How to Find Probability from a Z Score
Every probability question about a z score is one of four shapes — left, right, between, outside — and each shape is one table lookup or one subtraction away from the answer. Learn to name the shape first and the arithmetic takes care of itself.
The four moves: Left of z: read Φ(z) directly. Right of z: 1 − Φ(z). Between z₁ and z₂: Φ(z₂) − Φ(z₁). Outside ±z: 2 × (1 − Φ(|z|)). The finder below performs all four and shades the region so the shape is visible.
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
Each shape, worked
- Left — P(Z < 0.60). Straight from the cumulative table: Φ(0.60) = 0.7257. About 72.6% of values fall below z = 0.60. No arithmetic at all — the table is a book of left probabilities.
- Right — P(Z > 1.10). The table never lists right tails in cumulative layout, so subtract from the whole: 1 − Φ(1.10) = 1 − 0.8643 = 0.1357. (If your course's table is the right-tail layout, this is the direct read instead — know which book you hold.)
- Between — P(−0.50 < Z < 1.50). Two cumulative reads and a subtraction: Φ(1.50) − Φ(−0.50) = 0.9332 − 0.3085 = 0.6247. The region between the two values holds about 62.5% of the distribution. The finder's between-two solver prints exactly these steps.
- Outside — P(|Z| > 1.50). Both tails together: 2 × (1 − 0.9332) = 0.1336. Symmetry does the doubling; this is also the shape of a two-tailed p-value, which is why the p-value calculator is this same move in formal dress.
Raw scores first, then shapes
Questions usually arrive in raw units — "the probability a candidate scores above 82, given μ = 70, σ = 8". Standardise first (z = (82 − 70) / 8 = 1.50), then name the shape (right), then move: 1 − 0.9332 = 0.0668. Two habits keep this reliable: always sketch or visualise the region (the shaded curve above exists for this), and always state the answer as a probability and a percentage — 0.0668, about 6.7% — because a decimal that "feels" wrong gets caught, while a bare number slips through. The area language behind all of this is developed on the area under the normal curve page.
Also asked as: how to calculate probability from z score · how do you find the probability of a z score · how to calculate z score probability · how to find probability in a table · how to find p score from z score (that last one is the p-value version — see the p-value calculator linked above).
Related questions
Frequently asked questions
What is the probability that z is greater than 0?
Exactly 0.5. The mean splits the curve into two equal halves — the anchor fact from which every right-tail subtraction (1 − Φ(z)) ultimately borrows its confidence.
My probability came out negative — what went wrong?
The between-shape subtraction ran backwards: it must be the larger cumulative minus the smaller, Φ(z₂) − Φ(z₁) with z₂ > z₁. Probabilities live in [0, 1]; a negative result or one above 1 is the arithmetic asking to be re-read.