I'm a scholar in the humanities trying to not be a complete idiot about statistics. I have a problem relevant to some philological articles I'm writing. To avoid introducing the obscure technicalities of my field I'll recast this as a simple fictional "archaeology" problem.

In the Valley of Witches there are 29 tombs. Each contains an assortment of coins and gemstones. Some of the coins are gold coins and some of the gemstones are saphires.

There is a hypothesis in the field which predicts that the proportion of gold coins to total coins should correlate positively with the proportion of sapphires to total gemstones. Let's call this Angmar's prediction.

I would like to test Angmar's prediction for the dataset below. If I run a straightforward Pearson correlation on all 29 data points I get a correlation very close to zero (0.01). This looks bad for Angmar - but is it the whole story?

Some of the data points are clearly better than others. Tomb 1 has 46 gems and 990 coins. That seems to be a much more solid data point than Tomb 29, which has only 4 gems and 80 coins. In the dataset below I've arranged the tombs in order of "size", defined as the geometric mean of total gemstones and total coins. Now, if we only look at the 13 largest tombs we get a correlation of 0.67. This looks good for Angmar after all. If we include 25 tombs, all but the 4 smallest ones, we still have a correlation of 0.37.

It looks reasonable to look only at large tombs or exclude small ones but there is no non-arbitrary way to decide where to put the cut-off. And it seems wrong to throw any data away.

**My question**: Is there a way to make use of all the data and calculate some sort of properly weighted correlation?

**My attempted answer**: There are functions for weighted correlation out there (I've used this) - but what should I weigh by? If I weigh by total gems I get 0.28. If I weigh by total coins I get 0.16. Either seems reasonable but ideally I would make use of both. If I weigh by the product of total gems and total coins I get a correlation of 0.47. Is this a legitimate method?

To be clear - it's not that I want to gin up as large a correlation as possible - I have publishable data any which way. I just want to get this right.

**Edit 1**: There is no particular reason to think that the relationship should be linear. A rank correlation solution might also make sense.

**Edit 2**: We've settled on a rank correlation but the weighting formula is still unclear to me. Summing the sample sizes gives an intuitively wrong result in the case where one sample size is much larger than the other. But the geometric mean of the sample sizes also gives an intuitively wrong result for big numbers. A hundred zillion coins should not weigh a hundred times as heavily as a zillion coins. What might intuitively work in a case like that would be to use the sum of the sizes of the confidence interval (assuming a binomial distribution). Or maybe simply the reciprocal of the sums of the reciprocals - like with parellel resistors. But that's something I just pulled out of my behind. I don't feel on solid ground yet and additional answers would be very much appreciated.

The dataset is as follows. It is based on real data:

$$\begin{array}{c|c|c|c|c|c|c} \text{Tomb number} & \text{Sapphires} & \text{Total gems} & \text{Sapphire ratio} & \text{Gold coins} & \text{Total coins} & \text{Gold ratio}\\ \hline \text{Tomb 1} & 44 & 46 & 0.96 & 33 & 990 & 0.03\\ \text{Tomb 2} & 35 & 41 & 0.85 & 3 & 761 & 0.00\\ \text{Tomb 3} & 21 & 25 & 0.84 & 13 & 558 & 0.02\\ \text{Tomb 4} & 23 & 25 & 0.92 & 12 & 368 & 0.03\\ \text{Tomb 5} & 14 & 18 & 0.78 & 2 & 426 & 0.00\\ \text{Tomb 6} & 13 & 17 & 0.76 & 6 & 350 & 0.02\\ \text{Tomb 7} & 12 & 14 & 0.86 & 3 & 418 & 0.01\\ \text{Tomb 8} & 8 & 13 & 0.62 & 3 & 318 & 0.01\\ \text{Tomb 9} & 11 & 12 & 0.92 & 4 & 269 & 0.01\\ \text{Tomb 10} & 6 & 6 & 1.00 & 17 & 503 & 0.03\\ \text{Tomb 11} & 9 & 10 & 0.90 & 8 & 286 & 0.03\\ \text{Tomb 12} & 4 & 6 & 0.67 & 3 & 454 & 0.01\\ \text{Tomb 13} & 9 & 10 & 0.90 & 10 & 255 & 0.04\\ \text{Tomb 14} & 7 & 10 & 0.70 & 12 & 250 & 0.05\\ \text{Tomb 15} & 7 & 7 & 1.00 & 6 & 351 & 0.02\\ \text{Tomb 16} & 9 & 9 & 1.00 & 8 & 218 & 0.04\\ \text{Tomb 17} & 6 & 7 & 0.86 & 3 & 251 & 0.01\\ \text{Tomb 18} & 7 & 7 & 1.00 & 5 & 246 & 0.02\\ \text{Tomb 19} & 5 & 5 & 1.00 & 7 & 304 & 0.02\\ \text{Tomb 20} & 4 & 4 & 1.00 & 10 & 336 & 0.03\\ \text{Tomb 21} & 4 & 4 & 1.00 & 15 & 274 & 0.05\\ \text{Tomb 22} & 6 & 6 & 1.00 & 3 & 175 & 0.02\\ \text{Tomb 23} & 5 & 6 & 0.83 & 5 & 174 & 0.03\\ \text{Tomb 24} & 4 & 4 & 1.00 & 4 & 174 & 0.02\\ \text{Tomb 25} & 4 & 4 & 1.00 & 5 & 150 & 0.03\\ \text{Tomb 26} & 1 & 2 & 0.50 & 15 & 218 & 0.07\\ \text{Tomb 27} & 2 & 2 & 1.00 & 8 & 201 & 0.04\\ \text{Tomb 28} & 1 & 3 & 0.33 & 2 & 108 & 0.02\\ \text{Tomb 29} & 4 & 4 & 1.00 & 1 & 80 & 0.01\end{array}$$