Z Table (Standard Normal Table)

How to read a z table

• Write your z score to two decimals, for example 1.96. The row is the first decimal, 1.9, and the column is the second decimal, .06. Where they meet you find 0.9750. So P(Z ≤ 1.96) = 0.9750: about 97.5 percent of values lie below 1.96.
• For a negative z, use the negative table. The cell for z = −1.00 shows 0.1587, so 15.87 percent of values lie below −1. The two tables mirror each other: Φ(−z) = 1 − Φ(z). That is why textbooks print a separate negative table, and why you can get a negative value from the positive table by subtracting from 1.
• Use the switch above the table to see the same numbers as a right-tail, 0-to-z or two-tail area. These are converted by code from the same function, not copied from another table.

Left tail, right tail, 0 to z and two tails

Most reading mistakes come from using the wrong area. The cumulative table gives the left tail. The right tail is 1 minus the table value, so P(Z > 1.96) = 1 − 0.9750 = 0.0250. The area between the mean and z is the table value minus 0.5, which is 0.4750 for 1.96. A two-sided p-value doubles the smaller tail: 2 × 0.0250 = 0.0500. Some books and the NIST handbook print the 0-to-z version, so check the sketch or the first row of your table, such as 0.0000 or 0.5000 at z = 0, before you copy a value.

Worked example: a z score and its probability

Scores on a test are roughly normal with mean 100 and standard deviation 15. What share of people score below 115? First standardize with z = (x − μ) / σ = (115 − 100) / 15 = 1.00. The table gives Φ(1.00) = 0.8413, so about 84.13 percent score below 115, and 0.1587 score above it. To find the share between two values, subtract two table readings: the share between 85 and 115 is Φ(1.00) − Φ(−1.00) = 0.6827.

The 68-95-99.7 rule and critical values

The empirical rule is three table differences in disguise. Between −1 and 1 lies 0.6827, between −2 and 2 lies 0.9545, and between −3 and 3 lies 0.9973. To draw these curves yourself, use the bell curve generator. For confidence intervals you go the other way, from an area to a z. The critical values are 1.645 for 90 percent, 1.960 for 95 percent and 2.576 for 99 percent confidence; the Critical values tab lists more, and the calculator turns any probability into a z.

When to use a t table instead of a z table

Use the z table when the population standard deviation is known, when you test a proportion with a normal approximation, or as an approximation for the sample mean when the sample is large enough that the t and z values are close. When you estimate the standard deviation from a small sample of roughly normal data, use the t table with the right degrees of freedom: the statistic follows a t distribution, which has heavier tails. As the sample grows the t values approach the z values above. For counts in categories you need a chi-square table. Table values assume a normal distribution; check your data with a histogram first.