# Estimating mean and SD given the median and IQR values

it is possible to estimate mean and SD given the median and IQR? I am involved in a meta-analysis where some trials show outcomes as mean and standard deviation but most show median and inter-quantile range.

• The median and IQR can be consistent with any mean. You need some assumptions or constraints on the support (possible values) to get anywhere. There are a number of relevant posts on site. A suitable use of the search bar should turn some up Commented Jul 15 at 23:58
• What sort of distributions are we talking about? When a choice is made to report median and IQR, then It is probably not a normal distribution, for which mean and standard deviation are more usual values to report. Without knowing the distribution or having good ways to make a good approximation, there is not enough information to answer the question. Commented Jul 16 at 9:13

If you assume a normal distribution, then the first quartile would have a standardized score of about $$-\frac23$$ and the third quartile would have a standardized score of about $$+\frac23$$.  This means there should be about $$\frac43$$ of a standard deviation between the first and third quartiles.  Doing a little bit of arithmetic, you can estimate the standard deviation as $$SD \approx \frac{Q_3-q_1}{\frac43} = \frac{3(Q_3-q_1)}4$$
And, if you wish to estimate the mean, you can use the fact that the mean and median are the same for a normal distribution.  I've seen other approximations used in meta-analyses where the average of the median (as reported) and the mid-quartile, $$\frac{q_1+Q_3}2$$, are used to estimate the mean.  (You could extend this further if you also have the min and max and can assume a symmetric distribution.)