Percentile of Normal Distribution Calculator

Common percentiles for this distribution
Percentile	z-score	Value
1%	-2.326	65.1048
5%	-1.645	75.3272
10%	-1.282	80.7767
25%	-0.6745	89.8827
50%	-6.9573 × 10^-17	100
75%	0.6745	110.117
90%	1.282	119.223
95%	1.645	124.673
99%	2.326	134.895

z-score(x − μ) / σ

PercentileΦ(z) × 100

Valueμ + zσ

What Is Percentile of Normal Distribution?

A percentile is the share of a distribution at or below a value. In a normal distribution you find it by converting the value to a z-score and reading the standard normal cumulative probability.

Choose your direction. Enter a value x to get its percentile, or enter a percentile to get the value that sits there. The page shows the z-score, the areas below and above, a shaded bell curve and a table of common percentiles for your distribution.

The answers are exact only when the data really follow a normal curve. For skewed data, use the percentiles of the sorted observations instead.

How to Use the Percentile of Normal Distribution Calculator

Enter the mean and standard deviation.
Choose whether you know the value or the percentile.
Read the percentile, z-score and the curve with the shaded area.

Percentile of Normal Distribution Formula

z-score

z=\frac{x-\mu}{\sigma}

Percentile of x

\text{percentile}=\Phi(z)\times 100

Value at a percentile

x=\mu+z_p\,\sigma

Percentile of Normal Distribution Variables Explained

Symbol	Meaning	Notes
μ	Mean of the distribution
σ	Standard deviation
x	The value
z	Standard score, (x − μ) / σ
Φ(z)	Standard normal cumulative probability

Percentile of Normal Distribution Worked Example

With the example values loaded in the calculator above, the steps are:

z = (x − μ) ÷ σ = (120 − 100) ÷ 15 = 1.3333.
Percentile = Φ(z) × 100 = 90.879%, so about 90.9% of values fall at or below x.

Result: Percentile of x = 90.879%; Value x = 120.

Worked example: IQ-style scale, μ = 100, σ = 15

For x = 120, z = (120 − 100) ÷ 15 = 1.3333, and Φ(1.3333) = 0.9088, so 120 is at the 90.9th percentile.

Going the other way, the 90th percentile has z = 1.2816, so x = 100 + 1.2816 × 15 = 119.22.

Percentile versus percentage

A percentile is a rank, not a score. Being at the 90th percentile means 90% of values are at or below yours. It does not mean you scored 90%.

Where Percentile of Normal Distribution Is Used

Test scores

Find what percentile a score is, given the test mean and standard deviation.

Setting cut-offs

Find the score needed to reach the 90th or 95th percentile.

Growth and measurement charts

Place a measurement within a normal reference distribution.

Common Percentile of Normal Distribution Mistakes to Avoid

Percentiles from this page are exact only if the data really follow a normal curve.
The 50th percentile is the mean, because the normal curve is symmetric.

Percentile of Normal Distribution Terms and Definitions

Percentile: The value below which a given percentage of observations fall.
z-score: How many standard deviations a value is from the mean.

Percentile of Normal Distribution FAQs

How do you find the percentile of a normal distribution?

Convert x to a z-score with (x − μ) ÷ σ, then read the standard normal cumulative probability for z and multiply by 100.

How do I find the value at the 90th percentile?

Find the z-score for 0.90, which is 1.2816, then compute μ + 1.2816σ.

What percentile is one standard deviation above the mean?

About the 84.1st percentile. One standard deviation below is about the 15.9th.

What is the 50th percentile of a normal distribution?

It equals the mean, which is also the median.

Does this work for skewed data?

No. For skewed data, use the observed percentiles from the sorted data.

What is the difference between a percentile and a quartile?

Quartiles are the 25th, 50th and 75th percentiles.