Z-Table (Standard Normal Distribution Table)

What this table reports

This is a left-cumulative standard normal table: Φ(z) = P(Z ≤ z). It reports the area below z for Z with mean 0 and SD 1. At z = 0, the result is 0.5; at a positive z, it is greater than 0.5.

Other tables report the upper tail or the area from zero to z. Check the convention before using a number. Standardizing a nonnormal variable does not justify treating it as standard normal.

Read a positive z-score

For z = 1.23, choose row 1.2 and column 0.03. The cumulative probability is 0.8907. That answers “below 1.23.” The probability above it is 1 − 0.8907 = 0.1093.

The printable grid below gives Φ(z), rounded to four decimal places. The interactive lookup above also handles negative inputs and different regions.

Positive z grid: P(Z ≤ z)

Negative z-scores and symmetry

Use Φ(−z) = 1 − Φ(z). For z = −1.23, P(Z ≤ −1.23) = 1 − 0.8907 = 0.1093. The area above −1.23 is 0.8907.

A minus sign changes the side of the mean; it does not produce a negative probability. Avoid reading row −1.2 and adding 0.03 as if that represented −1.23: it represents −1.17 under ordinary addition. Use the negative lookup or the symmetry rule explicitly.

Choose the region before calculating

For −1 < Z < 1.5, use 0.9332 − 0.1587 ≈ 0.7745. For |Z| > 1.96, use 2(1 − 0.9750) ≈ 0.0500. Use unrounded values when available; differences in the final digit can come from subtracting rounded table entries.

Critical values and percentiles

The 95th percentile and the upper cutoff of a central 95% interval are different. The latter leaves 2.5% above the upper cutoff and 2.5% below the lower cutoff.

Worked example from an original measurement

Assume X follows a normal model with mean 100 and SD 15. For x = 130, z = (130 − 100)/15 = 2. The table gives P(X ≤ 130) ≈ 0.9772; P(X > 130) ≈ 0.0228.

For a mean of n independent observations, use the standard error σ/√n, not the SD of individual outcomes. If σ is estimated from the same sample, a mean test generally uses the t distribution.

Practice

A normal model has mean 50 and SD 8. Find the probability above 62.

Reference and related tools

NIST’s normal distribution reference defines the cumulative normal function. Read the z-score lesson, explore the normal distribution, or use the z-score calculator. Browser print includes this reference grid; interactive controls are omitted from the printed version.