Calculate the Percentile from Mean and Standard Deviation

The most typical case when finding percentiles is the case of finding a percentile from sample data . In that case, the percentile can only be estimated.

But when we have population information that determines exactly the population distribution, the percentiles can be computed exactly. More specifically, this calculator shows how to compute percentiles when the population mean (\(\mu\)) and standard deviation (\(\sigma\)) are known, and we know that the distribution is normal.

Standard Deviation Percentile Calculator

The procedure is simple in this case. For a given percentage value value, expressed as a decimal \(p\), which is a number between 0 and 1, we find using Excel or a normal probability table a z-score \(z_p\) so that

\[ p = \Pr(Z < z_p) \]

Then, once we have found \(z_p\), we use the following formula:

\[\text{Percentile} = \mu + z_p \times \sigma\]

Example: How to find the 80th Percentile with given Mean and Standard Deviation

Assume that the population mean is known to be equal to \(\mu = 10\), and the population standard deviation is known to be \(\sigma = 5\)

First, the requested percentage is 0.80 in decimal notation. Then we find using a normal distribution table that \(z_p = 0.842\) is such that .

\[ \Pr(Z < 0.824) = 0.80 \]

Therefore, we find that the 80-th percentile is

\[P_{80} = \mu + z_p \times \sigma = 10 + 0.842 \times 5 = 14.208\]

Does it have to be a normal distribution

Yes. This procedure, with z-scores and all that, assumes you are working with a normal distribution . If the distribution is not normal, you still can compute percentiles, but the procedure will likely be different.