Allan Hancock College
Physics Lab
Measurements and Uncertainty
The study of Physics is a quantitative science. We can discuss physical principles and systems
qualitatively, but in order to investigate and solve problems, we must make quantitative
measurements---that is to say, we take data. We then analyze the data, looking for patterns or
relationships to find our answers. However in order to know how reliable the answers are, we
need to assess how good our data is.
There is an uncertainty or error associated with any measurement. The word error does not
mean that the experimenter made a mistake, rather it is a measure of the resolution or accuracy
of the measuring device. For instance a meter stick is broken down into 100 centimeters, and
each centimeter is divided into 10 millimeters. However no finer measurement is possible with a
meter stick, so the resolution of the device is on the order of millimeters. For instance, the long
side of an envelope is measured to be between 21.1 cm and 21.2 cm. The best estimate of the
length might be 21.15 cm with an uncertainty or error of 0.005 cm. This is written as
x = 21.15  .005 cm.
This means that actually there is a range of answers, the envelope length falls in the range from
21.1 cm to 21.2 cm.
This is random error and must be taken into account whenever a measurement is made. A rule
of thumb is that the error in measurement is half the resolution increment of the measuring
device. Other factors may affect this estimate of error.
Another type of error is statistical error. This is the measurement of the variation in a large
experimental group. For instance if 10 envelopes were measured and the lengths varied from 21
cm to 21.3 cm, then the best estimate of the envelope length would be the average length and
the error would be the standard deviation. The average length,
xavg = (xi) / N,
where N is the number of envelopes measured. The standard deviation is
x = [ (xi – xavg)2 / (N-1)]1/2
where xi represents an individual measurement.
A third type of error is systematic error. Systematic error occurs when a measuring tool is not
properly calibrated. It becomes apparent if measurements are consistently too small or too large.
Purpose:
To become proficient with various measurement tools and explore how to handle
experimental error.
Apparatus:
Meter sticks, vernier calipers, micron calipers, screws, and tables.
Procedures:
Each student should make there own measurements. Make a schematic
diagram of the experimental set-up, including the measurements to be made.
Make a table of your data, be sure to include columns for the random errors.
Experiment 1:
Measure the perimeter and area of a lab bench and calculate the random error in
these measurements.
Using a meter stick measure the length (x) and width (y) of your lab bench. What
are the errors, x and y, associated with your measurements? Remember,
there is an error associated with each measurement you make, so when you
move the meter stick, you are making another measurement and adding
additional uncertainty to the total measurement. So the error in the length and
width are
x = [x12 + x22 + ---- +xn2]1/2 and
y = [y12 + y22 + ---- + yn2]1/2.
The perimeter P = 2x + 2y, so the error in P, P = 2x and 2y. The area is
length x width, but A  (x)(y).
What is the range in Perimeter and the range in Area?
Experiment 2:
Measure the length of 10 screws and calculate the statistical error in their
lengths.
Using either the micron calipers or the vernier calipers measure the length of 10
similar screws. What is the average length, xavg, and the standard deviation, x?
x = [di2/(N-1)]1/2,
where d = xi – xavg
What is the best estimate for the length of the screws? The best estimate of the
length should be written as xavg  . Statistical theory states that if your measure
another screw, there is a 68% probability that it will be within 1 standard deviation
of the mean, a 95% probability it will be within 2 standard deviations of the mean,
and a 99.7% probability it will be within 3 standard deviations of the mean.