A Measurement Dictionary
Physics attempts to provide a logical, quantitative, description of natural phenomena. This
requires that our observations of physical phenomena include quantitative measurements. Physics
is totally dependent upon these measurements. As a practical matter, however, it is impossible to
measure any quantity exactly. In fact, it is also theoretically impossible to simultaneously measure
some combinations of variables exactly. In order to discuss the problems that result from this
dilemma, language has been adopted to describe measurements and their significance. The
following is a short dictionary of these terms.
Replicable or Repeatable- Physicists are looking for general laws of nature. These laws
should have wide applicability, and similar circumstances should produce similar results and
experimental data. Experiments that cannot be repeated and measurements that cannot be
consistently performed are not considered trustworthy. (We will always assume that many repeated
measurements of a given experimental quantity are available.)
Accurate- if the average of a set of repeated measurements of a quantity is close to the true
value the measurement is said to be accurate. Example: A shotgun may scatter pellets all over a
target, but if the center of the spread is close to the center of the target then the shotgun is accurate.
Precise- if the individual measurements in a set of repeated measurements are close to each
other, the measurement is said to be precise. Example: A shotgun that spreads pellets all over a
target is not precise-a rifle in a bench stand should always hit the target in the same place-and is
Systematic error- the difference between the average value and the true value is called the
systematic error. Example: If the sights on a rifle are poorly adjusted, then even if it is locked into
a bench stand it will not hit the center of the target.
Random error- (statistical uncertainty) the deviation of the individual measurements from
their average is called the statistical uncertainty or random error. Example: A shotgun firing
pellets has a large random error.
Least Count- The smallest division on the measuring scale being used is called the least
count. This imposes a lower limit on the precision and accuracy possible with that scale. Example:
Meter sticks are usually calibrated to the nearest millimeter.
Agreement- an experiment agrees with theory if the experimental result accurately matches
the theoretical prediction to within the precision of the experiment.
Discrepancy- The difference between the results of two sets of measurement, or the
difference between the results from one set of measurements and an accepted value, is called the
Reliable-trustworthy, dependable, or replicable-Example: National Institute of Standards
and Technology measurements are more reliable than most private web pages.
Validity-Correctly derived from accepted premises and correctly measured. Example: A
measurement of the acceleration due to gravity that failed to consider the flotation of the test body
in air, or the viscous drag, or the rotation of the body as it rolled down a plane would be invalid at
some level of precision.
Absolute vs. Relative error-A result may be quoted giving the uncertainty as a number, or
as a percentage of the result. The former is an absolute statement of error, the latter is a statement
of the relative error. Example: Absolute 5 1 ; relative 5 20% .
Three groups, doing three different experiments have found values for the acceleration due to
Group 1 A Modified Atwood Machine Experiment
10 1 m/s 2
10 10% m/s 2
In this experiment a cart was subjected to the gravitational force pulling on a small mass, and both
the small mass and the cart were accelerated. The graph of the acceleration as a function of the
ratio of the force to the inertial mass was linear and the slope of the linear fit to the data produced
this accurate but not very precise value for the acceleration due to gravity. Several possible
experimental complications were ignored in the analysis. In particular the inertia of the pulley, the
friction in the pulley, and the inertia and friction in the wheels of the cart were all neglected, as was
the mass of the line connecting the masses.
Group 2 Galileo’s Experiment
7.0 0.5 m/s2
7.0 7% m/s 2
By rolling a ball down an inclined plane and repeatedly timing its arrival at different distances from
its starting point it was established that the speed of the ball was a linear function of time (good
correlation), and that the position was a quadratic function of time. By measuring the acceleration
for different angles of the inclined plane and extrapolating to a vertical plane a value of the
acceleration due to gravity was obtained. The value obtained, while reasonably precise, does not
agree with the accepted value. Given the large amount of data we collected, and the goodness of
the quadratic fits of position as a function of time, we believe that the experiment is reliable, and is
producing a valid result, but that the value we are obtaining is not a valid result for the acceleration
due to gravity, probably because of some effect that we have not yet encountered. (Perhaps the fact
that the ball rolled down the plane rather than slid is important!)
Group 3 Behr Free Fall Apparatus
9.777 0.005 m/s2
9.777 0.05% m/s2
Spark gap timing provided 32 positions as functions of time over a period of three-quarters of a
second for a body in free fall. By fitting the data to a quadratic curve we measured the acceleration
due to gravity directly. The quadratic fit to the data was excellent, showing an RMS average
discrepancy of only a few millimeters between the predicted and measured values of the position as
a function of time. This agrees with our estimate, based on the variability of the spark path, of the
experimental uncertainty in the position measurement. We achieved remarkable precision in our
value for the acceleration due to gravity. Unfortunately our result does not agree with the accepted
value. Perhaps there is some other effect of which we are not aware, such as air resistance. Maybe
we could do the experiment in vacuum. [Editorial note-the effect is the correct size to be due to the
weight of the air displaced by the falling object; a buoyancy force.]
The Requirements of Honesty
If someone were to tell you that Lock Haven was in Ohio it would be a lie. On the other hand, if
you were told “Lock Haven is in Ohio or a neighboring state”, that would be the truth. The clause
"or a neighboring state" is a statement of uncertainty. Notice how it affects your reaction to the
statement. In order to judge the value and meaning of an experimental result the reader of an
experimental report must be told both the accuracy and precision of the experiment. We do not live
in Ohio; it is similarly dishonest to hide the uncertainty in results from the reader. Whenever a
quantity is measured an estimate of the uncertainty in that measurement must be made and
recorded. Uncertainty in measurements will naturally result in uncertainty in the final result of the
experiment. This uncertainty must be calculated and reported!
Almost no experiment agrees with theory precisely. The degree to which we trust the results, and
therefore trust the theory that is being tested is always limited by our finite ability to measure
precisely and to include all relevant effects. In fact, when we get a discrepancy between
experiment and theory, we have either erred in our analysis, or perhaps we have discovered some
new physics. The former is cause for shame. The latter should be cause for rejoicing!