Z table (standard normal table)

How to read a z table

Each table gives Φ(z), the area to the left of z under the standard normal curve (mean 0, standard deviation 1). The z-score is split in two: the row is z to one decimal place and the column is the second decimal place.

Worked lookup for z = 1.53:

1. Split 1.53 into 1.5 and .03.
2. In the positive table, go down to row 1.5.
3. Go across to column .03. The cell reads 0.9370 (highlighted below).
4. So 93.70% of the area lies to the left of z = 1.53, and 1 − 0.9370 = 0.0630 lies to the right.

For z = −1.53, use the negative table: row −1.5, column .03, which gives 0.0630. The two answers add to 1 because the curve is symmetric about 0.

Negative z table

Area to the left of z for z from −3.49 to −0.00. The row is z to one decimal; read the column as the second decimal, so row −1.5, column .03 is z = −1.53.

Positive z table

Area to the left of z for z from 0.00 to 3.49. Row 1.5, column .03 is z = 1.53, highlighted. Beyond 3.49 the area to the left rounds to 0.9998 or more; the lookup box handles any z.

Left, right and between areas

The table gives only the area to the left. The other two areas follow from it:

left of z: Φ(z) right of z: 1 − Φ(z) between z₁ and z₂: Φ(z₂) − Φ(z₁)

Between example: the area between z = −1 and z = 1.5 is Φ(1.5) − Φ(−1) = 0.9332 − 0.1587 = 0.7745, so about 77% of a normal distribution lies in that range. For a symmetric range, the area between −z and z is 2Φ(z) − 1; for z = 1.53 that is 2 × 0.9370 − 1 = 0.8740.

From a score to a percentile

IQ scores are normal with mean 100 and standard deviation 15. What percentile is a score of 123?

1. Standardise the score: z = (x − μ) / σ = (123 − 100) / 15 = 1.533, which rounds to 1.53 for the table.
2. Look up z = 1.53: row 1.5, column .03 gives 0.9370.
3. Convert to a percentile: 0.9370 × 100 = 93.7, so a score of 123 is at about the 94th percentile.

Rounding z to two decimals costs a little precision: the unrounded z = 1.5333 gives 0.9374. The z-score calculator does the first step for you, and the lookup box above takes z to as many decimals as you like.

Common z values

A few z-scores come up so often that they are worth knowing without the table. They are the critical values for one-sided and two-sided tests and the multipliers for confidence intervals.

The 68–95–99.7 rule comes from the same table. The area within one standard deviation of the mean is Φ(1) − Φ(−1) = 0.6827, within two it is 0.9545 and within three it is 0.9973. The empirical rule calculator turns these into actual ranges for your mean and standard deviation. To go from an area to a cut-off on your own scale, use the inverse normal distribution calculator.

The tables use the standard normal cumulative distribution function described in the NIST/SEMATECH e-Handbook of Statistical Methods (section 1.3.6.7.1), evaluated to better than 1 part in 10 million and rounded to 4 decimal places.