Z-Score to Percentile Calculator

To convert a z-score to a percentile, find the area under the standard normal curve to the left of z and multiply by 100. This z-score calculator does that and more. It finds a z-score from a value, mean and standard deviation. It turns z into left-tail, right-tail, middle and two-tailed probabilities, and finds the probability between two values. In reverse, it gives the z-score (and raw value) for any percentile or confidence level. Every answer is shaded on a bell curve, and a full z-table follows below.

How to use the z-score calculator

1. Pick what to find. Use the first menu for a percentile from z, z from a value, a probability between two values, or the inverse direction.
2. Enter the numbers. Enter z directly, or enter x, the mean μ and the standard deviation σ. For plain z-scores in the “between” and inverse modes, leave μ = 0 and σ = 1.
3. Choose the area. Options are left (the percentile), right, middle, two-tailed, or between and outside a range.
4. Read the result and the curve. The shaded region is the probability. The dashed lines mark your z-values.

Z-score formula: how to find a z-score

A z-score standardizes a value: z = (x − μ) ÷ σ. OpenStax puts it this way: the z-score tells you how many standard deviations the value x is above or below the mean. Positive z-scores lie above the mean and negative ones below it. OpenStax’s example is a normal distribution with mean 5 and standard deviation 6, where x = 17 gives z = (17 − 5) ÷ 6 = 2, two standard deviations above the mean. Standardizing lets you compare values from different scales, like an SAT score and an ACT score, on one common curve.

If you have raw data rather than a known mean and SD, calculate them first with the average and standard deviation calculator. Its deviations table also lists the z-score of every value in your data.

Z-score to percentile

The percentile is the cumulative probability Φ(z) × 100, the share of the standard normal distribution below z. OpenStax writes this as P(X < x), the shaded area to the left of x. For a continuous distribution, P(X < x) equals P(X ≤ x), so “below” and “at or below” give the same answer. Some anchors: z = 0 is the 50th percentile, z = 1 is about the 84th, z = 2 about the 97.7th, and z = −1 about the 16th.

Z-score probability: left, right, between and two-tailed

Every normal probability is an area under the curve, and the total area is 1. The four common questions are:

• Left tail, P(Z < z) = Φ(z): the percentile.
• Right tail, P(Z > z) = 1 − Φ(z): the complement, as in OpenStax’s P(X > x) = 1 − P(X < x).
• Between, P(a < Z < b) = Φ(b) − Φ(a). With a = −z and b = z this is the middle area, the basis of a confidence level.
• Two-tailed, P(|Z| > |z|) = 2 × (1 − Φ(|z|)): the area beyond ±z in both tails, used for two-sided p-values.

Worked example from OpenStax §6.2: exam scores are normal with mean 63 and SD 5. What share score above 65? z = (65 − 63) ÷ 5 = 0.4 and P(Z > 0.4) = 0.3446, matching the textbook’s 0.3446. So about 34% of students score above 65. For the share between 60 and 70, z₁ = −0.6 and z₂ = 1.4, giving Φ(1.4) − Φ(−0.6) = 0.6450.

Percentile to z-score (the inverse)

Going backwards means finding the z whose left-tail area equals the percentile. On a TI calculator this is invNorm, and OpenStax uses it to find the 90th percentile of the exam scores: k = 69.4. The calculator gives z = 1.2816, so x = 63 + 1.2816 × 5 = 69.41. Choose “central area” to get the ±z cut-offs for a confidence level, or “two-tailed area” to enter a significance level α directly.

The empirical rule (68–95–99.7)

For normally distributed data, OpenStax states that about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. The exact areas are 68.27%, 95.45% and 99.73%. The rule is a quick sanity check: a z-score beyond ±3 is rare under a normal model, about 2.7 values in 1,000. In real data, it is often the first sign of a measurement error or a distribution that is not normal.

Z-table (standard normal table)

The table gives Φ(z), the area to the left of z, to four decimals. Find the row for the first decimal of z, then the column for the second decimal. For z = 1.96, go to row 1.9 and column .06 to read 0.9750. The NIST e-Handbook table lists the area from 0 to z instead. To convert, add 0.5 for positive z (NIST’s own example: 0.5 + 0.43699 = 0.93699 for z = 1.53). For negative z, use symmetry: Φ(−z) = 1 − Φ(z).

How the calculation works and how accurate it is

The standard normal CDF has no closed form, so every calculator approximates it. The NIST handbook gives the density and cumulative distribution but notes that the CDF must be computed numerically. Here Φ(z) = ½·erfc(−z/√2). Near the centre, erf is summed from a power series whose terms are all positive, which avoids cancellation. In the tails, a continued fraction for erfc is used, so tiny tail areas like P(Z > 8) keep full relative precision instead of rounding to zero. Compared with a reference implementation across z from −10 to 10 in steps of 0.001, the largest error is below 10⁻¹⁵, far inside the 10⁻⁷ needed for any printed table.

We also checked all 410 entries of the NIST table (z = 0.00 to 4.09). All but one match to NIST’s five decimals. At z = 1.36 the true area is 0.4130850…, which NIST prints as 0.41308 and we round to 0.41309. The inverse uses bisection on the same function, so it is accurate to about 15 digits.

When a z-score percentile is (and isn’t) valid

A z-score percentile assumes the values follow a normal distribution. That is reasonable for many physical measurements, standardized test scales built to be bell-shaped, and averages of large samples. It is a poor fit for skewed data such as incomes, reaction times or waiting times. Always look at the data first. A box plot quickly shows skew and outliers. If the model doesn’t fit, rank the value directly against the data with the percentile rank calculator, which needs no distribution assumption.

Common mistakes

• Using a right-tail table as if it were cumulative. Tables differ. Check whether yours shows the area to the left, the area from 0 to z (like NIST’s), or the upper tail.
• Forgetting to double for two tails. A two-sided p-value is twice the single-tail area.
• Mixing up sample and population SD. A z-score uses the population σ. If you only have a sample, s is an estimate. For small samples, t-procedures with n − 1 degrees of freedom are the right tool.
• Reading a percentile as percent correct. The 84th percentile means a score higher than 84% of the reference group, not 84% of the questions right.

Frequently asked questions

Sources & method

• NIST/SEMATECH e-Handbook §1.3.6.7.1 — Cumulative distribution function of the standard normal distribution (z table)
• OpenStax Introductory Statistics 2e §6.1 — The standard normal distribution (z-score formula, empirical rule)
• OpenStax Introductory Statistics 2e §6.2 — Using the normal distribution (areas to the left and right, percentiles, invNorm)
• NIST/SEMATECH e-Handbook §1.3.6.6.1 — Normal distribution (probability density and cumulative distribution functions)

Results are estimates for general information. Found an error? It helps everyone — see our methodology.