We have seen that scale uncertainties (scale errors) limit the precision of a measurement due to the limitations of the instrument used to make the measurement. Scale uncertainties provide an absolute limit to the precision of the measurement, that is, the range of values in which the "true value" of the measurement lies cannot be smaller than the scale uncertainty. However, there are often other factors that affect the precision of a measurement more than the scale uncertainty does.

Approximation Errors (Approximation Uncertainties)

The
length of a simple pendulum is the distance from the support to the
center of mass of the pendulum bob. For a symmetrical, homogeneous
mass, the center of mass will be in the geometrical center of the
object - but how precisely can the center of the pendulum bob be
located? Within 1 or 2 millimeters? Many pendulum bobs are not
precisely symmetrical - how accurately can the center of mass be
located in this case? This type of "judgment call" is present in
almost all measurements, to one degree or another. They are called
**approximation errors** or **approximation
uncertainties**.

In the case of the simple pendulum, the length of the pendulum will also be affected by the "stretchiness" of the string. In fact, the string may actually be longer when the pendulum is moving than when it is at rest. The pendulum is certainly longer when vertical (due to the stretch of the string) - did you hold the pendulum horizontally to make it more convenient to measure? If so, how hard did you pull on the mass to stretch the string taut?

If your task is to measure the height to which a ball bounces, the approximation uncertainty may be several centimeters if you must make the measurement instantaneously by eye!

The Problem of Definition

Approximation
errors are closely related to the "problem of definition", which
refers to the fact that it is often unclear exactly what a
measurement *really means*. For instance, suppose that you
need to measure the length of a plank. The measurement that you get
may depend on whether you measure the right edge, the left edge, or
the middle of the plank, or even which side of the plank you measure!
Of course you can make several measurements and average them, but is
this really **the** length of the plank? Does the phrase
"length of the plank" really have a precise meaning?

Random Errors (Random Uncertainties)

The relevant conditions of an experiment will vary to some degree
from run to run, and even at different times within the same run. In
fact, it is pretty much impossible to determine even what all of the
relevant conditions actually are, let alone control them. It is
impossible to start a ball from exactly the same spot every time, or
to return a knob to exactly the same position for each run. These
unpredictable and uncontrollable effects are called **random
errors** or **random uncertainties**. They affect
the precision of just about all measurements.

Random uncertainties are generally considered to follow what statisticians call a Gaussian, or normal distribution, which for our purposes means that small random deviations from the mean are much more likely than large ones.

As an example of a Gaussian distribution, suppose you catch all of the grass clippings the next time you mow your yard. Now, measure each clipping (!) and calculate the mean length. Then construct a graph showing the number of clippings versus deviation from the mean (clipping length - mean length). You will get a graph like the one shown here. Most clippings will be approximately the same length - near the mean, but there will be a relatively small number of very long or very short clippings. Random experimental uncertainties are often like that - most of the measurements cluster close to the mean value, with fewer measurements at large distances from the mean.

Mathematicians can tell you a lot about Gaussian distributions - for instance, about 2/3 of all values will fall within one standard deviation from the mean, which will provide us with a very useful and convenient tool.

In our work, we will generally begin our analyses by assuming that our measurement uncertainties are random. There are two reasons for this:

- It is often the case, and
- It provides us with some powerful and convenient mathematical tools for analyzing the data.

You need to be aware, however, that data often contain systematic errors, which we will discuss in more detail later. Often, what passes for random uncertainty is really a systematic drift of values - the pendulum string may actually be getting longer as the experiment goes on.

last update August 17, 2006 by JL Stanbrough