Z-Table: Negative and Positive Standard Normal Table

How to read the z-table

Split your z-score into two parts. The row is the z-score to one decimal, the column is the second decimal. The cell where they meet is Φ(z), the probability that a standard normal value is less than z.

For z = 1.53, go to row 1.5 and column 0.03: 0.9370. For z = −1.53, go to row −1.5 in the negative table and the same column 0.03: 0.0630. The two add up to 1, because the curve is symmetric around 0. That is also why one half of the table is enough in a pinch: Φ(−z) = 1 − Φ(z).

Your value is not a z-score yet? Convert it first with z = (x − μ) ÷ σ, where μ is the mean and σ the standard deviation. The z-score calculator does the conversion and the lookup in one step.

Negative z-table (z from −3.49 to 0)

Areas to the left of negative z-scores, all below 0.5. Row −1.2 with column 0.05 is z = −1.25. Swipe the table sideways on a phone; the first column stays in place.

Positive z-table (z from 0 to 3.49)

Areas to the left of positive z-scores, all 0.5 or more. Row 1.9 with column 0.06 is z = 1.96.

Worked examples

All five use one made-up case: a machine fills bottles with a mean of 500 ml and a standard deviation of 4 ml, and the fill amounts follow a normal distribution.

Left tail: how many bottles are under 494 ml?

z = (494 − 500) ÷ 4 = −1.50. Row −1.5, column 0.00 of the negative table: 0.0668. About 6.68% of bottles hold less than 494 ml.

Right tail: how many are over 507 ml?

z = (507 − 500) ÷ 4 = 1.75. The table gives the area to the left, 0.9599, so the area to the right is 1 − 0.9599 = 0.0401, or 4.01%. Same answer the other way round: Φ(−1.75) = 0.0401.

Between two values: from 497 to 505 ml

Convert both ends: (497 − 500) ÷ 4 = −0.75 and (505 − 500) ÷ 4 = 1.25. Look up both and subtract the smaller area from the larger: 0.8944 − 0.2266 = 0.6678. About 66.8% of bottles land in that range. Subtracting rounded table values can leave the last digit off by one; unrounded, the answer is 0.6677.

Backwards: the fill level only 5% of bottles fall below

Now you know the area and need z. Search the body of the negative table for the value closest to 0.0500. Row −1.6 has 0.0505 in column 0.04 and 0.0495 in column 0.05, so z sits halfway, at about −1.645. The NIST/SEMATECH handbook lists the same critical value, −1.645 for p = 0.05. Back in milliliters: 500 − 1.645 × 4 = 493.42 ml.

Two tails: a p-value for z = 2.05

A test statistic of z = 2.05 counts as extreme on either side, so double the tail: 2 × Φ(−2.05) = 2 × 0.0202 = 0.0404. That is below 0.05, so the result is significant at the 5% level in a two-sided test. The A/B test significance calculator runs this two-proportion z-test for conversion rates, computes z from your visitor counts and gives the p-value without a table.

Three kinds of z-tables

Tables in textbooks and handbooks do not all show the same area. Before you trust a number, check which one you are holding.

The NIST/SEMATECH e-Handbook of Statistical Methods uses the from-the-mean layout. Its own example: for z = 1.53 the table shows 0.43699, and adding 0.5 gives 0.93699, the same 0.9370 as above with one more decimal. A quick test for any table: the cell for z = 0 reads 0.0000 in a from-the-mean table and 0.5000 in the other two, and the cell for z = 1 tells those two apart.

Numbers worth knowing

Subtracting the left areas of −z and z gives the share within z standard deviations of the mean:

That is the 68-95-99.7 rule. The NIST handbook also lists the critical values most significance tests use. Each one cuts off the stated area in one tail:

For the negative side, flip the sign: a left area of 0.025 belongs to z = −1.960.

Between table values and beyond the edges

A z-score with three decimals falls between two columns. Linear interpolation is close enough for most work. For z = 1.534, take Φ(1.53) = 0.9370 and Φ(1.54) = 0.9382, and go 0.4 of the way from one to the other: 0.9370 + 0.4 × 0.0012 = 0.9375. The exact value is 0.93749, so the interpolated result matches to four decimals. Rounding z to 1.53 instead and reading the cell directly gives 0.9370, off by about five ten-thousandths, which is fine for homework and not for a calculation you build on.

Past ±3.49 the table runs out of digits: Φ(−3.49) already rounds to 0.0002. Real tails keep shrinking fast. Φ(−4) is 3.167 × 10⁻⁵, and Φ(−5) is 2.867 × 10⁻⁷. For those, use the z-score calculator, which keeps its precision far out in the tails instead of rounding to zero.

From your data to a z-score

The table only covers the standard normal distribution, with mean 0 and standard deviation 1. Any normal distribution maps onto it once you standardize, which is why the NIST handbook tells you to subtract the mean and divide by the standard deviation before looking anything up. If you start from raw numbers, get the mean with the average calculator and the standard deviation with the standard deviation calculator. Use the population version, which divides by n, when the numbers are the whole group, and the sample version, which divides by n − 1, when they are a sample from something larger.

Then check the shape. The table assumes a bell curve. If your data is skewed or has two peaks, a z-score is still a valid distance from the mean, but the percentile read from the table can be far from the real share of your data below that point.

How this table was computed

The normal CDF has no simple closed formula; as the NIST handbook puts it, it is computed numerically. Every cell here comes from Φ(z) = ½ × (1 + erf(z ÷ √2)), with the error function from W. J. Cody's rational approximations, published in 1969 and distributed as CALERF in the SPECFUN library on netlib, which the netlib listing says approximate erf and erfc to at least 18 significant digits. The z-score calculator uses the same code. Each value was rounded once, from the full double-precision result to four decimals.

We then compared all 700 cells from −3.49 to 3.49 with the five-decimal NIST/SEMATECH table. Every cell agrees within 0.00001. If you round NIST's five-decimal numbers to four yourself, 33 of the 700 cells come out one ten-thousandth higher than here. That is double rounding, not an error in either table. For z = 2.00, NIST shows 0.47725, so 0.97725, which rounds up to 0.9773; the unrounded value is 0.9772499, which rounds to 0.9772.

Frequently asked questions

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.7.1: Cumulative Distribution Function of the Standard Normal Distribution (area from 0 to z, reading examples, critical values)
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1: Normal Distribution (density, standard form, and why the CDF is computed numerically)
• CALERF, SPECFUN library on netlib: W. J. Cody's rational approximations of the error function (Mathematics of Computation, 1969), used for every value in the tables

Written by Tobias Kurth, designer. Updated October 20, 2026.