Interquartile ranges of 2 groups in meta analysis I am performing a meta analysis of mutliple trials (Intervention vs. Placebo) and wanted to describe the baseline characteristics in a table.
Some trials report continous variables as medians and interquartile ranges of the 2 groups seperately and not the total trial population. Is there any way to estimate the overall median and interquartile range for these continous variables?
 A: Suppose the groups have these characteristics:
\begin{align}
\text{Interventions:}&\ \ 
N = 45,\ \text{median} = 35,\ IQR = 25\\
\text{Placebo:}&\ \  
N = 55,\ \text{median} = 40,\ IQR = 30\\
\end{align}
Then you might guess that the groups are normally distributed, with standard deviations equal to the interquartile ranges divided by 1.35 (which is the interquartile range for a standard normal).
If those guesses are correct, then you can analyze the mixture distribution by trial and error, e.g.:
$$.45\, P\left[N(35, \frac{25}{1.35}) < x \right] + 0.55\, P\left[N(35, \frac{30}{1.35}) < x \right] = \frac14 \text{ for } x = 23.7$$
$$.45\, P\left[N(35, \frac{25}{1.35}) < x \right] + 0.55\, P\left[N(35, \frac{30}{1.35}) < x \right] = \frac12 \text{ for } x = 37.5$$
$$.45\, P\left[N(35, \frac{25}{1.35}) < x \right] + 0.55\, P\left[N(35, \frac{30}{1.35}) < x \right] = \frac34 \text{ for } x = 51.5$$
in which case the median is $37.5$ and the interquartile range is $51.5-23.7=27.8$.
You can apply a similar procedure with any number of groups (assuming that you know the group sizes), but there is no nice formula, and the result is only as reliable as the guess of normal distribtions.
