For example: a simple observation is that the Sun "goes down"
sometimes—but, after giving us time for a nap, it
"comes up" again on the other side of the house.
A theory based on several such observations
would be something like "*every* time the Sun goes down, it will
come back up on the other side of the world." This can be
tested by watching. Better observations would include
measurements of the times between sunset and sunrise,
using some sort of clock, and measurements of the time that
the sun is "up." The theory could then be improved to provide
predictions of when the Sun would rise and set. You can see how
this understanding can get more precise, as we make measurements
from various locations on Earth, and at various times of "year."
Hmm, what is a "year"? This observation, that the days get
longer, then shorter, and that this cycle repeats too, can lead
to a whole new theory.

Crucial to this process is the understanding that no measurement
is exact: there is *always* some uncertainty in a measurement,
and a statement of the result of a measurement in incomplete
without a statement if its uncertainty. In fact, as we have
already seen, it is usually harder to establish the uncertainty
of a measurement than to make the measurement itself. Some parts
of the uncertainty have to do with the quality of the measurement
tools and our use of these tools; these effects can be explored
and quantified by making a set of repeated measurements. Other
sources of uncertainty have more to do with our understanding
of the measurement technique, and can be very difficult to evaluate.

The words "uncertainty" and "error" are used interchangeably in this context. This is a misuse of the word "error", because we never mean "mistake". Mistakes, like confusing the inches scale with the cm scale, can be avoided and need to be rectified before determining the experimental error.

First, we need a few definitions:

**Accuracy:**This is the amount by which your measurement is*in fact*different from the true value. In any interesting situation, you will not know the "true" value, so there will be no way to absolutely establish the measurement's accuracy.**Precision:**This is the extent to which you can specify the exactness of a measurement. For example, a measurement given as 12.14 +/- 0.01 cm is more precise than one given as 12 +/- 1 cm. Higher precision does not necessarily imply higher accuracy.**Statistical (random) Uncertainty:**This is the inescapable fact that every time you repeat a measurement, you will get a slightly different value. The values will be distributed about the mean (average) value, and the way they are distributed can be used to establish the statistical uncertainty of the measurement. We will explore techniques for handling statistical uncertainty (random errors) in this lab.**Systematic Uncertainty:**Everything else that causes your measurement to lose accuracy. This includes instrumental effects, not taking things into account, and gross (stupid) errors.

Zeros can cause confusion. Leading zeros are not significant
figures: 0.00004 has one significant figure. Trailing zeros
without a decimal point may or may not be significant: 400 may
have one, two or three significant digits. This can be cleared
up by an explicit statement of the uncertainty (e.g. +/- 10),
or by putting in a decimal point: 400. is conventionally
taken to mean that there are three significant digits.
Expressing the number in powers of ten notation makes it easier
to tell which zeros are significant: 4 x 10^{2} has
one significant figure; 4.00 x 10^{2} has three.

The uncertainty is usually expressed as a single digit (sometimes two), of the same order of magnitude (i.e. same decimal place) as the last significant digit of the value. For example:

8.45 +/- 0.03

10.0 +/- 1.5

5 +/- 2

*The following numbers are all incorrectly written: write the
correct expression next to them.*

83.45 +/- 0.023815

100.0 +/- 2

5 +/- 0.5

0.00034 +/- 0.0001

When you are combining numbers to get a result, as we did for the parallax measurements, you can keep an extra digit for the computations to avoid rounding errors. Your calculator keeps lots of extra digits, of course. But be sure to trim off the meaningless digits when you express the result. In particular, converting from one unit to another does not change the uncertainty: if you measure a length to be 15.5 +/- 0.5 feet and want to convert it to cm, the value should be written as something like 470 +/- 15 cm (0.5 feet is about 15 cm), even though the calculator says 472.44. Notice that practically every single container in the grocery store gets this wrong when they convert from fluid ounces to ml.

The *variance* of a set of measured values is
the average of the squared deviations from the mean:

variance = (sum of (X_{i} - X_{mean})^{2}) / N

and the *standard deviation* SD is the square root of the variance.

If the errors are truly random, and a fairly large number of measurements are taken, they will scatter symmetrically about the mean value, with more of them close to the mean and a smaller number farther from the mean. This distribution is called a Gaussian or normal distribution. In a normal distribution, 68% of the measurements will lie within one standard deviation of the mean and 95% of them will be within two standard deviations. This means that if you make one more identical measurement, it has a 68% probability of being within one standard deviation of the previously calculated mean, and a 95% probability of being within two standard deviations.

*Calculate the mean and standard deviation of the
following set of numbers. Write them below.*

74, 75, 79, 77, 74, 65, 64, 78, 75, 74

The rule is that when you add or subtract two or more measured values, the absolute error in the result is the square root of the sum of the squares of the individual absolute errors. And when you multiply or divide two values, you do the same thing but using the fractional errors: the fractional error of the result is the square root of the sum of the squares of the individual fractional errors.

*Given two measurements X = 10.0 +/- 0.7
and Y = 3.1 +/- 0.4, what are the uncertainties in
computed values A = X + Y and B = X / Y?*

Now copy the tabulated measurements, and for each dimension compute the mean and SD, and SDM. Compute the volume of the brick in cubic cm, and the value of the uncertainty in the volume. Which dimension is most important to measure precisely?

Consider possible sources of systematic error in measuring the
volume of the brick. Not dumb mistakes, but real possibilities
that could cause the result to be systematically too large or
too small. List two of these, and suggest for each a way to
evaluate the error, or reduce its impact.

mickey@ifa.hawaii.edu

Last modified: April 7, 2005
`http://www.ifa.hawaii.edu/users/mickey/ASTR110L_S05/measurement.html`