1.6.1 - The Magnitude
and Reliability of the Measurement

Since
all experimental measurements have uncertainties, we need to know how much we
can trust the measurements. Let's illustrate the idea by considering the mass
of an object measured on two different balances. Let's assume that the two balances
are accurately calibrated. One measurement is taken on a "crude" balance, and
the other is taken on a "sophisticated"
balance. The table below shows us how to interpret the two masses.

crude balance

sophisticated
balance

Measurement

12.4 g

12.4536 g

On a crude balance, we
are able to measure to 1 decimal place. On a sophisticated balance, we are
able to measure to 4 decimal places. The digits that have no uncertainty are
the digits in blue. In other words, the digits
that are in blue are the digits that are CERTAIN.
The last digit of a measurement that is shown in red
always carry an uncertainty due to estimation.

In scientific work, we
must always be careful to write down the quantity of measurements properly
and to report and calculate quantities that reflect on the accuracy of the
measurements. For these reasons, it is important to indicate the margin of
error in a measurement by clearly indicating the number of significant
figures.

We can determine the number
of significant figures for any measurement.

# of sig fig = # of digits that are CERTAIN + 1 final uncertain digit

Points to note:

12.4539 g is a measurement
with 6 significant figures. (All digits except for the the final digit '9'
are CERTAIN. The final digit '9' is the uncertain digit.)

12.4 g is a measurement
with 3 significant figures. (All digits except for the the final digit '4'
are CERTAIN. The final digit '4' is the uncertain digit.)

Given that the measuring tool
is correctly calibrated, in general, a measurement that
has more significant figures is the more accurate or reliable measurement.