# How Much Precision?

## The Precision Problem

Once you realize that you need to be concerned with "how confident am I in this measurement?" - precision, in other words, you have 2 problems:

• How do I decide "how confident" I am in a measurement?
• How do I express this confidence?

## Scale Error (Scale Uncertainty)

Thankfully, (luckily, perhaps) when you reported your measurement you didn't say "exactly 25.45 centimeters"! If you had, you would have made yourself an example of what physicists call an "idiot". Why? First of all, 25.45 cm is a measurement - made by a real measuring instrument, in this case a meter stick. A typical meter stick has marks at 1 millimeter intervals - your 25.45 cm measurement is pictured at left. The arrow appears to be half way between the 25.4 cm mark and the 25.5 cm mark - therefore, you were justified in calling the length 25.45 cm.

Notice that the last digit of the measurement is an estimate, though. Certainly, there is nothing wrong with this. In fact, it is clearly more reasonable to call the length 25.45 cm than to call it 25.4 cm or 25.5 cm.

However, how could you justify saying that the measurement was exactly 25.45 cm? You could certainly judge the measurement to the nearest 0.25 mm, and a strong case could be made, probably, for an estimate to the nearest 0.1 mm in this case. But exactly? That's just an unjustified guess! A careful estimate is not the same as a wild guess!

In the same way, you would not be justified in calling your measurement 25.452 cm. Doing this would mean that you claim to be able to divide a single millimeter into 100 equal parts - accurately - by eye! (You are claiming that the measurement is 52/100 of the distance between 25.4 cm and 25.5 cm.) If you make a claim like this, you had better be ready to back it up!

Apparently, the precision of a measurement is limited by the markings on the measuring instrument. The best that a competent physicist can be expected to do is estimate one digit between the finest markings on the measuring scale. (Physicists are expected to be able to do that, by the way.)

This limitation on the precision of a measurement is commonly called scale error. Actually, this is unfortunate, since "error" carries the connotation of "mistake" - but

Scale errors are not mistakes.

For this reason, I prefer the term scale uncertainty, although "scale error" is more commonly used. The scale uncertainty is ultimate limit to the precision of a measurement, but it might not be the largest factor affecting the precision of a measurement.

At this point, you should realize that:

• The precision of any measurement is limited by the instrument used to measure it. This inherent limitation is called the scale error or scale uncertainty of the instrument. To get more precision, you need a more precise (more expensive and usually more difficult to use) instrument.
• The scale uncertainty of an instrument is determined by the finest divisions on the instrument's scale. The best that you can do in a measurement is estimate one digit between the finest markings on the instrument's scale.
• No measurement can be "exact". This would require a measuring instrument with marks infinitely close together - which is clearly impossible.

last update April 10, 2004 by JL Stanbrough