1.4. Measurement and Recording Data http://www.ck12.org
a. 120.5 lbs
b. 120.53 lbs
c. 120.534 lbs
In all likelihood you chose answer A. Good! Why didn’t you choose B or C?
Most likely, you did not select B or C because the finest measure on most bathroom scales is to the pound. At best
we can estimate to one decimal place.
This example highlights when and why significant digits are used in science, and especially in physics. When an
experiment is conducted, data measurements can only be as precise as the equipment used to make the measurement.
Significant Digits—Reporting Data Correctly
The length of the pen shown in theFigure1.7 is measured as 14.58 cm. It’s easy to see that the pen is clearly larger
than 14 cm and that it is past the 14.5 cm mark, as well. Where does the 8 (at the end of this measurement) come
from? The 8 represents an estimate by the person measuring the pen. The end of the pen is somewhere between the
fifth and sixth millimeter. If it had appeared to have fallen exactly in the center of this interval, the final digit would
have been reported as a 5. In fact, depending upon who read the ruler, the final digit may have been reported as a 7
or 9, giving us a measurement of 14.57 or 14.59.The last digit of any measurement is significant but uncertain
- but notcompletelyuncertain! It’s reasonable to argue based on varying individual perspectives about whether
the last digit is a 7, 8, or 9 (you may even say 14.60) but no one will argue, based on what they can see, that the
reading is between 14.50 and 14.55. There is a difference, however, in reading the ruler as best as you can, and the
actualuncertaintyof the ruler. Mass-produced measuring tools typically have a precision of no better than±^12 of
the smallest interval of the device. Therefore, 14.58 cm is recorded as 14.58±0.05 cm, [14.53-14.63].
FIGURE 1.
A pen measured in metric units having 4
significant digits.Note: we will not cite our results using a range of values when working on problems in later chapters. This method
should, however, be used in reporting laboratory data.
What if someone had used the same ruler inFigure1.7 to measure the pen and stated it was 14.573 cm? How
believable is it that they could read a measurement of 0.003 cm with this ruler? It’s not believable at all! (Think of the
example above.) The 7 was a reasonable guess, but there’s nothing reasonable about claiming the 3 is readable—it’s
a physical impossibility with this ruler. The ruler measures a maximum of 2 significant figures to the right of the
decimal- no more and no less!
What about zero?
Zero can be a source of confusion when it comes to recording significant digits. Let’s see if we can understand the
source of this confusion.
Using the ruler inFigure1.8, a student measures a calculator’s length as 18 cm. What’s wrong with how this
measurement is reported? A measurement must include a last digit, which is always an estimate. This final estimated
number is uncertain but reasonable (significant). This last digit informs us of the precision of the device used.
Since 8 is the last digit, it must be an estimate. Therefore it is impossible for the tool to measure numbers like 18.2 or