Φ(z) for z from 0.00 to 3.49
f(z) for z from 0.00 to 3.49 (symmetric: f(−z) = f(z))
How to use these tables
Standard Normal Distribution
The standard Normal distribution has mean μ = 0 and variance σ² = 1. Its density function is:
\( f(z) = \dfrac{1}{\sqrt{2\pi}}\,e^{-z^2/2} \)
And its cumulative distribution function Φ(z) = P(Z ≤ z) has no closed-form expression, so it is looked up in tables.
Φ(z) table — Cumulative distribution
Each cell shows the probability that a standard Normal variable is less than or equal to z. For negative z, remember that Φ(−z) = 1 − Φ(z), or look up the z < 0 tab directly.
- Φ(0) = 0.5000 (the mean splits the distribution into two equal halves).
- Φ(1.96) ≈ 0.9750 → 95% of values fall between −1.96 and 1.96.
- Φ(2.576) ≈ 0.9950 → 99% fall between −2.576 and 2.576.
f(z) table — Density function
Each cell shows the value of the density at the point z. Remember that density is not a point probability; it must be integrated to obtain probabilities. It is symmetric: f(−z) = f(z).
Key percentiles of the standard Normal
- z₀.₉₀ = 1.282 → P(Z ≤ 1.282) = 0.90
- z₀.₉₅ = 1.645 → P(Z ≤ 1.645) = 0.95
- z₀.₉₇₅ = 1.960 → P(Z ≤ 1.960) = 0.975 (two-sided 95%)
- z₀.₉₉ = 2.326 → P(Z ≤ 2.326) = 0.99
- z₀.₉₉₅ = 2.576 → P(Z ≤ 2.576) = 0.995 (two-sided 99%)
Worked example: finding a critical Z value
To build a two-sided 95% interval you need the value that leaves 97.5% accumulated in the standard normal. In the normal table, Φ(z)=0.975 corresponds approximately to z=1.96; for 99%, the cumulative 0.995 corresponds approximately to z=2.576.