Normal Distribution: P(Z ≤ 3.0) — Three Sigma
P(Z ≤ 3.0) = 0.9987, meaning only 0.13% of normally distributed values exceed three standard deviations above the mean.
How to use this tool
- Enter z-score in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your p(z ≤ z) — cumulative probability and the full breakdown beneath it.
Find the cumulative normal probability at z = 3.0 (three sigma), the threshold used in quality control to define process capability.
Frequently asked questions
- What does Φ(z) represent?
- Φ(z) is the cumulative distribution function of the standard normal distribution — the probability that a standard normal random variable is less than or equal to z. For example, Φ(1.96) ≈ 0.975, corresponding to the 95% confidence interval.