Unit Normal Table

The unit normal table is the psychology layout of the standard normal table: for every z score it prints the proportion in the body (between the mean and z), the proportion in the tail (beyond z), and — reading across — the proportions on both sides. If your textbook is Gravetter & Wallnau or any psychology statistics text, this is your table.

How to read a cell: for z = 1.00 the cell shows 0.3413 / 0.1587 — the body holds 34.13% of the distribution and the tail holds 15.87%. The two always sum to 0.5000, because together they are exactly one half of the curve. Click any cell in the finder tables on our other pages if you also want the cumulative value.

The three proportions, explained once and properly

Psychology texts ask three shapes of question, and the unit normal table is built to answer them without subtraction. The proportion in the body is the area between the mean and your z score — "how much of the distribution lies between average and this score?" The proportion in the tail is the area beyond your z score, on the far side from the mean — "how extreme is this score; what percentage did even better (or worse)?" The third shape — everything on the mean's side of z, the cumulative proportion — is not printed, but it is one addition away: body + 0.5000. For z = 1.00 that is 0.3413 + 0.5000 = 0.8413, the familiar cumulative value.

For negative z scores, the unit normal table works by symmetry, which is why it only prints positive values. The body and tail proportions depend on the distance from the mean, not the direction: a z of −1.00 has the same body (0.3413) and the same tail size (0.1587) — but the tail is now on the left. So for z = −1.00: proportion below it is 0.1587, and proportion above it is 0.8413. Sketch the curve, mark the mean, and put the tail on the correct end: that single habit prevents nearly every unit-normal error.

Worked examples, the psychology way

Unit normal table vs z score table — same numbers, different questions

Students meeting both tables in the same week often assume one must be wrong. Neither is: the unit normal table is the standard normal table re-sliced for the questions psychology asks. The cumulative z table answers "below this score?" directly and makes you subtract for body and tail questions. The unit normal table answers "in the body?" and "in the tail?" directly and makes you add 0.5000 for cumulative questions. Our standard normal distribution table page shows the conversion formulas, and the main z score table has a layout switcher that flips between the cumulative, mean-to-z (body) and right-tail views of the same data — use whichever matches your textbook, and use the other to check your answers.

One more column appears in some printings: the ordinate — the height of the curve at z. It is used in a handful of advanced formulas (for example, selection-ratio calculations in personnel psychology). Heights are not areas, so the ordinate can never be read as a proportion; we note it here so the column does not surprise you, and we leave it out of the table above to keep body and tail unmistakable.

A note on rounding in printed unit normal tables

Printed editions round each proportion to four decimals independently, so a body and tail pair occasionally sums to 0.4999 or 0.5001 instead of exactly 0.5000 — a rounding artefact, not an error in the underlying values. This page computes both from the same function at render time, so pairs here always sum cleanly. If your textbook's pair is off by one in the last place, trust the textbook's individual entries as printed for coursework, and know the discrepancy is in the printing, not in your reading.

Frequently asked questions

Work both ways until it clicks: try the worked examples, then the practice problems — several are written in body-and-tail language on purpose.