Chapter Two
Significant Figures,
Propagation of Error,
Graphs and Graphing
E
very measurement has an error associated with it. If you were to put an object on
a balance and weight it several times you will not get the exact same answer.
Generally, there are three kinds of errors. Each of these will be explained below.
Personal Error
Sometimes you make a mistake. Students want to report these mistakes as sources of
error in an experiment. Refrain from doing so. If you make a mistake, fix it. Your
mistakes are not considered a source of error and neither are the concentrations or
composition of the reagents you use. While you are not responsible for the quality of the
reagents you use, your instructor is, and if an error is found, it can be fixed. Therefore,
errors in composition or mistakes you make during an experiment like spilling a reagent
or missing an endpoint are never reported as sources of error.
Method Error
Sometimes when you do an experiment the method you used produces a consistent
error in the data. For example, if you mix two solutions together you can produce a
solid. One way of retrieving the solid would be to filter the entire solution and collect
the solid on the filter paper, but to weigh the solid, you must scrape it off of the filter
paper onto a weigh boat. It is obvious that you will never be able to scrape all of the
solid off of the filter paper so what you weigh will always be a little bit light. The
method used to do this experiment produces a consistently low value for the mass of the
solid produced. This is called a method error.
Method errors are always unidirectional, that is, they are always too large or too small.
As such, it is possible to figure out how much too large or too small the error is and
account for them. As a consequence, method errors are not usually considered as
sources of error when an experiment is being done.
Random Error
Random errors arise as random fluctuations in the measurement of your data. As
explained earlier, if you weigh an object several times you will get several different
answers. The values will fluctuate around some average value. Sometimes the value
will be too high and other times it will be low. There is no consistent direction for these
kinds of errors. They will be random. If many measurements are made (many = 20 or
more measurements) then statistically the average measurement is considered to be the
actual value.
Usually, we do not run an experiment 20 times or take 20 measurements. Usually, we
take three or maybe four measurements. The question then is, how well do we know the
value that we have just measured? We begin by taking an average and we expect the
average to be very close to the actual value, but random errors produce fluctuations in
the data and to be completely accurate we need to tell the reader the size of these
fluctuations. But as we will soon find, even the average is not necessarily a good
measure of the actual value. Just how accurate is this data and how precise is it? Some
of you might be saying, “What’s the difference?”
Accuracy and Precision
Accuracy and precision are very different from one another. Precision measure how
close one measured value is to another. Accuracy measures how close the average value
is to the accepted value. You should note that scientists do not use the term “true”
value. True values do not exist in science. Scientists do experiments that confirm values
that then become accepted but a “true” value never exists.
Consider the following dartboards. In the first case the darts are very close to one
another but they do not hit the bulls eye. If the bull’s eye represents the accepted value
then the darts are not very accurate, but since the thrower is able to cluster the darts
close to one another the thrower is being precise if not accurate. In the second case the
darts are both accurate and precise since they are clustered close to one another in the
bull’s eye. It is possible to be accurate and not precise as shown in the third case. Here
we see that the darts are scattered around the dartboard but the average of all these
throws is in the bull’s eye. So while the thrower was not very precise, the average was
very accurate. In the last case we see darts that are neither precise nor accurate. The
average of the throws are not in the bull’s eye and the darts are not very close to one
another. This is the worst possible case for a scientist. Science strives for both accuracy
and precision but this can sometimes be difficult to determine. When determining a
value that has never been measured we hope that we are being both accurate and
precise. Often though, we are only being precise and only time will tell if our data is
Precise but not Accurate
Accurate and Precise
Accurate but not Precise
Not Accurate or Precise
also accurate. When enough people have done our experiment and confirmed that they
have measured the same value then the value that we measured may eventually become
the accepted value. Only by comparing our value to an accepted value can we
determine if we are being accurate.
Significant Figures
One way of indicating how well a number is known is by using significant figures. In
general, significant figures mean counting the number of digits in a number. For
example, look at the number of significant digits in each of the following numbers
Number
34.8
1.0086
125.6
187,123
67
Sig. Figs.
3
5
4
6
2
Counting the number of digits in any number is relatively easy but zeros are always a
problem. There are two cases when zeros present a challenge to science students and
both share something in common; the zeros are placeholders. Lets take each case one at
a time.
Zeros at the End of Numbers without Decimal Points
Number
100
40
12000
1850
3568700
Sig. Figs.
1
1
2
3
5
To figure out the number of significant figures in each of the following numbers given
above, start on the left hand side and begin counting numbers that are not zeros. The
zeros in each of these problems are placeholders. How else do you write 100 than with
two zeros? The zeros are placeholders.
Zeros at the End of Numbers with Decimal Points
Number
100.0
40.0
12000.0
1850.00
3568700.00
Sig. Figs.
4
3
6
6
9
If a number has a decimal point after a bunch of numbers then all the numbers,
including the zeros, are significant. The decimal point makes all the difference. It tells
you that you know that these are zeros and they are not just placeholders. It is the
difference between saying that you have about $100 and that you have exactly $100.00.
If you say that you have about $100 then nobody would be surprised if you actually had
$109.57. But if you said that you had exactly $100.00 but actually had $109.57, then you
would be lying. You can see that having about $100 and having $100.00 is really two
very different statements. In the first number, $100 the zeros are simply placeholders
and you are only approximating $100, but with $100.00 the extra zeros have meaning
and you are trying to say that you know, to the penny, how much money you have.
Therefore, if you know that you have $100.00 and you wrote down that you had just
$100, you would be wrong. You would be conveying the wrong information about the
amount of money you had.
In science it is important to write down all the numbers that you know, even if those
numbers are zero. The zeros give the reader extra information. If you weigh something
and find that it weighs 24.650 grams, you are saying that you know the last digit is zero
so it is important to write it down. If you decided to write 24.65 grams when the
number is really 24.650 grams then you would lose some information about the mass
you just weighed. Truth is, 24.650 grams ≠ 24.65 grams. In the first number, you know
the last digit is a zero. In the second number, you only know that the last digit is a 5 and
you have no idea what the next number might be. You just lost some information about
the mass you just weighed. ALWAYS write down all the number a balance gives you,
even if they are zeros.
No Numbers Before the Decimal Point
Number
0.01005
0.001
0.1287
0.050060
0.00018789
Sig. Figs.
4
1
4
5
5
In the case where there are no numbers before the decimal point (except a zero) then all
the zeros that appear before numbers begin to appear are not significant and can be
ignored. So starting on the left, you continue moving right until you hit the first nonzero number and begin counting (in bold in the example above). This gives you the
significant figures in each number.
Addition and Subtraction
Now that we have established how to count significant figures we can now turn our
attention on how to use them in our work. In addition and subtraction, the result is
rounded off to the last common digit occurring furthest to the right in all components.
Another way to state this rule is as follows: in addition and subtraction, the result is
rounded off so that it has the same number of decimal places as the measurement
having the fewest decimal places (or digits to the right). For example,
100 (assume 3 significant figures) + 23.643 (5 significant figures) = 123.643
which should be rounded to 124 (3 significant figures). Note, however, that it is possible
two numbers have no common digits (significant figures in the same digit column).
When combining measurements with different degrees of accuracy and precision, the
accuracy of the final answer can be no greater than the least accurate measurement. This
principle can be translated into a simple rule for addition and subtraction: When
measurements are added or subtracted, the answer can contain no more decimal places
than the least accurate measurement.
(a)
150.0 g H2O
+ 0.507 g salt
150.5 g solution
You will notice in (a) that the first number has 4 significant figures and the second has 3
significant figures. The answer has 4 significant figures because the least accurate
number is actually the 150.0 because it is only known to the tenths place while the
second number is known to the thousandth. We therefore report our answer to the
hundredth place which results in a number with 4 significant figures.
(b)
56.0 g H2O
+ 55.5 g salt
111.5 g solution
In case (b) we have two numbers with 3 significant numbers each. They are both known
to the tenths place so our answer must be reported to the tenths place. Doing so
produces a number with 4 significant figures. Therefore, it is possible to increase the
total number of significant figures when adding numbers.
150.0 g salt + weigh boat
- 62.507 g weigh boat
87.493g salt = 87.5 g salt
In similar fashion to (b) above, when subtracting two numbers it is possible to lose a
significant figure. In this case we have two numbers that have 4 significant figures each
but when one is subtracted from the other and reported to the tenths place (the least
accurate of the two numbers) the result is a number with only 3 significant figures.
Multiplication and Division
In multiplication and division, the result should be rounded off so as to have the
same number of significant figures as in the component with the least number of
significant figures. For example,
3.0 (2 significant figures ) × 12.60 (4 significant figures) = 37.8000
which should be rounded to 38 (2 significant figures). This rule applies to more
complicated examples,
23.6 x 14.503 = 11.5734 => 12
8.215 x 3.6
Since the number 3.6 has the least number of significant figures in this problem (2
significant figures) the answer must be to 2 significant figures also. So the answer is
rounded to 12.
Rounding Numbers
There are two methods used to round numbers. The simpler, and more common, of the
two uses the following rule,
a) Round down if the last digit is 0, 1, 2, 3, or 4.
b) Round up if the last digit is 5, 6, 7, 8, or 9.
Therefore, from the previous example,
11.5734
11.573
11.57
11.6
12
10
6 Significant Figures – unrounded original number
5 Significant Figures – 4 rounds down
4 Significant Figures – 3 rounds down
3 Significant Figures – 7 rounds up
2 Significant Figures – 6 rounds up
1 Significant Figure – 2 rounds down
Banker’s Method
There is another method for rounding that is called the “Banker’s Method” that is more
accurate than the simple method of rounding shown above but it is used less often. This
method addresses how we round the number 5. Since the number 5 sits in the middle of
our number line, it is argued that rounding it up puts a greater emphasis on the
numbers that are 5 or greater and leads to answers that are slightly too large. To
compensate, a rule dealing just with then number 5 has been formulated that deals with
the error caused by always rounding up. The rule is a simple one,
Banker’s Rule – If the digit to be rounded is 5, make the
preceding digit an even number.
It is best to give an example of how this is applied. Suppose you have the following two
numbers, 3.215 and 3.225. Using the Banker’s Rule, both numbers would round to 3.22.
In the first case, 3.215, the digit previous to the 5 is odd (it is 1) so the 1 gets rounded up
to 2, making it an even number. In the next number, 3.225, the digit previous to the 5 is
even (it is 2), so the number is not changed and is just left as a 2. The result is that 3.225
actually gets rounded down to 3.22. So we see that using the Banker’s method, both
numbers, 3.215 and 3.225, get rounded to 3.22. Statistically, using this method, half of
the numbers would be rounded up and the other half would be rounded down so the
error that arises from always rounding up at 5 has been fixed.
While this method generally leads to better answers, it is not commonly used. Most
people simply round up at the number 5. Your instructor will tell you which method he
or she prefers.
Exact Numbers
In the lab and elsewhere you may be asked to find the average of a set of numbers. The
process is simple, and up the numbers and then divide by how many numbers have
been added. For example,
25.12
24.83
25.26
75.21
75.21 = 25.07 = Average
3
But the question comes up, how many significant figures should we report? The
tendency of a student is to look for the least number of significant figures in the problem
and report the answer to that number of significant figures. In this problem, the number
with the least number of significant figures is the division by 3 when taking the average.
But rounding your answer to just 1 significant figure would be a great mistake.
The number 3 does not represent just one significant figure. It is an exact number. That
is, it is actually 3.0000… for as many digits as you need. Numbers like these are exact.
In this case you have exactly 3 numbers that you are averaging. There is no such thing
as 3.1 numbers that you are averaging. You have either 3 numbers or 4 numbers but not
3.1. So the number 3 in this example is exactly 3 to as many significant figures as you
need. So your answer is limited by the 75.21 (4 significant figures) and not by the exact
number, 3, so the answer should be rounded to 4 significant figures (25.07).
We use exact numbers all the time. A dozen eggs, 5 people, and a six-pack of beer all
represent exact numbers. Exact numbers never limit the number of significant figures
you report.
A Practical Example Using Significant Figures
We began this section by talking about the errors that arise when measurements are
made and how to report numerical values based on these measurements. In general, we
report answers to problems based on the least number of significant digits in our
problem. As a simple example of how this might work in a problem involving some
kind of chemistry, why don’t we see how we might report the density of water using
some common glassware found in a lab.
Buret
Graduated
Cylinder
50.00 mL
50.0 mL
Erlenmeyer
Flask
50 mL
Suppose that we want to measure the density of water using each of these pieces of
glassware so we fill each of them to their 50 mL mark and then pour that water out into
another container and weigh it. These data are shown in the following table,
Density of Water using Various Glassware
Glassware
Buret
Grad Cylinder
Erlenmeyer Flask
Sig. Figs
4
3
1
Volume
50.00 mL
50.0 mL
50 mL
Mass
49.982 g
49.871 g
55.236 g
Mass/Volume
0.99964 g/mL
0.99742 g/mL
1.10472 g/mL
Reported Density
0.9996 g/mL
0.997 g/mL
1 g/mL
Each of these pieces of glassware can be read to varying degrees of accuracy. The buret
is the most accurate measure of volume so we write 50.00 mL to indicate that it can be
read to 4 significant figures and 2 digits past the decimal point. The marks found on an
Erlenmeyer flask are only approximate volume measurements. They can be off by as
much 20 mL or more so we must indicate this by writing our volume as 50 mL which is
only 1 significant figure. As a consequence, using an Erlenmeyer flask to measure an
accurate volume is not advised. Burets and pipets are the best way to measure volume
and so are graduated cylinders. Other glassware is far less accurate and should not be
used unless the exact volume is not important.
You will notice that our mass readings are all 5 significant figures because a balance is
capable of giving you this many digits in its reading. Always write down all the digits
given to you by a balance. Don’t ever round and never leave off zeros if they are the last
digit (53.3 grams is NOT the same as 53.300 grams). The zeros are significant and they
convey information about how well you know your number. Always write down every
digit given to you by a balance.
When we calculate the density of water we divide the mass of the water by its volume.
Your calculator will give a bunch of numbers but not all of these numbers are significant
so they must be rounded to the proper number of significant figures. In each of the
three examples, the least number of significant digits is found in the measurement of the
volume of water being weighed. For the buret, this means that we must write our
density to 4 significant figures so 0.99964 g/mL gets rounded to 0.9996 g/mL, and this is
the density we would report. Similarly a graduated cylinder can be read to just 3
significant figures so, even though the mass has been measured to 5 significant figures,
we can only report 3 digits in our final answer. In this case, for the graduated cylinder,
we would round to 3 significant figures since that is the best we can read the volume. In
this case, using a graduated cylinder, our reported density would be rounded to 0.997
g/mL. Finally, since an Erlenmeyer flask can only be read to 1 significant figure, our
reported density would be just 1 g/mL to reflect this fact.
Significant digits give us a quick way of determining the number of digits we should
write down when doing a calculation, and while the answers we get are always close to
the right answer, sometimes they are not. More importantly, significant figures have an
important theoretical basis that cannot be seen using the simple methods employed
here. Therefore we must reexamine significant figures in light of the actual error found
in each measurement and using statistical methods, determine the number of significant
figures to report in an answer. These statistical methods only apply to random errors.
Let us investigate random errors further.