Significant Figures in the Laboratory
by
Dr. David Summers,
Quantitative
observations in chemistry laboratory work routinely include measurements of
such quantities as masses, volumes, and temperatures.
Despite the apparent diversity of these measurements, they all share one
general feature that’s important to notice.
They all involve the reading of a
scale – the markings on a graduated cylinder, a meter stick, a beam
balance, an optical scale, etc. Many
instruments in the modern laboratory have digital readouts. You should not be lulled, however, into
thinking that they are absolutely accurate, since somewhere, at some time, the
digital scale was calibrated by a human being reading a scale.
The thermometer at
the right will serve for our present discussion.
Read this thermometer and write
your reading down here: _________°C. (Do
it NOW!)
Without being able
to peek over your shoulder, it’s hard to discuss your actual measurement, but a
safe bet would be that your answer is “24pointsomething.” Assuming that the marks on the thermometer
were correctly placed, we can say that the “2” and the “4” are known with total certainty.
However, more can
be said about the temperature. You would
probably agree that the reading is not as high as 24.9 °C, and it is certainly
higher than 24.0 °C, or something in this neighborhood. Maybe the best answer would be 24.3 ± 0.1 °C, indicating the apparent doubt we
have about how far the mercury is between 24 °C and 25 °C. Our best guess for the number is 24.3, and the uncertainty is
about ± 0.1.
Before continuing,
an important definition: We will say that the Celsius temperature is
known to three
significant figures.
The number of
significant figures is found by looking at the position of the first digit in
the uncertainty (the tenths position in this example) and then counting all the
digits in the number (from the left) up to and including this position.
Sometimes significant figures are explained
by saying that we count all the digits whose certainty is known (“2” and “4”)
plus the first uncertain digit (the “3”). This can be misleading, however,
because if the reading is 24.9 ±
0.1, then the “ones” digit is not certain (it is “4” in 24.8 and 24.9, but it
is “5” in 25.0), but the “best guess,” 24.9, still has three significant
figures.
Chances are that
you know your weight to three significant figures also. “I weigh about 152 pounds” suggests that your
actual weight might be 150154 pounds or so.
The uncertainty is about ± 2, in the ones position. The number 152 has 3 digits up to and
including the ones position, or 3 significant figures.
If this is where
things ended with significant figures, all that we would have is a definition
to play with, but there is more –
General Assumptions Regarding Significant Figures
1. Notice that the number of significant
figures is determined by the person doing the measuring. You need to decide the uncertainty in your
scale readings. Naturally, however, if
the instrument is out of adjustment, there will be “builtin” error or
uncertainty that you might not be aware of.
In some cases, this might reduce the number of significant figures you
can claim to have, just as a badlysighted rifle would reduce target practice
scores. In Chemistry, we assume
“wellsighted rifles,” so the uncertainty is presumed to be something between
you and the scale you are staring at.
This assumption is not always justified – in some cases,
you may want to blame the instrument or procedure itself for errors you
find. Fine! Just supply evidence and reasons!
2. Notice
that in everyday life, you tend to follow an unspoken, intuitive rule:
Stop a numerical expression when you reach the position
of the first digit of uncertainty. A
person says “I weigh about 155 pounds.”
He doesn’t say, “I weigh about 155.2386439 pounds.” You followed this
intuitive rule with your thermometer reading by stopping at the tenths
position. All we ask is that you continue to follow
this unspoken guideline in chemistry. Stop a numerical expression when you reach
(roughly speaking) the first uncertain digit.
3. Notice, also, in everyday life people assume
when they hear a numerical expression that the last digit is the only
uncertain one. If you are told that
the road you are looking for is 4.6 miles ahead on the right, you would be a
little upset to miss the turn, only to discover that it was 3.6 miles instead. If the person had said “about four miles,”
you might have started looking for the road earlier. Since the tenths were expressed, you assumed
that the “4” was known with certainty – just because that’s the way we
intuitively operate. Again, all we ask
is for you to make the same assumption.
The first uncertain digit is the last one expressed. If you say the magnesium ribbon is 18.46
centimeters long, we will take the “1” and “8” and “4” to be certain, but will
assume the “6” is a little doubtful – therefore we declare there are “four
significant figures.”
Multiplication
and Division
Now, let’s look at
what happens when we want to multiply or divide numerical measurements. A student measures a rectangular piece of
tile, and reports the measurements:
Length: 15.1 cm; Width: 3.2 cm.
What, then, is the area of the tile surface?
Here is the way one
student worked the problem on a calculator:
Area = (Length) x (Width). A = (15.1 cm) x (3.2 cm) = 48.32 cm^{2}.
The student was
quite happy with “48.32 square centimeters.” This answer, however, is not only
incorrect, but downright dishonest!
Remember that the
last digit reported is the uncertain one.
The honest length is 15.1 ± 0.1 cm, and the honest width is 3.2 ± 0.1
cm. (The uncertainties might even be
larger than this.) The area of the tile could
conceivably be (15.2 cm) x (3.3 cm) = 50.16 cm^{2}. At the other extreme, the area could be (15.0
cm) x (3.1 cm) = 46.5 cm^{2}.
So what are we
certain about in this answer? The area
could vary from over 46 to 50 cm^{2}, simply due to the doubt that is
always built into judging distance between scale markings. The calculated number is 48.32 cm, but the
uncertainty is about ± 1.8 cm. Since the
first position in the uncertainty is the ones position, that is where the value
of the area should “stop,” therefore the correct answer must be “48 cm^{2}.” Everything after the doubtful 8 must be
chopped off. It is dishonest to report
48.32 cm^{2}, simply because that would imply that the first uncertain
digit is in the hundredths place. You
just don’t know anything approaching this much certainty about the number. If you want higher precision, the only way to
get it is to make more precise measurements.
Carrying out things to greater extremes on paper when you calculate is
simply selfdeception and a waste of time.
Another example:
7 sf 3 sf 3 sf
(21.00000) x
(3.00) = 63.0 (NOT 63.00000!!)
The same rules
holds true for division:
21.00000 / 3.00 =
7.00
(Our answer is limited
to three significant figures by the 3.00.)
If you are multiplying and dividing together, things still work the same
way:
[(2.000) x (4.1) x (10.0000000)] / 3.00000 =
27 (NOT 27.3333333!!)
Here we are
limited by the two significant figures in 4.1!
Zeros
as Significant Figures
We have been
implying that the number 10.0 has three significant figures. If the person went to the trouble of putting
the zero after the decimal point, there must have been a reason for it; this
just happens to be the first uncertain digit, we assume. This is fine.
(Of course, many students just add zeros to add zeros, so we have to
follow the rules so we know that 10.0 does indeed have
three significant figures.)
How many
significant figures are there in 10,000 yr?
If this represents the estimated time since the last Ice Age, it’s
fairly clear that there are five significant figures here! In other words, we are uncertain by more than
± 1 year – in fact, the uncertainty is probably more than hundreds of years,
maybe even thousands of years.
How would you
express 10,000 yr to indicate that this might be uncertain by, say, at least a
few hundred years? You can’t chop the
last two zeros – they are needed to represent the size of the number. This is the best way: 1.00 x 10^{4}
yr. In scientific notation we simply
leave those zeros that we wish; we are showing three significant figures in
1.00 x 10^{4}; 10,000 to two significant figures would be 1.0 x 10^{4}.
Now work this
problem: (10.0) x (10.0) = ??? Well, the
answer should have three significant figures, and the answer is obviously “one
hundred.” But if you write 100, you are
leaving the reader to wonder whether any of the zeros are significant.
Solution: 1.00 x 10^{2}.
How many
significant figures are there in 0.00033?
Before you think too far here, put the number in scientific notation:
3.3 x10^{}^{4}. Now how many significant
figures are there? Both expressions have
two significant figures. Apparently the
only function of those lefthand zeros in 0.00033 is to keep the decimal point
where it belongs. Zeros to the left of
all other numbers are never significant; they are only
placeholders.
To summarize, here
are some examples of significant figure determinations:
4.00030 
6 significant
figures 
0.00300 
3 significant figures 
1.02050 
6 significant
figures 
10,000.0 
6 significant
figures 
10,000 
? significant
figures (ambiguous; could be 15) 
10.000 
5 significant
figures 
If you fail to
understand any of these examples, check with your instructor.
Rounding Off
If we wish to
express the number 326.337 to four significant figures, we would write
326.3. To make it have 3 significant
figures, since the first nonsignificant figure is
less than 5, we simply drop it. How
would you express the same number to two significant figures? 320?
330? 3.2 x 10^{2}? 3.3 x 10^{2}? Here, the first nonsignificant
figure we are dropping is greater than 5 – the number is closer to 330 than
320. But “330” fails to tell us whether
the zero is significant. 3.3 x 10^{2}
is the clearest answer.
Suppose we wish to
express the number 37.52 to two significant figures. Here the first nonsignificant
figure being dropped is 5 and it is followed by a nonzero digit, 2. Since the 2 is the second nonsignificant
figure, it contributes nothing in helping us make our decision. In fact, we treat 37.52 as though it were
37.50. So, do you round up or round
down? Example: Express 37.50 to two significant
figures. It is clear there is not really
a best way of operating here, since 37.50 is just a close to 37 as it is to
38. Many scientists decide in cases like
this to round to the even digit. This
rule simply guarantees that over a large number of “roundoffs” about half the
time we would round up, and half the time round down. Thus 38.5 to two significant figures would
also be 38. However, 39.5 would be
rounded up to 40, so that the last digit of the number, the zero, is an even
number. Using this rule, you will never get an ODD number when you
round off a number ending in 5. It is
expected that you will follow this rule for any rounding that you do.
Final Comments on Significant Figures
1. Learn the rules – they save time – but
remember that the only justification for all of this business is to avoid
saying more about a calculated value than you can possibly know. It should be admitted that “significant
figures” are only a simple shortcut for indicating the uncertainty of numbers;
if you pursue this topic further, you will discover
cases where “significant figures” give you incorrect estimates of
uncertainty. This should not diminish
their routine utility; as long as you “think
significant figures” you will avoid gross misstatements of uncertainty
(and save yourself a lot of lost
points). You should not have to ask “How
far do I carry my answers out?” Stop
when you reach the first position of uncertainty.
2. Textbook authors and problem writers
routinely violate significant figure rules.
You will probably notice this more than once in this course. Be assured that we fully intend to
consistently follow our own advice, and we will expect you to do the same –
particularly on laboratory calculations and on homework and tests.
3. What is the “rule” for adding and
subtracting numbers with significant figures? Well, there really isn’t any, if you insist
on referring to significant figures.
But, there is a simple way of knowing how to round off your addition and
subtraction results:

2.34000 
= 
2.3 

0.001 
= 
0.0 

4.3 
= 
4.3 

10.82 
= 
10.8 
+ 
3.37 
= 
3.4 

? 

20.8 
Round off each
term to the first position of uncertainty in any number that contains an
uncertain digit. Then add.
(Or add the numbers first, and then round back to the least amount of
decimal places in any numerical value that was included in the calculation.) In either case, you have avoided extending
things beyond the first uncertain digit.
4. Please keep in mind that this entire
discussion has been limited to numbers involved with scalereading type
measurements. There are other kinds
of numbers – particularly counting numbers or defined numbers –
which can lead you astray if you blindly follow rules.
For example, there are exactly 12 in a dozen. The number “12” has what amounts to an
infinite number of significant figures: 12.00000000000……. and so on.
There are exactly 1000 milliliters in a liter, because we define
it to be that way.
If you count 23 people in a room, that is assumed to be exactly
23.
These kinds of numbers can never
limit the number of significant figures in any calculated result from
multiplying or dividing. Thus: If I count exactly 144,000 nails, then
that is 12,000 dozen nails, exactly, even though you might think that
144,000/12 should be carried out to only two significant figures. Similarly, in converting 1.0612 liters to
milliliters I would calculate (1.0612 L)(1000 mL/L) =
1061.2 mL The
number of significant figures (5) in the answer is limited by the number of
significant figures in 1.0612 L, NOT by the number 1000, which in this
example is an exact number.
Lab Instructor Comment:
Please review Chapter 2 in your Lecture 1025 textbook for the
discussion of Significant Figures. See Chapter 1 in CHM 1032C. Dr
Summers has an excellent handout and he has given us permission to use any of
his handouts here at FSCJ.