How to Read a Z Score Table

To read a z score table: find the row for your z score's whole number and first decimal, find the column for its second decimal, and read the value where they meet. That value is the proportion of the distribution to the left of your z score. Everything else — right tails, between values, percentiles, p-values, reading the table backwards — is built from that one move, and this guide works through all of it with real numbers.

Practise on this page: type 1.09 in the finder and watch the table jump to row 1.0, column 0.09, cell 0.8621 — the lookup this guide now explains by hand.

The basic lookup, step by step

1. Round your z score to two decimal places if it has more (1.0934 becomes 1.09 — round once, at the end of your calculation, never mid-way through it).
2. Split it. The part up to the first decimal (1.0) chooses the row; the second decimal (0.09) chooses the column.
3. Run your eye along the row, down the column. Where they cross for z = 1.09 sits 0.8621.
4. Say what the number means before you use it: 86.21% of the standard normal distribution lies to the left of z = 1.09. Naming the area out loud is the habit that stops wrong-tail errors, because the sentence immediately sounds wrong when it is.

The four question shapes (and the move each one needs)

1. Area to the left — read it straight off

"What proportion scored below…?", "P(Z ≤ z)?", "what is the percentile?" — one lookup, no arithmetic. Below z = 1.09: 0.8621, the 86.21st percentile. Roughly five of every six homework questions are secretly this shape in different words.

2. Area to the right — subtract from 1

"Scored above", "greater than", "exceeds" — the table never stores this directly in cumulative layout, so compute 1 − entry. Above z = 1.09: 1 − 0.8621 = 0.1379. Notice the same pair of numbers (0.8621, 0.1379) keeps reappearing in this guide: left and right are complements, and fluency is mostly recognising which of the pair a question wants.

3. Area between two z scores — two lookups and a subtraction

Read both ends cumulatively and subtract the smaller from the larger. Between z = −0.50 and z = +1.00: 0.8413 − 0.3085 = 0.5328, so 53.28% of values lie in that band. The rule is sign-proof: it works for two positives, two negatives, or one of each, because the cumulative table accumulates from the far left and the difference isolates exactly the slice between. The one combination that tempts a different method — two negatives on a positive-only table — is worked in Example 3 below.

4. Area between the mean and z — subtract 0.5 (or read the other layout)

Between the mean (z = 0, cumulative 0.5000) and z = 1.09: 0.8621 − 0.5000 = 0.3621. Textbooks that print the mean-to-z layout store 0.3621 directly, and psychology's unit normal table calls the same number the "proportion in the body". Three names, one slice of curve. Our standard normal table page converts between the layouts with a switch, and the unit normal table prints body and tail side by side.

Reading the table backwards (probability → z score)

Half of real statistics runs the table in reverse: you are given an area and need the z that produces it — for percentiles, cutoffs and critical values. Find the area inside the body of the table, then read the row and column labels of the closest cell. The 95th percentile: hunt for 0.9500 in the body; it sits at row 1.6, column 0.045 — between the printed 1.64 (0.9495) and 1.65 (0.9505) — and the exact z is 1.645. Two habits matter here. First, when the target area falls between cells, say so and interpolate or report the exact value if your course allows software: pretending 0.9495 "is" 0.9500 misstates the cutoff. Second, for areas below 0.5000 the z you find is negative — the bottom 5% is z = −1.645, the mirror of the top. The finder's probability box at the top of this page does reverse lookups exactly, which makes it the right way to check your manual reads.

Worked examples

Seven mistakes that cost marks

1. Taking the wrong tail. A cumulative table always answers "to the left". If the question asks "higher than", the extra step (1 − entry) is not optional.
2. Mixing up layouts. Cumulative rules on a mean-to-z table produce answers off by exactly 0.5. Identify any new table at z = 1.00 first: 0.8413 cumulative, 0.3413 mean-to-z, 0.1587 tail.
3. Mishandling negative z scores. There is no negative area. On a positive-only table, look up the absolute value, then decide — from a sketch — whether the question wants the entry or its complement.
4. Doubling wrongly for two-tailed p. Find the single tail beyond |z|, then double that. Doubling the cumulative entry of z = 2.17 gives 1.97, and any "probability" above 1 announces its own error.
5. Rounding z too early. Look z up at two decimals. Rounding 1.0934 to 1.1 instead of 1.09 changes 0.8621 into 0.8643 — small, and exactly the size that flips borderline significance calls.
6. Reading the column as the row. The second decimal lives in the columns (0.00–0.09). Reading 1.09 as row 1.0 column "9" works; reading it as row 1.09 does not exist, and row 1.9 column 0.00 gives 0.9713 — a different answer that looks just as plausible.
7. Treating z as a probability. z is a position measured in standard deviations. Probabilities come only from areas at that position. If an answer sentence says "the probability is 1.09", stop.

Interpolation: when your value falls between cells

The grid steps in hundredths; life does not. A z of 1.645 falls between the 1.64 cell (0.9495) and the 1.65 cell (0.9505). Linear interpolation takes the proportional share of the gap: 1.645 is halfway, so the value is halfway — 0.9500. The method is sound because Φ is nearly straight over a single hundredth in most of the table; it is least reliable in the far tails, where the curve bends and the cells barely change anyway (so the error is tiny in absolute terms). Examination boards differ: many accept the nearest cell, some expect interpolation for critical values, and software-based courses skip both. Know your board's rule before the exam. On this site you never need it — the finder evaluates the function itself — but the printed table in your exam hall will not, which is why the skill is taught here rather than assumed.

The same lookups in Excel, Python and R

Software returns the cumulative area directly: Excel =NORM.S.DIST(1.09, TRUE) gives 0.8621, and =1-NORM.S.DIST(1.09, TRUE) gives the right tail. Reverse: =NORM.S.INV(0.95) gives 1.645. Two-tailed p: =2*(1-NORM.S.DIST(ABS(z), TRUE)). Python (SciPy): norm.cdf(z), norm.sf(z), norm.ppf(p). R: pnorm(z), qnorm(p). And to standardise a raw value in Excel: =STANDARDIZE(x, mean, sd). The logic is identical to the table; only the reading is automated, and computed values can differ from the table in the fourth decimal because nothing was rounded on the way.

Checking someone else's lookup

Marking, tutoring and group work all involve auditing a lookup you did not perform, and there is a fast order to it. First, the bounds: is the claimed probability inside 0 and 1, and — for a "left of" claim — on the correct side of 0.5 for the sign of z? Second, the anchors: does the claimed value sit sensibly between the landmark cells (a claim of 0.93 for z = 1.20 fails instantly, because Φ(1.00) = 0.8413 and Φ(1.50) = 0.9332 bracket it — 0.93 belongs near 1.48, not 1.20)? Third, the tail: was the question's region the one actually computed? Most wrong lookups pass the first check and die on the second or third, which is why the order matters: bounds, anchors, tail.

Frequently asked questions

Ready to apply it? Work the ten graded examples, then sit the practice set — every question shape from this guide appears in both.