Z-Table (Standard Normal Table)

How to Use the Z-Table (Standard Normal Table)

1. 1. Pick the table type. The cumulative table — the area to the left of z — is the standard; the right-tail and mean-to-z versions match the tables some textbooks print.
2. 2. Look up a z-score. Type it in the first box: its cell is highlighted, and all three areas appear to five decimals.
3. 3. Or find z for an area. Type an area such as 0.975 in the second box to get the z that cuts it off, with no hunting through the table.
4. 4. Print it. The print button prints only the tables, without the rest of the page.

How to Read a Z-Table

Split the z-score into its first decimal and its second. The row gives the units and tenths, the column gives the hundredths, and the cell where they cross is the area. For z = 1.96, go down to the row 1.9 and across to the column .06: the cumulative table reads 0.9750, so 97.5% of the standard normal distribution lies below 1.96.

Negative z-scores have a table of their own, in which row −1.9 and column .06 mean z = −1.96: the hundredths move further from zero. By symmetry Φ(−z) = 1 − Φ(z), so the area below −1.96 is 0.0250.

The fourth row shows what four-decimal rounding costs: the exact area within one standard deviation is 0.6827, and the lookup box gives every area to five decimals. Any normal value becomes a table lookup once you standardise it with z = (x − μ) / σ — the Z-Score Calculator does that step, and the Normal Distribution Calculator skips the table altogether.

Left-Tail, Right-Tail and Mean-to-Z Tables

Textbooks print three versions of the same information, and reading the wrong one is the most common z-table mistake. Check the shaded picture or the heading on your table before using it.

• Cumulative (left-tail). The area from −∞ up to z, Φ(z). It is what most modern textbooks, AP Statistics and statistical software use, and it runs from 0 to 1.
• Right-tail. The area beyond z, 1 − Φ(z), which is the one-tailed p-value for z read directly. P(Z > 2.33) = 0.0099.
• Mean to z. The area between 0 and z, Φ(z) − 0.5, printed for z ≥ 0 only and never above 0.5. For z = 1.50 it reads 0.4332; add 0.5 to get the cumulative area.

Frequently Asked Questions

How do you read a z-table?

Find the row for the units and first decimal of z (such as 1.9) and the column for the second decimal (.06); the cell where they meet is the area. In a cumulative table that area is P(Z < z): 0.9750 for z = 1.96.

How do I find the area to the right of a z-score?

Subtract the cumulative value from 1: P(Z > z) = 1 − Φ(z). For z = 1.28 that is 1 − 0.8997 = 0.1003. Or switch the table above to "Area to the right of z" and read the answer directly.

How do I find a z-score from a probability?

Look for the area inside the body of the table, then read off its row and column. When it falls between two cells, take the closer one or interpolate: 0.95 lies between z = 1.64 and 1.65, giving the familiar 1.645. The "Find z for an area" box does this exactly — 1.6449.

What is the z-score for a 95% confidence level?

1.96. A 95% interval leaves 2.5% in each tail, so you need the z with a cumulative area of 0.975 — row 1.9, column .06. For 90% confidence it is 1.645, and for 99% it is 2.576.

What about z-scores beyond ±3.49?

The table stops there because so little area remains: P(Z < −3.49) is only 0.0002. Type a larger z into the lookup box for its exact area — P(Z > 4) = 0.00003, for instance.

Why does my textbook's z-table look different?

It is probably a right-tail or mean-to-z table rather than a cumulative one, or it lists only positive z. All of them describe the same curve: for positive z the cumulative area is 0.5 plus the mean-to-z area, and it is always 1 minus the right-tail area.