Statistics Reference
How to Read a Z-Table
A z-table condenses the entire standard normal distribution into a grid, and reading it takes exactly three moves: split the z-score, find the cell, and decide what the number means for your question. This guide walks through each move with real lookups from the z-table on this site, then covers reverse lookups and the mistakes that flip answers.
What the Table Contains
Every cell of a cumulative z-table answers one fixed question: what fraction of a standard normal distribution lies below this z-score? Rows carry the z-score's sign, ones digit, and tenths digit; columns add the hundredths digit; the cell where they meet holds P(Z < z). Because the table is cumulative from the left, values run from near 0 (far left tail) through 0.5000 at z = 0.00 to near 1 (far right tail).
One number, three equivalent readings: the cell for z = 1.00 reads 0.8413, which means 84.13% of values fall below one standard deviation above the mean, a random value has probability 0.8413 of landing there, and z = 1.00 sits at the 84th percentile.
The Three-Step Lookup
Example: find P(Z < 1.96)
- Split the z-score after the tenths digit. 1.96 becomes 1.9 (row) and 0.06 (column).
- Find the cell. Run down the left edge to the row labeled 1.9, then across to the column headed 0.06.
- Read the value. The cell reads 0.9750: 97.5% of the distribution lies below z = 1.96.
Negative z-scores work identically on the negative half of the table: for z = −1.25, the row is −1.2, the column is 0.05, and the cell reads 0.1056 — about 10.6% of values fall below −1.25. Note that the column always contributes the hundredths digit only; a z of −1.25 does not use a “−0.05” column.
Turning the Table Value Into Your Answer
The table always hands you a left-tail area. Most questions need one of three quick conversions:
| Question | Rule | Worked lookup |
|---|---|---|
| Below z (left tail) | table value | P(Z < 1.5) = 0.9332 |
| Above z (right tail) | 1 − table value | P(Z > 1.5) = 1 − 0.9332 = 0.0668 |
| Between two z-scores | larger − smaller | P(−1 < Z < 1) = 0.8413 − 0.1587 = 0.6826 |
The between-values result is the familiar 68% of the empirical rule — reading it straight off the table (0.6826 rather than the exact 0.6827) shows the rounding that two-decimal tables introduce.
A Complete Example: From Raw Score to Percentile
Table lookups usually start one step earlier, with a raw value that must be standardized first.
Exam scores are normal with mean 70 and SD 10. What percentile is a score of 85?
- Standardize: z = (85 − 70) / 10 = 1.50.
- Look up: row 1.5, column 0.00 → 0.9332.
- Interpret: a score of 85 beats 93.32% of scores — roughly the 93rd percentile. The right-tail reading says 6.68% of students scored higher.
The z-score calculator performs the standardizing step and the lookup together if you want to check your arithmetic.
Reading the Table in Reverse
Percentile questions run the lookup backwards: given an area, find the z. Scan the body of the table for the target probability, then read the z-score off the headers. For the 90th percentile, the nearest cell to 0.9000 is 0.8997, sitting in row 1.2, column 0.08 — so z ≈ 1.28. The next cell over (0.9015 at z = 1.29) brackets the true value, which is why finer sources quote 1.2816.
Reverse lookups are where the standard critical values come from: 0.9500 falls midway between z = 1.64 and 1.65 (hence the quoted 1.645), 0.9750 lands exactly at z = 1.96, and 0.9901 at z = 2.33. For exact critical values at any level, the critical value calculator computes the inverse directly.
The Mistakes That Flip Answers
- Wrong table convention. A cumulative table reads 0.9332 at z = 1.5; a “from the mean” table reads 0.4332 for the same z. If your positive-z values are all below 0.5, you are holding the second kind — add 0.5 to convert.
- Forgetting the complement. “What percentage scored above?” needs 1 − the table value. An answer of 0.9332 to a top-tail question should feel wrong on sight: more than 93% cannot be above an above-average score.
- Standardizing with the wrong spread. Individual values standardize with σ, but sample means standardize with the standard error σ/√n. Using the wrong one shifts z dramatically.
- Sign slips on negative z. P(Z < −2) = 0.0228, not 0.9772. A quick sanity check: negative z-scores always have left-tail areas below 0.5.
Try the Z-Table (Standard Normal Distribution)
Look up cumulative probabilities for any z-score from −3.49 to 3.49 in the full printable table, with worked examples and common critical values.
Frequently Asked Questions
What does the number inside a z-table actually mean?
Each cell is a cumulative probability: the fraction of a standard normal distribution that lies to the LEFT of that z-score. The cell for z = 1.96 reads 0.9750, meaning 97.5% of values fall below 1.96 and only 2.5% above it. Equivalently, the cell is the z-score's percentile rank written as a decimal.
How do I read a negative z-score in the table?
Use the negative half of the table the same way: row for the ones and tenths digits, column for the hundredths. The value for z = -1.25 is 0.1056, the area below -1.25. If your table only prints positive z-scores, use symmetry: P(Z < -z) = 1 - P(Z < z), so P(Z < -1.25) = 1 - 0.8944 = 0.1056.
How do I find the area to the right of a z-score?
Subtract the table value from 1, because the table gives the area to the left and the two areas must sum to 1. For z = 1.5 the table reads 0.9332, so the right-tail area is 1 - 0.9332 = 0.0668. Right-tail areas are what one-sided hypothesis tests and 'top X percent' questions need.
What if my z-score has more than two decimal places?
A printed table resolves z to two decimals, so either round (z = 1.647 to 1.65) or interpolate between the neighboring cells for a closer answer. The error from rounding is at most a few thousandths of probability. When that matters, skip the table and compute the exact value with a normal distribution calculator, which evaluates the same function without the rounding.
How do I use the z-table in reverse to find a z-score from a percentage?
Search the body of the table for the probability and read the z-score off the row and column headers. For the 90th percentile, the closest cell to 0.9000 is 0.8997 at z = 1.28 (the next cell, 0.9015, sits at 1.29). This reverse lookup is how the standard critical values arise: 1.645 for 95% one-sided, 1.96 for 95% two-sided, 2.576 for 99% two-sided.
Are all z-tables cumulative from the left?
No, and this is the most common source of wrong answers. Most modern tables are cumulative (left-tail), but some textbooks print 'cumulative from the mean' tables showing the area between 0 and z (those cells never exceed 0.5), and a few print right-tail tables. Check the table's picture or header first: for positive z, a left-tail table reads above 0.5, a from-the-mean table reads below 0.5.