Download Significant Figures and Measurement Techniques What are they?

Significant Figures and Measurement Techniques
What are they?
· Scientists make a lot of measurements. They measure the weight of things, the length of things, times, speeds,
temperatures, etc.
·
When they report a number as a measurement the number of digits and the number of decimal places tell you how
exact the measurement is
o For example: 121 is less exact than 121.5
o The difference between these two numbers is that a more precise tool was used to measure the 121.5.
o If a scientist reports a number as 121.5 they are saying that they were able to measure that quantity up to the
tenths place.
o If a scientist reports a number as 121 they are saying that they were able to measure that quantity up to the ones
place.
o The total number of digits and the number of decimal points tell you how precise a tool was used to make the
measurement.
Why do I have to know about significant figures?
1. Reporting measurements:
o You need to know how to report measurements you take in the lab so that you are not only reporting a
measurement but also stating how good of a tool you used.
2. Math with sig figs
o You need to know how to do math with significant figures so that you don’t end up reporting a number that is
more exact than the numbers you used to calculate it.
1. Reporting measurements:
a. There are 3 parts to a measurement:
i. The measurement
ii. The uncertainty
iii. The unit
b. Example: 5.2 ± 0.5 cm
i. Which means you are 100% sure the actual length is somewhere between 4.7 and 5.7
c. No measurement should be written without all three parts.
d. The last digit in your measurement should be an estimate
i. If the smallest marks on your tool are .001 apart (as they are on a meter stick that has millimeters marked)
then your last digit should be in the ten­thousandths place (i.e. 0.0001)*
ii. Logic:
0
5
10
15
20
25
30
35
In the above, you would report the length of the bar as 31 ± 2 cm. You know the bar is longer than 30 cm and the
last digit is your best guess. You are 100% sure the actual bar length is between 30 and 33 cm.
0
5
10
15
20
25
30
35
In the above, you would report the length of the bar as 31.0 ± 0.5 cm. The bar appears to line up with the 31st mark
and you know it’s more than ½ way from the 30 mark and less than ½ way from the 32nd mark. So you can be 100%
sure the actual length of the bar is between 30.5 and 31.5 cm.
e. The uncertainty is ½ the amount between the smallest hash marks. Notice in the above three examples that this
is the case. This is a good rule to go by but really, making measurements is a lot more difficult than just
memorizing that rule.
i. This rule may change depending on the book you look at or the teacher you work with, even the tool you
work with. Estimating your uncertainty is an estimate and it is really up to you to be as accurate as
possible about how uncertain you are. As long as you are thinking about each situation and about how
sure you are of each digit in your measurement – than you’re probably doing the right thing.
ii. Some uncertainties are determined by the manufacturer. (e.g. electronic balances, probes)
iii. Some uncertainties are standards (e.g. Burets are always ± 0.02 mL)
iv. Some times you have to deal with fluctuations in the tool and your uncertainty is likely to be much larger
v. Some times you have to deal with instruments in which the markings are very far apart in which case your
uncertainty is likely to be less than half the distance between the smallest marks.
vi. Sometimes you have double the uncertainty if you have to measure something from two ends – like using
a ruler. The starting point has just as much uncertainty as the ending point.
f.
Measurements can sometimes be difficult to determine. The following are some important
techniques.
i. When measuring liquids that have a curve at the surface, measure from the bottom of the
meniscus. The meniscus is the curve formed at the surface of a liquid due to attraction of
the liquid for the sides of the container (adhesion). Measuring from the bottom – you
should get 5.5 ± 0.1 mL (assuming marks represent milliliters).
ii. Sometimes the measurement on an electronic balance will fluctuate. Start with the
numbers that are not fluctuating and then make your best guess as to what the next digit
would be. Say for example you are weighing something on a balance and you get the
following readings:
1. 12.345
2. 12.320
3. 12.349
4. 12.357
5. 12.327
I would report this measurement as 12.34 ± 0.05
7
6
5
4
3
2
1
*This is true for measurements that don’t fluctuate. If the tool you use fluctuates than your estimated digit will probably
not be smaller than the smallest hash mark on the tool but should indicate how sure you are of the exactness of your
measurement.
Remember: when reporting measurements, you need to do at least two things
1. give the measurement
2. tell how good a tool you used to measure it (this is given by the number of significant figures)
And it’s even better if you also give your uncertainty! Otherwise everyone will assume ± 1 in the last decimal place, and
that’s ok – if it’s true.
Counting Significant Figures
When you add two measurements together, you can end up with a number that makes
it look like you a much better tool than you actually did.
For example: let’s say you make the following measurements for the mass of something:
· Mass of empty container: 2.3 g
· Mass of container with copper: 22.54 g
What is the mass of the copper? 22.54 – 2.3 = 20.24 g
Answer to report: 20.2 g
Why 20.2 g and not 20.24g?
Since you only measured the container to the tenths place than the 3 is really an
estimate. Perhaps the actual value was 2.2 or 2.4… than the mass of copper could be
20.34 or 20.14 g. As you can see the difference in the tenths place is far more significant
than the hundredths place. So the mass you should report is 20.2 g.
Instead of going through this logic each time you solve an equation (especially since this
was a simple example and things get a lot more complex when you multiply and divide)
there are some rules to follow
Rule
Example:
#of Significant
Significant figures
Figures
are in bold
1. All non­zero digits are significant
1234.5667
8
2. Zeroes after a decimal point AND after a non­zero
12.0
3
digit are significant
0.0020
2
3. Zeroes between non­zero digits are significant
102
3
4. Zeroes at then end of numbers punctuated by a
120.
3
decimal point or line are significant.
12Ō0
3
1200
2
5. When adding and subtracting, your answer needs to
12.0 + 5.23 =17.2
1 dec. place
have the same number of decimal places as the
14.56 – 0.02 = 14.54 2 dec. place
number with the fewest decimal places
75 – 5.5 = 70.
0 dec. place,
but zero is
significant
6. When multiplying and dividing, your answer needs to
12 x 2 = 20
1
have the same number of significant figures as the
5.00 x 7.0 = 35
2
number with the fewest significant figures
2.00/6.0 = .33
2
7. Exact numbers can be treated as if they have an
3.2 x 2 = 6.4
2
infinite number of significant figures. (example, you
know you have 5 quantities of something you are
adding together)
8. When doing more than one calculation, do not round
13.2 x 2 / 5 = 5
numbers until the end.
(not 6)
*Note: I didn’t record uncertainties here or show you how to handle uncertainties when
numbers are mathematically manipulated to simplify things. There are other documents
that discuss what to do with uncertainties.