Is there a good way to systematically increase the accuracy and precision of a measuring tool using only mathematical means? For example, average 10 measurements can create a better one. Or using two independent tools. I don't know if measuring theory or statistical quality control or other subject can help.
I don't know much about measuring theory, but I do know a lot about how the tools work and accuracy and precision. Accuracy, is how close you are to the actual answer. Suppose a length is 10 cm and you get 9.8, then you are accurate. Precision, on the other hand, is the measure of how steady you are in taking measurements, in other words how well can you carry out the same experiment multiple times and still get the same result. For example, you can do 10 experiments and if you're measurements are all very close every time, you are precise, your accuracy is a different matter.
To answer your question, accuracy depends on the quality of your tools and your methods, if you have a bad scale with wrong calibration, there is nothing you can do mathematically to make your answers accurate. For precision, multiple experiments will help you find your answer with more precision, so yes that can be helped. Using different tools will help with accuracy, if either of the tools are well calibrated, but you could just use the one that's better and get the answer. So yeah, maths can't help your accuracy, but can help precision.
this is a standard topic treated in any college-level statistics or SQC class.
You can in practice increase the precision of a poorly-repeatable tool by increasing the sample size, which increases your confidence in the validity of the average of those measurements.
However, there is no mathematical trick to improve the accuracy of an out-of-calibration tool.