# Beta Distribution Parameters

Hi all!

I would like to establish beta distribution by giving Mean and Standard deviation. However, I only had Median and Standard error. Can I use Median and Standard error instead?

Thanks for helping!

• I would not use the median and SE.

It seems a bit weird that the median and SE were reported and not the mean and SE or the median and IQR, but anyway:

First, if you're sampling from the Beta for individuals, and assuming that the SE you refer to is the standard error of the mean, then you'll want the SD = SEM*sqrt(n) where 'n' is the sample size of the study from which the median and SEM were estimated.

There is no closed form for the median of the Beta distribution but it can be approximated from its alpha and beta parameters as median ~ (alpha - 1/3)/(alpha + beta - 2/3). This assumes that alpha and beta are both > 1. You could run a mini calibration in Excel trying out different alpha and beta values until you find the pair that, when plugged into the formula above,  comes closest to the observed median, Then the mean is alpha/(alpha + beta). With the mean and SD in hand you can set up your Beta distribution in Treeage.

I would sample from it some large number of times and check that the median of the samples is close to the observed median from the study.

• Hi David,

The Mean and Standard error of the mean mentioned above were from the sample group, not from the Beta for individuals.

If I changed to use Median and IQR, can they be typed in the Mean and SD part to calculate the alpha and beta by the software?

• I think you are on the right track. The mean from your sample can be entered in as the mean of the beta distribution. Since your data is from individuals (rather than different cohort-level estimates), I think it is appropriate to use the standard error of the mean (SE) as the standard deviation for the beta distribution.

As David wrote: SE = StdDev/Sqrt(sample size). By using the SE, you have aggregated the individual-level variance to the level of how far off the mean value could be for the cohort. I believe this is appropriate.

Andrew