Consider an annular region of space. We divide this annulus into four equal regions and we take some measurements in each of the four sectors.
Imagine we are measuring something like a particle's energy which enters a given sector. The quantities are all strictly greater than zero. Measurements from one sector have weak to no correlation to other sectors so we just assume independence between sectors. The problem is that the number of measurements between sectors vary by orders of magnitude. One sector can return thousands of measurements, another can return hundreds, the third may return only tens, and the fourth may have even fewer. Furthermore, the measurements themselves in a given sector vary across orders of magnitude. For example, A may have a thousand measurements and their max and min might be 4 orders of magnitude apart.
Main question: What is a reasonable way to estimate a reasonable measure of central tendency for the entire annulus?
The measurements within a sector seem to be log-normally distributed. Here are some typical distributions for all four sectors.
The logs are all base ten here. The panels on the left show the distributions of the actual data in all four sectors. The right panels show the distribution of the data after I take the log for all four sectors. Within each bin, I am convinced that the "best" measure of central tendency is the median (in all of my cases, the median and the geometric mean are almost identical...the mean is always at like 75th percentile or higher). But then the question is, how can I get a measure of central tendency for the entire annulus? I can
- take the average of the four medians
- take the weighted mean of the four medians where the weights are inversely proportional to the number of measurements in a bin. So A gets the lowest weight and D gets the highest weight. This "makes sense" because I don't want A to influence the entire annulus too much. After all it is only a quarter of the entire annulus, just as valuable as D.
- take the weighted mean of the four medians where the weights are directly proportional to the number of measurements in a bin. So A gets the highest weight and D gets the lower weights. This "makes sense" because A has the most number of measurements. It is the "cleanest" data set with the best statistics so it is the "best" representation of what is actually happening in the entire annulus.
- merge all four datasets together and then take the median. This "makes sense" because the actual population in the entire annulus is highly skewed so median is a good measure for the entire annulus.
Which one of these is "best"? This is brand new stuff so we don't have much theory available. We have no idea what it is supposed to be theoretically. The end goal is just descriptive statistics as well as plugging the "average" energy into an equation and reporting numbers. At the end of the day, for now, we just want to be able to say, "The typical energy in the given annulus under these circumstances is ________ which plugging into equation 1 gives us Billy's constant as ________."
Assume that experiments are expensive and extremely time consuming and cannot be repeated so we have the data that we have. Does this have a name? Are there any books or papers that someone can recommend which we read, use, and cite? That'll be perfect.