Z-Table Calculator

What is a z-table calculator?

A z-table calculator finds the area under the standard normal curve for a z-score, or the z-score for a given area. For example, z = 1.25 gives Φ(z) = 0.8944, so 89.44% of values fall below it and 10.56% above it. Use it both ways, with a shaded curve and the full table to check by hand.

Here, Φ(z) is the cumulative area to the left of z — the proportion of the standard normal distribution below that point. A z-score measures how many standard deviations a value sits from the mean, so the table turns any standardised value into a probability or a percentile. Printed tables list Φ(z) to four decimal places, with rows for the first decimal place and columns for the second. This page adds what a printed table cannot: a two-way lookup (z to area and area to z), a shaded bell curve, and a full clickable table with the cell highlighted for every calculation. If you are learning the printed table by hand, our guide to how to read a z-table walks through the same lookup step by step.

How do I read a z-table?

Reading a z-table takes four steps. First, write your z-score to two decimal places. Second, find the row for the first decimal place. Third, find the column for the second decimal place. Fourth, read the cell where the row and column meet — that number is Φ(z), the area to the left.

1. Split the z-score. For z = 1.25, the row is 1.2 and the column is 0.05.
2. Find the row. In the positive table, look down the first column until you reach 1.2.
3. Find the column. Move across to the column headed 0.05.
4. Read the cell. The cell reads 0.8944, so Φ(1.25) = 0.8944 and 89.44% of the distribution lies below z = 1.25.

For a negative z, use the negative table, or use the symmetry rule Φ(−z) = 1 − Φ(z). The calculator above highlights the exact cell for you, so you can check a hand lookup in seconds. If you are starting from a raw value rather than a z-score, work it out first with our z-score calculator, and the normal distribution calculator handles distributions with any mean and standard deviation.

How do I find a z-score from an area?

To find a z-score from an area, use the inverse lookup: pick the tail your area describes, then find the z whose cumulative area equals it. Choose “area → z” in the calculator above, enter the area, and choose the tail. For a left-tail area of 0.975, z = 1.96; for 0.95, z = 1.645; for 0.90, z = 1.282; and for 0.99, z = 2.326. Those four values cover most confidence intervals and critical values you will meet. If the area is a right-tail probability, the calculator flips it first (a right-tail area of 0.025 also gives z = 1.96). For the reverse of a probability, you can also turn a probability into a z-score with our dedicated tool, or use the p-value calculator from a z-score when you are testing a hypothesis.

What is the area between two z-scores?

The area between two z-scores is the difference of their cumulative areas: P(a < Z < b) = Φ(b) − Φ(a). Between z = −1 and z = 1 it is Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827, which is the 68% of the 68–95–99.7 rule. Between −2 and 2 it is 0.9545, and between −3 and 3 it is 0.9973. Choose “Between” in the calculator, enter the positive z value, and the tool shades the middle of the curve and gives the same result. The middle area is also what left- and right-tail probabilities leave over: for z = 1, the two tails together are 1 − 0.6827 = 0.3173. Read more about the curve itself in our article on the standard normal distribution.

Z-table formula

The z-table is the cumulative distribution function (CDF) of the standard normal distribution:

Φ(z) = (1 / √(2π)) ∫_−∞^z e^−t²/2 dt

There is no closed-form answer, so the calculator evaluates the integral through the error function, erf, and the table stores the result to four decimal places. The inverse lookup, z = Φ⁻¹(p), finds the z whose cumulative area is p. The percentile is simply Φ(z) × 100. If you already have a z-score and only need the percentile, the z-score to percentile converter does that step on its own.

Worked examples

Example 1: z = 1.25

Row 1.2, column 0.05 → Φ(1.25) = 0.8944. The right-tail area is 1 − 0.8944 = 0.1056, and the percentile is the 89.44th. The area between −1.25 and 1.25 is 0.8944 − 0.1056 = 0.7887.

Example 2: z = −1.25

Row −1.2, column 0.05 in the negative table → Φ(−1.25) = 0.1056. By symmetry, that is 1 − Φ(1.25). Only 10.56% of values fall below z = −1.25, so it sits at the 10.56th percentile.

Example 3: area 0.975, left tail

Φ⁻¹(0.975) = 1.96. This is the upper critical value of a 95% confidence interval: 95% of the area lies between −1.96 and 1.96, with 2.5% in each tail.

Example 4: between −1 and 1

Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827. About 68.27% of a normal distribution falls within one standard deviation of the mean.

Common z values to remember

z	Φ(z)	Percentile	Right-tail area
−1.96	0.0250	2.50th	0.9750
0	0.5000	50th	0.5000
1.00	0.8413	84.13th	0.1587
1.645	0.9500	95th	0.0500
1.96	0.9750	97.50th	0.0250
2.326	0.9900	99th	0.0100
2.576	0.9950	99.50th	0.0050

These are the values most exams and confidence intervals reuse. Click any cell in the full table above to see the same lookup for that z-score.

When should I use a z-table instead of a calculator?

Use the printed table when an exam requires it, or when you want to check that you understand the lookup. Use the calculator when you need a value the table does not list — a z-score with more than two decimals, an area such as 0.973, or a two-way lookup in either direction. The tool is also the faster route inside a larger analysis, for example when a one-proportion z-test calculator has already given you a z-score and you need its tail area.

Z-table calculator FAQs

What is a z-table?

A z-table lists Φ(z), the area to the left of each z-score under the standard normal curve, to four decimal places. For example, z = 1.25 gives Φ(z) = 0.8944, and z = 0 gives 0.5000.

How do I find the p-value for z = 1.96?

For z = 1.96, Φ(z) = 0.9750, so the one-tailed p-value is 1 − 0.9750 = 0.0250 and the two-tailed p-value is 2 × 0.0250 = 0.0500.

What z-score is the 95th percentile?

The 95th percentile is z = 1.645, because Φ(1.645) = 0.95. The 97.5th percentile is z = 1.96 and the 99th percentile is z = 2.326.

What is the area between z = −1 and z = 1?

The area between z = −1 and z = 1 is 0.6827, or 68.27%. That is the 68% of the empirical rule: Φ(1) − Φ(−1) = 0.8413 − 0.1587 = 0.6827.

How do I use a negative z-table?

A negative z-table lists Φ(z) for z below zero. By symmetry, Φ(−z) = 1 − Φ(z), so Φ(−1.25) = 1 − 0.8944 = 0.1056. This page has a negative table built in.

What is the difference between a z-table and a t-table?

A z-table is for the standard normal distribution, used when the population standard deviation is known or the sample is large. A t-table is for the t-distribution and needs degrees of freedom; it has heavier tails, so its critical values are larger.

Can a z-score be greater than 3?

Yes. A z-score can be any value, but areas beyond z = 3 are tiny: Φ(3) = 0.9987, so only 0.13% of the area lies above 3. Printed tables usually stop at 3.4 or 3.49; this calculator works beyond that.

Is this z-table calculator free?

Yes. This z-table calculator is free, with no sign-up and no download. You can use it for homework, exams and research as often as you like.

Keep going: you can browse all the calculator tools on one page, or read how to calculate a z-score and what a p-value is. The standard normal values on this page follow the NIST/SEMATECH e-Handbook of Statistical Methods.