Type a z-score to get the area to its left, right, between two values or in both tails, or type a probability to find the matching z. The bell curve shades the area, and the full positive and negative z-table below is clickable, printable and downloadable.
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Quick picks
P(Z < 1.96)0.9750
97.50% of the area
Full z-table
Read the row for the first decimal and the column for the second: z = 1.96 is row 1.9, column .06. Click any cell to use that z above.
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How to use the Z-Table (Standard Normal Table)
Choose what to find: a probability from a z-score, or a z-score from a probability.
Pick the area (left of z, right of z, between two z-scores, both tails or from the mean) and type the z-score(s).
Press Calculate (results also update as you type) to see the area, the shaded curve and the table cell to read.
For a reverse lookup type a probability like 0.95 or 95% and choose percentile, right tail, central or two tails.
Use the full table below: switch between negative and positive z, click a cell to use it, or print / download it.
What the z-table shows
A z-table (standard normal table) lists areas under the standard normal curve, the bell curve with mean 0 and standard deviation 1. The usual cumulative table gives Φ(z) = P(Z < z), the share of the distribution below z. Any normal value x can be looked up after standardising it: z = (x − μ) / σ.
Left of z: Φ(z), read straight from the table.
Right of z: 1 − Φ(z).
Between a and b: Φ(b) − Φ(a).
Both tails outside ±z: 2 × (1 − Φ(|z|)), the two-tailed p-value.
Symmetry: Φ(−z) = 1 − Φ(z), which is why some books print only the positive half.
Worked example
Exam scores are normal with μ = 70 and σ = 8. What share of students score between 62 and 86? Standardise both: z₁ = (62 − 70) / 8 = −1.00 and z₂ = (86 − 70) / 8 = 2.00. From the table, Φ(2.00) = 0.9772 and Φ(−1.00) = 0.1587, so the share is 0.9772 − 0.1587 = 0.8185, about 82% of students. Choose “Between z₁ and z₂” above with −1 and 2 to see the shaded area.
Critical z-values worth memorising
Confidence level
Two-tailed α
Central critical z
One-tailed z (same α)
80%
0.20
±1.282
0.842
90%
0.10
±1.645
1.282
95%
0.05
±1.960
1.645
98%
0.02
±2.326
2.054
99%
0.01
±2.576
2.326
99.9%
0.001
±3.291
3.090
Tips for table lookups
Round z to two decimals before using a printed table; if your z falls halfway between cells, averaging the two neighbours is a good approximation.
When an exact probability isn't in the table during a reverse lookup, take the closest cell. For 0.95 the cells 0.9495 (z = 1.64) and 0.9505 (z = 1.65) are equally close, which is where 1.645 comes from.
Beyond z ≈ 3.5 the table rounds to 0.9998–1.0000; use the calculator above for small tail areas such as P(Z > 4) = 3.17 × 10⁻⁵.
Check which kind of table your course uses: cumulative from the left (most common), from the mean (0 to z) or right tail. The table menu above switches between all three.
Frequently asked questions
How do you read a z-table?
Split the z-score into its first decimal and its second decimal. Find the row for the first part (for z = 1.96, row 1.9) and the column for the second part (column .06). The number where they meet, 0.9750, is the area to the left of z in a cumulative table.
What is the difference between the positive and negative z-table?
They are two halves of the same cumulative table. The negative table lists Φ(z) for z below 0, so all its values are under 0.5; the positive table lists z of 0 and above, so its values are 0.5 or more. Because the curve is symmetric, Φ(−z) = 1 − Φ(z).
How do I find the area to the right of a z-score?
Subtract the table value from 1: P(Z > z) = 1 − Φ(z). For z = 1.5, Φ(1.5) = 0.9332, so the right-tail area is 0.0668. Choose “Right of z” above and the tool does this for you.
How do I find the probability between two z-scores?
Look up both values and subtract the smaller from the larger: P(a < Z < b) = Φ(b) − Φ(a). For example, P(−1 < Z < 1) = 0.8413 − 0.1587 = 0.6827, the familiar 68% within one standard deviation.
How do I find a z-score from a percentile or probability?
Do a reverse lookup: find the area closest to your probability inside the table and read its row and column. The 95th percentile is about z = 1.645, and the central 95% lies between z = −1.96 and +1.96. This tool computes the exact inverse instead of the nearest cell.
What is a 0 to z (from the mean) table?
Some textbooks print the area between the mean and z instead of the area to the left. That value equals Φ(z) − 0.5 for positive z, so 1.96 gives 0.4750 rather than 0.9750. Pick “From the mean, 0 to z” in the table menu to match that style.
How accurate are these values?
The cumulative probabilities are computed directly with a series expansion and, in the far tails, a continued fraction, so they are accurate to many more decimals than the 4 shown in printed tables. Table cells are rounded to 4 decimals the usual way.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.