Standard Normal Distribution Table

The standard normal distribution table in all three layouts used by textbooks — cumulative, mean to z, and right tail — with one switch that converts between them. If you have ever been handed a table that "looked wrong", it was probably a different layout of these same numbers. This page ends that confusion permanently.

Use the layout switch above the table. The values underneath never change — only which area of the curve each entry reports. That is the whole secret of the three layouts.

The three layouts, side by side

Every standard normal table in print is one of these three. They store the same information with a different starting point, and each can be rebuilt from the others with one line of arithmetic. Here they are for the values students actually test with:

The 10-second layout test: look up z = 1.00 in any unfamiliar table. If the entry is 0.8413 it is cumulative; 0.3413 means mean-to-z; 0.1587 means right tail. Do not use z = 0 for this test — cumulative and right-tail tables both read 0.5000 there, so the test tells you nothing.

Converting between layouts

Call the cumulative entry A, the mean-to-z entry B and the right-tail entry C, all for the same positive z. Then A = 0.5 + B and C = 1 − A, and everything else follows:

• Area to the left of a positive z: A, or 0.5 + B, or 1 − C.
• Area to the right of a positive z: 1 − A, or 0.5 − B, or C.
• Area to the left of a negative z (look up the positive value): 1 − A, or 0.5 − B, or C.
• Area to the right of a negative z: A, or 0.5 + B, or 1 − C.
• Area between the mean and z: A − 0.5, or B, or 0.5 − C.
• Area outside ±z (both tails): 2 × (1 − A), or 1 − 2B, or 2C.

Check it at z = 1.30: the cumulative entry is 0.9032, the mean-to-z entry is 0.4032 and the tail entry is 0.0968. The area to the left is 0.9032 — and also 0.5 + 0.4032, and also 1 − 0.0968. Three routes, one answer. When an exam question uses an unfamiliar table, convert once with this list instead of relearning the method.

Which layout does your course use?

Most modern courses, AP Statistics (Table A), and all statistical software use the cumulative layout, because P(Z ≤ z) is the form probability statements are written in. Older textbooks and many engineering references print the mean-to-z layout, historically because tables were computed outward from the centre of the curve. Some hypothesis-testing texts print right-tail values because test statistics are judged by what lies beyond them. Psychology courses use the unit normal table, which is really the mean-to-z and right-tail layouts printed together with extra columns. None of them is more correct: they are three windows onto the same curve, and the switcher on this page exists so that your homework, your textbook and this site always agree.

What "standard normal" actually means

A normal distribution is fixed by two numbers: its mean μ and its standard deviation σ. The standard normal distribution is the special case μ = 0 and σ = 1 — what you get after every value has been converted to a z score with z = (X − μ) / σ. That conversion is why one printed table serves every normal problem ever set: heights, exam marks, IQ scores, temperatures and measurement errors all collapse onto this single curve, and the area under it between any two z scores is the proportion of cases between them. The total area is 1, the curve is symmetric about 0, and the tails never quite touch the axis — the technical word is asymptotic — which is why extreme z values have tiny but never quite zero probabilities.

The same table in software

Software removes the reading, not the logic — and it removes table rounding, so a computed p-value can differ from a table value in the fourth decimal place. The equivalents are: Excel =NORM.S.DIST(z, TRUE) for the cumulative area, =1-NORM.S.DIST(z, TRUE) for the right tail, =NORM.S.INV(p) to go from a probability back to z, and =STANDARDIZE(x, mean, sd) to compute a z score. In Python (SciPy) they are norm.cdf(z), norm.sf(z) and norm.ppf(p); in R, pnorm(z) and qnorm(p). The reading guide shows each formula beside the table step it replaces.

The function behind the table: Φ(z)

Underneath the grid sits one mathematical object: the cumulative distribution function of the standard normal, written Φ(z) = P(Z ≤ z). It has no closed-form formula in elementary functions — it is defined by the area under the density curve φ(z) = (1/√(2π)) · e^(−z²/2), accumulated from the far left up to z — which is precisely why tables of it had to be computed once and printed forever. Every layout on this page is a slice of Φ: cumulative prints Φ itself, mean-to-z prints Φ(z) − 0.5, and the right tail prints 1 − Φ(z). The density φ is also why the table's steps behave as they do: where the curve is tall (near 0), Φ climbs quickly, gaining about 0.004 per hundredth of z; out at z = 3 the curve is nearly flat, and a whole row of cells changes only in the fourth decimal. Knowing this shape explains the table's texture — dense information in the middle, near-silence in the tails — and why interpolation matters near the centre and hardly at all near the edges.

One table, many names — a field guide

Depending on the book in front of you, this exact table is called the z score table, the z table, the standard normal table, the standard normal distribution table, the normal probability table, the cumulative normal table, Table A (AP Statistics), or the unit normal table (psychology). Search data shows all of these in heavy use, plus descriptive variants like "z table normal curve" and "normal distribution z table" — which have been growing fastest of all recently. The names differ because each field met the table through its own doorway, but the entries do not: if the grid has z in rows and hundredths in columns, you are looking at Φ, whatever the heading says. When a name does signal a real difference, it is the unit normal printing (extra body and tail columns) — covered on its own page — and the rare "standardised score" tables of educational measurement, which re-scale z to a friendlier mean and spread and are conversions, not new mathematics.

Frequently asked questions

Prefer the split textbook view? Use the positive and negative tables. Need the answer without the table at all? The z table calculator computes any area directly.