Standard Normal Distribution Table (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.

How to read a row and column

First choose the tab for the result you need: Φ(z) for a cumulative probability or f(z) for the height of the density. Then select z ≥ 0 or z < 0. In both tables, the first column gives the integer part and first decimal place of z, while the top header gives its second decimal place. The cell where the two meet is the result.

For example, to read z = 1.96, find row 1.9 and column .06. In the cumulative table, their intersection shows Φ(1.96) = 0.9750: 97.50% of the area lies to the left of 1.96. For z = −1.96, select the negative table and cross row −1.9 with column .06; the result is Φ(−1.96) = 0.0250.

In the f(z) tab, combine the row and column in the same way, but the cell represents a density rather than an area or probability. You can click any cell and the page will confirm the complete z and its value. If you start with a probability and need z, find the closest value in the body of the Φ(z) table and reconstruct z by combining its row label with its column header.

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.

Related tables and calculators

• Binomial table
• Normal distribution calculator
• Student's t table
• Student's t distribution calculator
• CI for the mean (known σ)

Frequently asked questions

• How do you read the standard normal table? Find the value of z: the integer part and first decimal in the row, and the second decimal in the column. The cell is Φ(z) = P(Z ≤ z), the cumulative probability.
• What is Φ(z) in the normal distribution table? It is the cumulative distribution function: the area under the normal curve to the left of z, that is, the probability P(Z ≤ z).
• How do you find the z value from a probability? Look for the probability inside the normal table and read the z from its row and column. For example, 0.975 corresponds to z ≈ 1.96, the value used in 95% intervals.