Percentile Calculator
Turn a z-score into a left-tail percentile under the standard normal curve, recover z from a percentile, or standardize a raw score first. The percentile is 100 × Φ(z): the percent of a mound-shaped population at or below that z.
z = 1 · above = 15.8655263832%
- The percentile is the left-tail area under the standard normal curve: 100 × Φ(z).
- Φ(1) = 0.841344736168, so the percentile is 84.1344736168%.
- The area above z is 15.8655263832%. The area between −|z| and |z| is 68.2689472335%.
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How to Use the Percentile Calculator
From z-score reads the standard normal table for you. From percentile inverts that table. From x, μ, σ standardizes first, then reads the table.
- Choose the knowns you have.
- Enter a finite z, or a percentile strictly between 0 and 100, or a raw score with a positive σ.
- Read the percentile (or the z), the upper tail, and the two-sided area between −|z| and |z|.
Getting z = (x − μ) / σ without a curve belongs on the Z-Score Calculator. Computing σ from a list belongs on the Standard Deviation Calculator. “84% of 200” belongs on the Percent Calculator.
Φ(z) and the Left Tail
Φ(z) is the area under the standard normal curve from −∞ to z. That area is the fraction of a mound-shaped population at or below that standardized score. Multiply by 100 to speak in percentiles: the 84th percentile means about 84% scored at or below you, not that you scored 84% on the test.
The inverse Φ⁻¹ undoes the table. A homework “95th percentile” is the z with left-tail area 0.95, about 1.645. The 0th and 100th percentiles would require infinite z; this page rejects them. The curve here is the standard normal only — mean 0, variance 1 — after any raw score has been turned into z.
Percentile Calculator Examples
z = 1
Left tail ≈ 84.1344736168%. About 15.87% sit above. Between −1 and 1 lies about 68.27% — the first empirical-rule band, now as a computed area instead of “about 68%.”
z = 0
The mean is the 50th percentile. Half the curve is on each side. “Average” on this scale is the 50th, not the 0th and not the 100th.
z = −1
Left tail ≈ 15.8655263832%. The lower tail at −1 matches the upper tail at +1. The middle band is the same 68.27%.
z = 2
Left tail ≈ 97.7249937113%. About 2.28% sit above two standard deviations. Between −2 and 2 lies about 95.45%.
The 95th percentile
z ≈ 1.64485349227. A printed table often rounds that to 1.64 or 1.65. This page keeps the inverse of Φ, not a one-decimal lookup.
The 0th percentile is rejected
No finite z puts area 0 to its left. The same is true of 100. Use a value such as 0.1 or 99.9 if you need a far tail.
The Empirical Rule, With Numbers
The sketch “68–95–99.7” is the two-sided area within 1, 2, and 3 standard deviations. This page’s “Within ±|z|” metric is that area from the curve, not the rounded slogan. For |z| = 1 it is about 68.27%, not 68. For |z| = 2 it is about 95.45%, not 95. For |z| = 3 it is about 99.73%.
Those percents assume a normal shape. A skewed homework list can have a z-score on the Z-Score Calculator and still not match these tails. This page does not test normality, and it does not compute a t-tail or a discrete rank percentile from a short list.
Common Percentile Mistakes
Calling z = 1 “the 1st percentile”
z = 1 is about the 84th percentile. The 1st percentile is a far lower tail, z ≈ −2.33.
Treating the percentile as a test score
The 84th percentile is a rank on the curve, not 84 points out of 100. Use the Percent Calculator for 84% of a total.
Reading the upper tail as the percentile
The percentile is the area to the left. For z = 1 that is ~84%, not the 16% above.
Entering 0.95 when you mean the 95th percentile
0.95 as a percentile is the 0.95th — a far lower tail. Type 95, not 0.95.
Using this as a p-value for a test
A two-sided test often wants both tails. This page reports a one-sided left tail and a two-sided middle band. It is not a hypothesis-test tool.
Percentile Calculator FAQs
What percentile is z = 1?
About the 84th: 100 × Φ(1) ≈ 84.1344736168%.
What z-score is the 95th percentile?
About 1.64485. Printed tables often show 1.64 or 1.65.
What percentile is the mean?
The 50th. z = 0 splits the standard normal in half.
Does this use a printed z-table?
It evaluates Φ(z) from the standard normal curve, which is what those tables approximate. Rounding will differ from a two-decimal table.
Is this the same as the Z-Score Calculator?
No. That page only computes z = (x − μ) / σ. This page turns z into a left-tail percentile, or the reverse.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
Disclaimer
This calculator is for general educational use. It reports a standard-normal left-tail percentile from an approximation to Φ(z). It is not a printed z-table, a hypothesis-test p-value, an IQ or clinical instrument, or a substitute for a statistics package.