How is a Z-score converted to a percentile?

The percentile equals the cumulative area under the standard normal curve to the left of the Z-score, computed with the normal CDF, then multiplied by 100. A Z of 0 gives the 50th percentile.

What formula does this use for the normal CDF?

It uses an accurate rational approximation of the error function (erf), which gives the standard normal CDF to within about 1e-7 — more than enough for percentile interpretation.

How is a percentile converted back to a Z-score?

It applies the inverse normal (probit) using the Acklam rational approximation, which inverts the CDF accurately for percentiles strictly between 0 and 100.

What does a Z-score of 2 mean?

A Z-score of 2 is two standard deviations above the mean and corresponds to about the 97.7th percentile, meaning roughly 97.7 percent of values fall below it in a normal distribution.

Can I enter the 0th or 100th percentile?

No exact Z-score exists for 0 or 100 percent because the normal curve extends infinitely in both directions. Enter a percentile strictly between 0 and 100, such as 0.1 or 99.9.

Z-Score & Percentile Calculator

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Move between Z-scores and percentile ranks

A Z-score says how many standard deviations a value sits from the mean of a normal distribution, while a percentile says what fraction of the distribution falls below it. They are two views of the same point on the standard normal curve. This calculator converts in both directions: Z to percentile using the normal CDF, and percentile to Z using the inverse normal.

How it works

To convert a Z-score to a percentile, integrate the standard normal density up to z — the cumulative distribution function:

percentile = 100 * Phi(z)
Phi(z) = 0.5 * (1 + erf(z / sqrt(2)))

erf is the error function, computed here with a high-accuracy rational approximation. To go back the other way, the tool applies the inverse normal (the probit function) using Acklam’s rational approximation to solve Phi(z) = p for z.

Tips and example

A Z-score of 1.5 gives Phi(1.5) ≈ 0.9332, so the percentile is about 93.3 — roughly 93 percent of values lie below it. Going the other way, the 90th percentile corresponds to a Z-score of about 1.2816. Symmetry helps as a sanity check: a Z of 0 is exactly the 50th percentile, and equal-and-opposite Z-scores (such as -1 and +1) give percentiles that sum to 100. Use this to interpret standardised test results where scores are reported as Z-scores or percentile ranks.