What is a z-score and how does it relate to a percentile?

A z-score measures how many standard deviations a value sits above or below the mean of a standard normal distribution (μ = 0, σ = 1). The percentile is the cumulative probability Φ(z) — the fraction of the distribution that lies at or below that z-score. For example, z = 1.6449 corresponds to the 95th percentile: exactly 95% of observations in a normal distribution fall below that point.

What is the formula for converting a z-score to a percentile?

The percentile is Φ(z) = (1/2)[1 + erf(z / √2)], where erf is the Gauss error function. Because erf has no closed form, this calculator uses the Horner-form rational approximation from Abramowitz & Stegun §26.2.17, which achieves a maximum absolute error below 7.5 × 10⁻⁸ — well beyond what any practical application requires.

How do I find the z-score for a given percentile (the inverse problem)?

Switch to "Percentile → Z-score" mode and enter the desired percentile. The calculator applies the Beasley–Springer–Moro probit algorithm, which inverts the normal CDF to 5 × 10⁻⁸ accuracy. For example, entering 97.5 returns z ≈ 1.9600 — the familiar 5% two-tail critical value used in hypothesis testing.

What is the difference between a one-tail and a two-tail p-value?

A one-tail (left-tail) p-value is simply Φ(z) — the area to the left of your z-score. The right-tail p-value is 1 − Φ(z). The two-tail p-value is 2 × min(Φ(z), 1 − Φ(z)), which measures how extreme the result is without specifying a direction. In hypothesis testing, z = 1.96 gives a two-tail p-value of exactly 0.05 (5%).

Why does the 68-95-99.7 rule come from z-scores?

The rule states that roughly 68% of a normal distribution lies within ±1σ of the mean (z between −1 and +1), 95% within ±2σ (z ≈ ±1.96), and 99.7% within ±3σ. These are just the "between −|z| and +|z|" values from the CDF: Φ(1) − Φ(−1) ≈ 0.6827; Φ(1.96) − Φ(−1.96) ≈ 0.9500; Φ(3) − Φ(−3) ≈ 0.9973. The calculator reports this symmetric coverage in the secondary stats.

Can I use this for non-standard normal distributions?

Yes — with a small extra step. If your data has mean μ and standard deviation σ, first convert your raw value x to a z-score using z = (x − μ) / σ, then enter that z-score here. The percentile result is the same because any normal distribution is just a rescaled standard normal. The companion z-score calculator on this site can do the standardisation for you.

Z-Score to Percentile Calculator

The z-score to percentile calculator converts a standard normal z-score to its cumulative probability (percentile rank) — and works in reverse: enter any percentile between 0 and 100 to get the exact z-score that cuts off that tail. The tool also reports the left-tail area, right-tail area, two-tail p-value, and the symmetric central coverage, all alongside a live bell-curve chart that shades the region for you. Every calculation runs in your browser; nothing is sent to a server.

What a z-score and a percentile actually measure

A z-score expresses a value’s distance from the mean in standard-deviation units. A z-score of 0 is exactly average; z = 1 is one standard deviation above the mean; z = −1.96 is 1.96 standard deviations below it. The percentile (or cumulative probability) answers the question: what fraction of observations in a normal distribution lie below this value?

The link between them is the standard normal cumulative distribution function (CDF):

Φ(z) = (1/2) [1 + erf(z / √2)]

where erf is the Gauss error function. Because there is no elementary closed form, the calculator uses the five-term Horner rational approximation from Abramowitz & Stegun §26.2.17 (Hart, 1968), which delivers a maximum absolute error below 7.5 × 10⁻⁸ — accurate to eight decimal places in all practical cases.

How the inverse (percentile → z-score) is computed

The inverse of Φ is called the probit function, written Φ⁻¹(p) or z(p). The calculator uses the Beasley–Springer–Moro algorithm, a three-region rational approximation that achieves accuracy of 5 × 10⁻⁸ across the full range (0, 1). The three regions are:

Central region (0.02425 ≤ p ≤ 0.97575): a 6th-degree rational function of the centred variable (p − 0.5)
Tail regions (p < 0.02425 or p > 0.97575): a 6th-degree rational function of √(−2 ln p), exploiting the known log-tail behaviour of the normal distribution

For the most common critical values — z = 1.6449 for the 5% one-tail test, z = 1.96 for the 5% two-tail test, z = 2.5758 for the 1% two-tail test — the probit algorithm agrees with published tables to all four decimal places.

Worked example

Suppose a student scores 730 on an exam with mean μ = 620 and standard deviation σ = 90. Their z-score is (730 − 620) / 90 = 1.2222. Entering z = 1.2222 into this calculator gives:

Metric	Value
Percentile	88.89%
Area below	0.88893
Area above (right tail)	0.11107
Two-tail p-value	0.22213
Symmetric ±1.22 coverage	77.79%

So the student outperformed about 89% of test-takers. In a hypothesis-testing context, a test statistic of z = 1.22 would give a two-tail p-value of 0.222, which does not reach the conventional α = 0.05 threshold.

For the reverse: a teacher wants to find the z-score that marks the top 10% of students (the 90th percentile). Entering 90 in “Percentile → Z-score” mode returns z ≈ 1.2816, matching the value in every standard statistical table.

Key critical values at a glance

The five most important z-scores for hypothesis testing:

Two-tail α	z (one-tail)	z (two-tail)
0.20	±1.2816	±1.2816
0.10	±1.6449	±1.6449
0.05	±1.9600	±1.9600
0.02	±2.3263	±2.3263
0.01	±2.5758	±2.5758

The full table of 13 commonly used critical values — from the 0.05th to the 99.95th percentile — is accessible inside the calculator under “Common critical values”.