The z table

A z table (a standard normal table) lists Φ(z), the area under the standard normal curve to the left of z: the probability that a standard normal value is at most z. Rows give z to one decimal place and columns add the second decimal, so Φ(1.96) is in row 1.9, column 0.06, and reads 0.9750. The table above is computed, not copied, and the full printable table is further down the page.

Every normal distribution is the standard one, stretched and shifted, so this one table serves them all. To use it for a value x from a normal population with mean μ and standard deviation σ, first convert to a z-score, z = (x − μ)/σ (the z-score calculator does this and shows the steps), then look up z^[1]. established

How to read it, with examples§

The table gives one kind of area, the area to the left. Every other area comes from it with a subtraction, because the total area under the curve is 1 and the curve is symmetric about 0.

1. Area to the left. P(Z ≤ 1.28): row 1.2, column 0.08, which reads 0.8997. About 90 % of a normal population lies below 1.28 standard deviations above the mean.
2. Area to the right. P(Z > 1.64) = 1 − Φ(1.64) = 1 − 0.9495 = 0.0505.
3. Area between two values. P(−1 ≤ Z ≤ 1) = Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6826, the “68 %” of the 68–95–99.7 rule. (The exact value is 0.6827; subtracting two rounded table entries can be off by one in the last digit.)
4. A negative z. Use the negative half: Φ(−0.52) is in row −0.5, column 0.02, and reads 0.3015. Or use symmetry with the positive half: Φ(−0.52) = 1 − Φ(0.52) = 1 − 0.6985.
5. Backwards, from an area to z. To find the 97.5th percentile, look in the body of the table for 0.9750: it sits at row 1.9, column 0.06, so z = 1.96. The 95th percentile falls between 0.9495 (1.64) and 0.9505 (1.65), so z ≈ 1.645.
6. A non-standard normal. On a test with mean 100 and standard deviation 15, a score of 130 has z = (130 − 100)/15 = 2.00, and Φ(2.00) = 0.9772: the score is at about the 98th percentile, and 2.28 % of test takers score higher.

The critical values people look up most§

Confidence intervals and significance tests use a handful of z values over and over. Two-sided values leave half of the remaining area in each tail; one-sided values leave all of it in one tail^[2].

For a mean estimated from a small sample, replace these with Student’s t critical values, which are larger; Chapter 5 explains why and by how much. established

Which area does your table give?§

Printed z tables come in three kinds, and most mistakes come from using one as if it were another. This page, like most modern textbooks, gives the cumulative area to the left of z, Φ(z). Some books print the right-tail area, 1 − Φ(z), and older ones often print the area between the mean and z, Φ(z) − 0.5, which runs from 0 to just under 0.5. A quick check: in a cumulative table the entry for z = 0 is 0.5000 and entries grow toward 1; in a right-tail table they shrink toward 0; in a mean-to-z table the entry for 0 is 0.0000.

The full table§

Each entry is Φ(z) to four decimal places. For negative z the column is subtracted: −1.96 is row −1.9, column 0.06, and its entry 0.0250 equals 1 − Φ(1.96) by symmetry. Beyond ±3.49 the areas round to 0.0000 or 1.0000 at four decimals; the figure at the top gives them to full precision, and the chapter on the normal distribution explains how they are computed^[3].