975?
Yes. ±1.96 leaves 2.5% in each tail, which is why 1.96 defines the 95% interval.
What is the difference between density and probability?
The density φ(z) is curve height, not a probability; probabilities come from areas under the curve.
Limits and interpretation
- Normal probabilities are meaningful only when a normal model is appropriate for the variable and parameters.
- Rounded tail probabilities lose relative precision in extreme tails, where specialist numerical methods may be required.