How to Read a Z Table (Left-Tail Areas, Step by Step)

What the table gives you

A standard normal table lists areas under the bell curve with mean 0 and standard deviation 1. In the most common layout, the value in the body of the table is the cumulative (left-tail) area: the proportion of the distribution at or below a given z. This is often written Φ(z) or P(Z ≤ z).

The row label gives the ones digit and the tenths digit of z. The column label gives the hundredths digit. You find the row, find the column, and read the number where they meet.

Worked example with a positive z

Find the area to the left of z = 1.96. Go down to the row labeled 1.9, then across to the column labeled 0.06. The entry is 0.9750, so Φ(1.96) = 0.9750. About 97.5% of the distribution lies at or below 1.96.

In the same way, Φ(1.00) = 0.8413 and Φ(0.50) = 0.6915.

Worked example with a negative z

Find the area to the left of z = −1.23. Use the negative half of the table: row −1.2, column 0.03. The entry is 0.1093, so Φ(−1.23) = 0.1093. A negative z always has a left-tail area below 0.5.

If your table only shows positive z, use symmetry: Φ(−z) = 1 − Φ(z). Here Φ(1.23) = 0.8907, and 1 − 0.8907 = 0.1093.

Right-tail, between-values and two-tailed areas

The table gives left-tail areas, so other areas come from subtraction. The total area under the curve is 1.

• Right-tail area: P(Z > z) = 1 − Φ(z). For z = 1.65, 1 − 0.9505 = 0.0495.
• Area between two values: P(a < Z < b) = Φ(b) − Φ(a). For a = −1.23 and b = 1.96, 0.9750 − 0.1093 = 0.8657.
• Two-tailed area beyond ±z: 2 × (1 − Φ(|z|)). For z = 1.96, 2 × (1 − 0.9750) = 0.05.
• Reverse lookup: find the area inside the table, then read the z from the row and column. An area of 0.9750 gives z = 1.96. An area of exactly 0.9500 falls halfway between Φ(1.64) = 0.9495 and Φ(1.65) = 0.9505, so the usual value is z ≈ 1.645.

Other table formats

Some textbooks print the area from 0 to z instead of the cumulative area. The NIST e-Handbook table, for example, lists the area under the curve from 0 to z. In that layout, z = 1.96 gives 0.4750, and you add 0.5 to get the cumulative area of 0.9750 for a positive z.

For a negative z in a 0-to-z table, look up |z| and subtract the result from 0.5 to get the left-tail area. Check the table title and the sketch printed above it before you start, because mixing up layouts is the most common reason an answer comes out off by 0.5.

Quick checks

Sketch the curve and shade the area you want before looking anything up. Then confirm your answer makes sense: a left-tail area for a positive z must be above 0.5, and any probability must fall between 0 and 1.