Metrology – Accuracy and Scale

No matter what is being measured, it is critical that the measuring instrument chosen be capable of providing an accurate and repeatable measurement of whatever needs measuring.  It sounds simple – – but isn’t.  In general, the larger the scale the less accurate an instrument becomes.  Let’s use weight as an example.

Weighing devices come in a vast array of sizes, shapes and capacities.  A precision laboratory scale can usually weigh things down to a small fraction of a milligram.  Such a scale can, for example, easily measure the weight of a single penny.  In most cases the laboratory scale would also be able to measure the weight of at least several pennies.  Assuming that all pennies are of approximately the same weight, it would be an easy matter to determine how many pennies were on the scale by dividing their total weight by the known weight of a single penny.  Granted, the result of the division will not likely come out to be an exact integer but a little interpolation will tell you if there are 7, 8, or 12 pennies in the pile being weighed.  Now, let’s say that instead of a few pennies we have several thousand pennies.  Most precision laboratory scales are not capable of weighing thousands of pennies.  So, instead, let’s assume we use a bathroom scale.  The problem is that bathroom scale, although it is capable of weighing our several thousand pennies doesn’t give us a very accurate reading.  If it is one of those scales with a spring and a dial you may be able to read it with an accuracy of plus or minus one pound (and that’s assuming that the scale is correctly calibrated).  The best digital scale may read to an accuracy of plus or minus 1/10 of a pound and we all know that those scales, at least the ones in our Doctor’s offices, are subject to some real calibration problems.  All of a sudden, our old scheme of dividing the total weight by that of a single penny to determine how many pennies there are altogether becomes pretty useless.  We can get an approximate number but I don’t think any of us would place any bet that we would hit the number precisely.

The same is true in whatever we measure.  The larger the scale, the less accurate the measurement is.  In most cases a measuring instrument’s rated accuracy is given as a percentage its total capacity or “full scale.”  If a bathroom scale is capable of measuring 300 pounds and is accurate to 1% of its total scale then we know that we can expect that something that registers as 150 pounds on the scale may actually weigh anywhere from 147 to 153 pounds.  Using a different, more sophisticated device with the same 300 pound maximum capacity but an accuracy of .01% of full scale would narrow the range of our 150 pound object to between 149.7 and 150.3 pounds.  In either case, the 300 pound capacity scale would not be able to accurately weigh a single penny with sufficient accuracy to do that division thing described above.

Practical Demonstration – While standing on the bathroom scale (a digital one works best for this) start picking up pennies and putting them in your pocket.  See how many pennies you have to put in your pocket before the scale jumps to the next number.  Then pick up a bowling ball!  The bathroom scale barely sees pennies but can, easily, detect a bowling ball in your pocket!

This basic example applies to any measuring task we undertake.  A 50 foot tape measure is not appropriate for measuring a few 1000ths of an inch nor is it appropriate for measuring the distance to the moon.  Those tasks require other instruments.

Finally, when taking measurements of any kind make sure you know the accuracy of the device being used and factor the possible inaccuracy into any conclusions drawn as a margin of error.

–  FJF  –