What Is Probability Z Score?
A z-score says how many standard deviations a value lies from the mean. For normally distributed data, each z-score matches a probability, which is the area under the bell curve.
Enter a z-score directly, or enter a raw score with the mean and standard deviation and the tool computes z first. You get the left-tail probability, the percentile, the right-tail probability, the area between −|z| and |z|, the two-tailed probability and how rare the value is.
Below are the formulas, a worked IQ example and answers on percentiles and p-values.
How to Use the Probability Z Score Calculator
- Choose z or raw score.
- Enter the values.
- Read left, right, between and two-tail probabilities.
- Check the shaded curve.
Probability Z Score Formula
Z-score
Left-tail probability
Probability Z Score Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| z | Standardised score | |
| x | Raw score | |
| μ | Mean | |
| σ | Standard deviation |
Probability Z Score Worked Example
An IQ of 115 with mean 100 and SD 15 gives z = (115 − 100) ÷ 15 = 1.0. Then P(Z < 1) = 0.8413, so the score is at the 84.13th percentile and 15.87 % of people score higher.
With the example values loaded in the calculator above, the steps are:
- Use z = 1.96 directly.
- Left tail P(Z < z) = Φ(1.96) = 0.9750021.
- Right tail P(Z > z) = 1 − 0.9750021 = 0.024997895.
- Between −|z| and |z|: 2Φ(1.96) − 1 = 0.95000421. Two tails: 0.04999579.
Result: z-score = 1.96; P(Z < z) left tail = 0.9750021; P(Z > z) right tail = 0.024997895.
How to Read a z-Score Probability
A z-score says how many standard deviations a value sits from the mean. An IQ of 115 with mean 100 and SD 15 gives z = 1.0. The left-tail probability is 0.8413, so the score is at the 84.13th percentile and 15.87% of people score higher.
The two-tailed probability counts both extremes: P(|Z| > 1) = 0.3173. Use the left or right tail for a one-sided question and the two-tailed value when direction does not matter. The normal model assumes a roughly bell-shaped distribution, so check that first.
Where Probability Z Score Is Used
Percentiles
Convert a z-score or a raw score into a percentile.
Test scores
Find the share of people scoring above or below a result.
Hypothesis tests
Get the tail probability for a z statistic.
Rare values
See how unusual a value is by its one-tail probability.
Common Probability Z Score Mistakes to Avoid
- Left tail means "less than". Right tail means "greater than". Pick the one that matches the wording.
- Standardise a raw score with z = (x − μ) ÷ σ before you look up a probability.
- The result is only as good as the normal assumption.
- Use the t distribution instead when σ is estimated from a small sample.
Probability Z Score Terms and Definitions
- Percentile
- The percentage of the distribution below a value.
Probability Z Score FAQs
How do you find probability from a z-score?
Look up the area under the standard normal curve to the left of z. For z = 1.0 it is 0.8413. The calculator does this directly.
How do I get a right-tail probability?
Subtract the left-tail value from 1. For z = 1.0, the right tail is 1 − 0.8413 = 0.1587.
What is the two-tailed probability?
It is the chance of being at least |z| from the mean in either direction. For |z| = 1 that is 2 × 0.1587 = 0.3173.
How do I turn a raw score into a probability?
First compute z = (x − μ) ÷ σ, then find the area to the left of z. For x = 115, μ = 100 and σ = 15, z = 1.0 and the probability is 0.8413.