Find Z Score from Probability
Given an area, a percentile or a tail probability, find the z score that produces it. This is the reverse lookup — the single most useful z table skill after reading the table forwards — with the tool, the by-hand method, and the cutoff problems it solves.
The slider starts at 1.96 — the most-used value in statistics: it marks the edge of the middle 95% of values.
Values generated from the standard normal cumulative distribution · Verified November 2026
The by-hand method
Printed tables are organised z-first, so reversing one is a search. Take the probability you are given and hunt for it (or its nearest neighbours) inside the body of the table — the four-decimal cells, not the row and column headers. When you find the cell, read its coordinates back out: row label plus column label is the z score. Looking for the z that leaves 0.975 to its left: the cell 0.9750 sits at row 1.9, column .06, giving z = 1.96. When the target falls between two cells — 0.95 sits between 0.9495 (z = 1.64) and 0.9505 (z = 1.65) — the honest answers are "between 1.64 and 1.65", the rounded convention 1.645, or the exact value from the reverse box above, 1.6449. Exams accept the convention; software wants the exact value.
Two adjustments cover most real problems before the search even starts. If your probability is a right-tail area, convert it: left area = 1 − right area, then search (a right tail of 0.05 means searching for 0.95, giving z = 1.645). If your probability is a central area — "the middle 90%" — split the remainder: (1 − 0.90) / 2 = 0.05 in each tail, so the boundaries are the z values for 0.05 and 0.95, namely ±1.645. That splitting step is the entire secret of confidence-interval z values, which is why the z table for confidence intervals page can be so short.
The problems this solves
- Cutoff scores. "Only the top 10% pass" → find z for 0.90 → 1.282 → raw cutoff = μ + 1.282σ. On a test with μ = 100, σ = 15, the pass mark is 119.2.
- Quartiles and percentiles. The 25th percentile is the z for 0.25: −0.674. Any "the value below which p% fall" question is this lookup.
- Critical values. A two-tailed test at α = 0.05 needs the z leaving 0.025 in each tail: search 0.975 → 1.96. The full standard set is tabulated on the critical z values page.
- Design targets. Manufacturers run it in reverse: "at most 1% may fall below the strength limit" sets the z for 0.01, −2.326, and therefore where the process mean must sit relative to σ.
Worked problem — the bottom 10% line
A course reserves a support programme for the bottom 10% of scores (μ = 500, σ = 100). Bottom 10% means a left area of 0.10, so search the negative table body for 0.1000: the nearest cell is 0.1003 at row −1.2, column .08 — z ≈ −1.28 (exactly −1.2816). Raw cutoff = 500 + (−1.2816 × 100) = 371.8, so scores of 372 and below qualify. The negative sign is the whole point of the exercise: a below-average region must produce a negative z, and checking that it did is the fastest error screen in statistics.
Frequently asked questions
What z score corresponds to a probability of 0.975?
z = 1.96 — the most famous reverse lookup in statistics. It is the upper boundary of the middle 95% of the distribution and the two-tailed 5% critical value.
How do I find z when the probability is not in the table?
Take the two bracketing cells and either report the midpoint convention (0.95 → 1.645, between 1.64 and 1.65) or interpolate. The reverse box above computes the exact value directly: for 0.95 it returns 1.6449.
What if my probability is greater than 0.5 but I need a negative z?
Then the probability you were given is probably a right-tail area or a "greater than" statement. Convert first: the z with 0.90 of the area to its right has 0.10 to its left, giving z = −1.282. Always restate the problem as a left area before searching — the table only speaks cumulative.
Is this the same as the percentile to z conversion?
Yes, with one translation step: a percentile is a probability × 100. The percentile to z score calculator page is this same lookup dressed in percentile language, with the quartile landmarks listed.