How to normalize the power of a QAM signal?
In $16$-QAM, four constellation points 
$\left(\pm\frac{d}{2},\pm\frac{d}{2}\right)$ are "interior" points while
the remaining $12$ are "edge" points or "corner" points for which
at least one of the coordinates is $\pm\frac{3d}{2}$. 
Thus, in
$n$ measurements $(X_i,Y_i)$ of successive transmitted symbols,
some of the $X_i$ will equal $\pm\frac{d}{2} + NI_i$ while
others will be $\pm\frac{3d}{2} + NI_i$.  Similarly,
some of the $Y_i$ will equal $\pm\frac{d}{2} + NQ_i$ while
others will be $\pm\frac{3d}{2} + NQ_i$.  Here, $NI_i$ and $NQ_i$
are independent zero-mean Gaussian random variables with variance
$\sigma^2$ representing the noise. Note that we can expect 
about $\frac{3}{4}$-th of the
samples to have at least one large coordinate. Now, consider
that if we want to estimate the value of $d$ from the measurements,
and want to use the average energy calculations, then
$\sum_i (X_i^2+Y_i^2)$ is one possibility, but we are weighting
equally the symbols with large means and the symbols with small
means, and the noise component is a more significant part of
the symbols with small means than it is of symbols with large
means.  It might be better to use a weighted sum to estimate
$d$ (or $d^2$), and one extreme possibility is to give zero weight to
interior points ($(X_i$ and $Y_i$ are both small) and to use
only the large values of $X_i$ and $Y_i$ (edge and corner
points) in the estimation.  If the number of samples is large,
this could be as simple as taking those (approximately) $3n/4$
samples in which either $X_i$ or $Y_i$ (possibly both) is
large, and averaging the absolute values of the largest $X_i$ and
$Y_i$ to get an estimate of $3d/2$.  

Once upon a time, when
analog circuits were in use, these kinds of calculations would
have been difficult to implement in hardware, and this effect
carries over into some present-day implementations that faithfully
follow the methods of the past that were originally devised
because of hardware limitations.
But these days, with digital computation being ubiquitous,
perhaps something new can be attempted.