# Calculating SD for normal distribution with only mean and 5% and 95% quantile values

As part of a Bayesian method to estimate the divergence times of species, priors have to be set with values based on previous literature or known fossil dates. These priors can have different distributions selected to model the uncertainity.

For example, if I had prior information that a particular species diverged between 5 and 10 million years ago, I might want to model this with a normal distribution with mean 7.5, the 5% quantile at 5, and the 95% quantile at 10.

However, I need to input the mean and standard deviation (SD) to describe this. Is there a way, other than trial and error, to find the SD of a normal distribution using only the mean and values for the 5% and 95% quantiles?

• if the 5th and 95th percentile's are in q.05 and q.95 then in R sd=(q.95-q.05)/(2*qnorm(0.95)) calculates the standard deviation; similar commands would apply in other languages. Jun 23, 2015 at 8:49
• Thanks Glen_b, this is the best method in terms of simplicity and precision Jun 23, 2015 at 11:55
• It's essentially the same method proposed by jaradniemi. Jun 23, 2015 at 12:26

Let $\mu$ be the mean and $\sigma$ be the standard deviation of a normal distribution. The 5% and 95% quantiles are $\mu - 1.645\sigma$ and $\mu+1.645\sigma$, respectively. So, $$\sigma = (\mu - q.05)/1.645.$$ With your values, $$\sigma = (7.5-5)/1.645 = 1.52.$$
• -/+ 1.96 are the 2.5% and 97.5% quantiles. I think you mean -/+ 1.645 (qnorm(0.95) in R for the exact number)