Z Score Table

Find any z value, probability or percentile in seconds — free, no sign-up. Type your z score in the finder and this z score table jumps straight to the cell, lights up the row and column in colour, and shades the area on the curve. Every table type is here: cumulative, mean to z, right tail, positive, negative and unit normal.

The table gives the cumulative probability P(Z ≤ z): the proportion of values below your z score in the standard normal distribution. Click or tap any cell to pin it, use the arrow keys to move around, or switch the layout above — many textbooks print the mean-to-z or right-tail version instead, and all three are the same table wearing different clothes.

What is a z score table?

A z score table — also called a standard normal table, a z table, or Table A in AP Statistics — lists the areas under the standard normal curve for every z score. The standard normal distribution is the bell-shaped curve with a mean of 0 and a standard deviation of 1. Because the total area under the curve is exactly 1, an area is also a proportion, a percentage and a probability at the same time, which is why one small table answers so many different questions.

A z score itself measures distance: it counts how many standard deviations a value sits from its mean. A z score of 1.96 means "1.96 standard deviations above the mean". The table converts that distance into the share of all values below it, above it, or between it and another point. Teachers set questions in all three forms, so this page gives you all three areas for every lookup instead of making you redo the arithmetic under exam pressure.

Three layouts of the same table circulate in textbooks, and mixing them up is the number one cause of wrong answers. The cumulative layout (the default here) gives the area to the left. The mean-to-z layout gives the area between the mean and z. The right-tail layout gives the area beyond z. You can tell them apart at z = 1.00: cumulative reads 0.8413, mean-to-z reads 0.3413 and right-tail reads 0.1587. The layout switcher above the table converts this page between all three instantly, and our standard normal distribution table page explains the conversion maths in full.

Three worked lookups, solved from the table

Every table type, one click away

The z values everyone looks up

These three values anchor hypothesis testing and confidence intervals. The critical z values page gives the full set by confidence level and significance level, and the z table for confidence intervals page shows the (1 + confidence level) / 2 method that produces them, so you can find the z for any level — 92%, 97%, 99.9% — not just the famous three.

The formula behind the table

The table only works on z scores, so raw values must be standardised first. An exam score of 85 in a class with mean 70 and standard deviation 10 gives z = (85 − 70) / 10 = 1.50, and the table turns that into 0.9332 — the student outscored 93.32% of the class. The z score formula page works through the population and sample versions with five examples, and how to calculate z score covers the reverse problem too: given a z score, find the original value X.

Z scores and percentiles

A cumulative table value is a percentile in disguise: multiply by 100. z = 0 is the 50th percentile, z = 1.00 is the 84.13th, z = −1.00 is the 15.87th and z = 1.96 is the 97.5th. The mini chart below covers the values students ask about most; the full z score percentile chart runs from −3.5 to +3.5 in 0.1 steps with a downloadable image.

Common mistakes this page is built to prevent

• Taking the wrong tail. A cumulative table always answers "to the left". "Scored higher than" needs 1 minus the entry. The finder prints both, side by side.
• Mixing up layouts. Cumulative rules applied to a mean-to-z table give answers off by exactly 0.5. Test any unfamiliar table at z = 1.00 before trusting it (0.8413 / 0.3413 / 0.1587).
• Using symmetry backwards. For a negative z, the left area is small. Φ(−1.25) = 0.1056, not 0.8944 — that larger number is the area to its right.
• Rounding z too early. Look up z to two decimal places. Rounding 1.0934 to 1.1 instead of 1.09 changes the answer from 0.8621 to 0.8643 — enough to flip a significance decision near a boundary.
• Treating z as a probability. A z score is a position on the axis, measured in standard deviations. The probability comes from the area at that position, which is what the table stores.
• Skipping the sketch. Nearly every error above survives because nobody drew the curve. Ten seconds with a pencil — mean in the middle, z marked, the wanted region shaded — makes the wrong tail and the off-by-0.5 layout error visible before they reach the answer line. Professional statisticians still sketch; it is not a beginner's crutch, it is the method's built-in audit.

Why there are separate positive and negative tables

Printed textbooks split the table in two because a single grid from −3.99 to +3.99 will not fit on one page. The split is a printing constraint, not a mathematical one: the normal curve is symmetric, so Φ(−z) = 1 − Φ(z), and either half can rebuild the other. Online there is no page to fit, so the full table lives above — but the dedicated positive and negative pages keep the textbook layout for students whose courses still teach it that way, each with its own worked examples.

A short history of this table

The standard normal table is older than the statistics course that teaches it. Abraham de Moivre derived the normal curve in 1733 as an approximation to the binomial distribution, working out by hand the areas that printers would later tabulate. Through the 1800s, astronomers and surveyors — Gauss and Laplace among them — made the curve the mathematics of measurement error, and Adolphe Quetelet showed that human measurements, heights first among them, follow the same shape. By the time printed tables of the cumulative function appeared in the early twentieth century, the z table had become the working tool of every science that measures people or instruments. Karl Pearson's biometric tables and later textbook printings fixed the layout students still meet today: rows for the first decimal, columns for the second. The genuinely recent change is not the numbers — they have been stable for a century — but the medium: a printed table forces you to the nearest hundredth, while this page's finder also evaluates the values in between, which is why z = 1.645 returns exactly 0.9500 here instead of a choice between two neighbouring cells.

Using this table in exams and coursework

Most exam boards hand you a printed version of this table — cumulative in AP Statistics (where it is literally called Table A), mean-to-z in some older syllabuses, body-and-tail in psychology. Three habits transfer directly from this page to the paper one. First, identify the layout before the exam, not during it: one look at the z = 1.00 entry (0.8413, 0.3413 or 0.1587) tells you which of the three you hold, and the conversion between them is two lines of arithmetic. Second, sketch the curve on your rough paper for every question and shade the region being asked about — markers award method marks for a correct setup even when a later digit slips. Third, learn the five anchor cells by heart (0 → 0.5000, 1.645 → 0.9500, 1.96 → 0.9750, 2.576 → 0.9950, and the symmetry rule Φ(−z) = 1 − Φ(z)): with those, you can reconstruct or sanity-check almost any answer in the paper, including spotting when you have read the wrong row under time pressure.

What the table cannot do

Honesty about limits is part of using a tool well. The table assumes your data follows a normal distribution after standardisation: for strongly skewed or bimodal data, the probabilities it returns describe a curve your data does not have. It stops being precise beyond ±3.99, where software takes over (though by then the remaining tail is under 0.003% anyway). It gives probabilities for z scores, not for raw values — the standardising step with z = (X − μ) / σ is always yours to do first. And it cannot choose your tail, your alpha, or your hypothesis for you: those are judgement calls that belong before the lookup, and the critical values page explains why fixing them in advance is what makes the lookup mean anything at all.

One number, three readings

Every cell in this table can be read three ways, and fluency is switching between them without noticing. As a proportion, 0.9750 says 0.975 of the distribution lies below z = 1.96. As a percentage, it says 97.50% of values. As a probability, it says a randomly drawn value has probability 0.9750 of falling below 1.96 — which is also, read as a percentile, the 97.5th. Questions in courses rotate through all four costumes ("what proportion…", "what percentage…", "what is the probability that…", "what percentile…") while asking for the same cell, and candidates lose time translating instead of answering. The finder prints the left area, right area, mean-to-z area and percentile together precisely so the equivalence stops being a translation exercise and becomes a glance.

Frequently asked questions

Keep learning

New to the topic? Start with what is a z score, then how to read a z score table for the full step-by-step method. Ready to test yourself? The worked examples and 20 practice problems with answers cover every question shape — left, right, between, and reading the table backwards.