6.03 Accuracy and measurement

Which of these three numbers is larger? 0.0025,\, 0.002\,82, \, 0.009 The answer is 0.009.

Now let's try again with these numbers:25, \, 28.2, \, 90 The answer is 90.

The second list is the same as the first, just multiplied by 10\,000. Still, it is much easier to tell by sight that the last number of the second list is the largest, compared with the first list.

We have a better intuitive sense of the size of numbers if they are between 1 and 1000, ideally less than 100. When selecting the correct scale to report a measurement, we want to lie somewhere in this range to compare it to others quickly and easily.

Examples

The aim of a measurement is to obtain the "true" value of a quantity: the height of a tree, the temperature of a room, the mass of a rock, or whatever we want to know.

But is a tree ever exactly 5 metres tall? Is a room ever exactly 22\degree ? Is a rock ever exactly 2 kg in weight?

We can carefully design a measurement procedure to make more and more precise measurements, which makes the number of significant figures in our measurement increase. But we can only ever report the closest marking, and at some point the object we are measuring will fall between the markings.

For any measurement tool, we say its absolute error is equal to half the distance of its smallest unit. Any measurement we make with that tool must be given as plus or minus the absolute error.

Any subsequent measurement that is more precise will fall within this range, but we can't know exactly where until we try with a better tool.

The appropriate unit of measurement to use makes the numerical value lie between 1 and 1000, ideally less than 100.

The absolute error of a measuring tool is equal to half its smallest unit.

Measurements with any tool should always be reported as: \text{Closest mark} \pm \text{Absolute error}

Examples