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.
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1$\begingroup$ 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 $\endgroup$– Glen_bCommented Jul 15 at 23:58
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1$\begingroup$ 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. $\endgroup$– Sextus EmpiricusCommented Jul 16 at 9:13
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The long answer is "no" this is not possible to determine without additional information about the distribution of the data.
The pragmatic answer is "yes", if you make assumptions about the distribution, then—for meta-analytic purposes—it may be possible to obtain estimates for the mean and standard deviation from the median and the IQR.
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.)
But again, the caveat here is that you must reasonably be able to assume a normal distribution to be able to use this approximations.
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1$\begingroup$ +1. Of course the key point is that one should be very careful about whether one can reasonably assume an approximate normal distribution for this to work. For instance, if M-SD < 0 for a variable that physically cannot be negative, then this is not consistent with approximating it as normally distributed. Similarly if the median or the mean is "far away" from the mean of the two quartiles. $\endgroup$ Commented Jul 16 at 6:42
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$\begingroup$ Similarly, if you can reasonably assume some other specific distribution, one could do similar computations based on approximating your data to that specific distribution. It doesn't have to be the normal distribution but you do need a good approximation to some specific well-known distribution. $\endgroup$– quaragueCommented Jul 16 at 10:48