Z Score Table: Positive and Negative

How to read a z table

Split your z value at the hundredths place. Find the ones-and-tenths part (for example 1.9) down the left column, then the hundredths part (0.06) across the top row. The cell where they meet is P(Z ≤ z). For z = 1.96 that cell reads 0.9750, so about 97.5 percent of a standard normal distribution lies at or below 1.96.

Which convention this table uses, and why it matters

Three incompatible z tables circulate, and they report different numbers for the same z. A cumulative-from-left table reports P(Z ≤ z), the area to the left. A "0-to-z" or from-the-mean table reports P(0 ≤ Z ≤ z), which runs from 0 up to about 0.5. A right-tail table reports P(Z ≥ z), the area to the right. Landing on the wrong one produces a wrong answer.

This page uses cumulative-from-left, P(Z ≤ z). It is the convention that statistical software uses by default, including scipy.stats.norm.cdf, R's pnorm, and Excel's NORM.S.DIST(z, TRUE). To convert a value here to the other two forms: for the from-the-mean area subtract 0.5, and for the right-tail area subtract the value from 1.

Negative z values

A negative z is a first-class input, not an error. It sits below the mean, so its area to the left is less than 0.5. The negative table gives these directly. Because the curve is symmetric, P(Z ≤ -z) = 1 - P(Z ≤ z): the value for -1.96 (0.0250) and the value for 1.96 (0.9750) add to exactly 1.

Reading confidence intervals from the table

The critical z values above are the ones a confidence interval or a two-sided test uses. A 95 percent interval uses z = 1.96, because 0.9750 of the area sits to its left and 0.0250 sits in the upper tail, leaving 0.95 in the middle. A 90 percent interval uses 1.645, and a 99 percent interval uses 2.576. The one-sided alpha column is the single-tail significance level that each critical value cuts off.