# accurate measurements with less accurate tools

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 from the same tool can create a better measurement. Or using two independent tools and average them. I don't know if measuring theory or statistical quality control or other subject can help.

• It does not even qualify as a measurement untill you have calculated your error bound. So no you can not even call a sigle shot value taking a measurement. – joojaa Aug 28 '19 at 4:49
• Is there a book or website or reference explains all measuring related topics? Such as error bound and accuracy improvement techniques? – user5425156 Aug 28 '19 at 5:41
• The way we've been doing this for the past 5000 years is to put one person in charge of it and keep them there until they die of old age. People are amazingly accurate given a few thousand hours of calibration time. You also end up with a lot of free expertise as far as how much accuracy is actually needed on any given day. – Phil Sweet Aug 28 '19 at 8:58
• – user16 Sep 4 '19 at 2:05

If you know that the errors in your measurements are truly random and have a normal distribution then you can improve the resolution of a measurement by taking several measurements and averaging them. This technique is called oversampling and is common in digital data acquisition systems where the hardware resolution may be low.

You mentioned accuracy, which is the difference between a measured value and the true value. This technique will not improve accuracy. If a measuring device is inaccurate it will still be inaccurate if you average a bunch of measurements. If you want accuracy you must calibrate your measuring tools against another tool that is known to have better accuracy...and so on back to NIST (in the U.S.)

The word precision usually implies repeatability. I don't think you can get rid of something like long-term drift or temperature affects just by averaging multiple samples.

I think the Kalman filter can answer your question (it may be an overkill of your case but the folowing tutorial show you many interesting cases)

https://www.kalmanfilter.net/default.aspx

Kalman Filter is one of the most important and common estimation algorithms. The Kalman Filter produces estimates of hidden variables based on inaccurate and uncertain measurements. As well, the Kalman Filter provides a prediction of the future system state, based on the past estimations.

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 from the same tool can create a better measurement. Or using two independent tools and average them. I don't know if measuring theory or statistical quality control or other subject can help.

The question is a little confusing first you make a simple question

systematically increase the accuracy and precision of a measuring tool using only mathematical means

to which the answer is obviously No then you proceed to talk about using more than one tool and doing averaging.

So really the question should be Can measuring theory or statistical quality control increase the accuracy and precision of measurements? To which again the answer is obviously No

However what mathematical means can do is predict the percentage error and so the reliability of the measured values and this can give you a theoretical (possible) answer using averages etc. However useful this may be it does not increase the accuracy and precision of the actual measurements. and unfortunately never will.

The only way to increase the accuracy and precision of the actual measurements it to use high precision instruments that are regularly re-calibrated against a known standard.

One method which may work depending on what you're measuring, is to measure mulitples of a thing instead of just 1. For example measuring the thickness of a piece of paper by measuring 100 pieces and dividing the measurement by 100. If done correctly, this effectively makes your tool 100 times more accurate. Also works well when weighing small objects.