Review of
Significant Figures
Dr. C. Yau
Spring 2014
(Loosely based on Sec 2.3)
from Jespersen 6th edition)
1
Accuracy vs. Precision
• Accuracy is how close a measurement
•
•
is to the correct value.
Precision is how close several
measurements are to each other.
When calibrated instruments are used
properly, the greater the number of
significant figures, the greater is the
degree of precision for a given
measurement.
2
Uncertainty in Measurement
When making a measurement, we record
all the digits that we are certain of and then
estimate one more digit.
What is the volume of the
liquid in the graduated cylinder
to the left? Ans. 21.3 mL?
21.4 mL?
to the right? Ans. 21.33 mL?
21.32 mL?
There is always uncertainty in the last digit we record.3
Uncertainty in Measurement
Another way of looking at this is to say that
we record to 1/10 th of the smallest division
of the measuring device.
Smallest division of the
graduated cylinder on the right
is _____ mL.
1/10 th of that would be ___ mL.
Its reading should therefore be
recorded to what decimal
place? Ans. ...to 2 decimal places
(21.32 mL)
4
Uncertainty in Measurement
Smallest division of the
graduated cylinder on the left is
_____ mL.
1/10 th of that would be ___ mL.
Its reading should therefore be
recorded to what decimal
place?
Ans. ...to 1 decimal place
(21.3 mL)
5
(Graduated cylinder on the left)
21.3 mL has 3 significant figures
(sig. fig.) and 1 decimal place.
(Graduated cylinder on the
right)
21.32 mL has 4 sig.fig. and
2 decimal places.
Know the difference between the
number of sig. fig. and decimal
places!
6
Errors Arise From A Number Of Sources Including:
• Errors - inherent error due to the equipment or
procedure (instrumental or method errors)
– Changing volume due to thermal expansion or
contraction (temperature changes)
– Improperly calibrated equipment
– procedural design allows variable measurements
• Mistakes-blunders that you know that you
have made. Do not use these data
– Spillage
– Incomplete procedures
– Reading scales incorrectly
– Using the measuring device incorrectly
7
Reducing Error:
• Errors can often be detected by making
repeated measurements
• Error can be reduced by calibrating
equipment
• The average or mean reduces data
variations: it helps find a central value
8
How many sig.fig. are in a number?
All digits are significant EXCEPT:
• Zeros to the left of the first nonzero digit (leading
zeros) are never counted as significant.
0.00045 has only 2 sig. fig.
• Zeros at the end of a number without a decimal
point are assumed not to be significant (Trailing
without a decimal place)
• 5900 is assumed to have 2 sig. fig.
• We don’t really know for sure.
• The 2 zeroes are ambiguous, and the number
should be expressed in scientific notation.
• 30.0 on the other hand has 3 sig.fig.
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• The zeroes are not ambiguous. Why not?
Learning Check: How Many
Significant Figures Are There In
the Following?
2.33
3
500.0
4
1000
.0500
Ambiguous, assumed to be 1
3
How many sig. figs. are there in the number
5 sig. fig.
010.010?
10
Numbers with trailing zeroes & without a
decimal are said to be ambiguous in terms
of sig. fig. and must be expressed in
scientific notation:
3400 in 3 sig. fig. ______________
1530 in 2 sig. fig. ______________
1400 x 103 in 3 sig. fig. _______________
3000 x 10-2 in 2 sig. fig._______________
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Exact Numbers
• Numbers that come from definitions are exact
and have no uncertainty.
• They can be assumed to contain an infinite
number of significant figures.
Example:
25 mph means 25 miles/hour
25 has 2 sig. fig.
Per hour means exactly 1 hour
They do not affect the sig. fig. in a calculation.
(1.000000000…. hour)
12
Measurements Limit The Precision
Of Calculated Results
Rules for combining measurements depend on
the type of operation performed:
• Multiplication and division
– The # of sig. fig. in the answer should be the
same as the # of sig. figs of the least precise
measurement.
3.14  2.751

13.49709375
=
13
0.64
(3 sig. figs.)  (4 sig. figs.)

(2
sig.
figs.)
(2 sig. figs.)
13
How many sig. figs. result from the
following: 12.33 x 0.00002?
Ans. ONLY ONE!
0.0002466 = 0.0002 = 2 x 10-4
14
Addition and Subtraction
The answer should have the same # of
decimal places as the number that is
least precise, with the fewest # of
decimal places.
3.247 ← 3 decimal places
41.36 ← 2 decimal places
+125.2
← 1 decimal place
169.807 ← ans rounded to 1 decimal place
169.8
How many sig. figs. result from the following:
10.33 - 0.0344?
Tricky! Do this carefully before blurting out the
10.2956 = 10.30
answer! Ans. Four!
Ans.
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Addition and Subtraction
What if the numbers do not have a
decimal points?
e.g. 7200 + 132 = ?
Mathematically, the answer is 7332, but
should we keep all 4 digits? NO
7200 is precise only to the hundred place.
It will limit the precision of the answer
Ambiguous
to the hundred place. 7200
132
zeroes
7332 = 7300
= 7.3x103
Think Carefully!
How many sig. figs. result from the
following?
(10.0 x 10.88)-12.2
A. 2
2.20
B. 3
10.0 x 10.88 = 108.8
C. 4
108.8 – 12.2 = 96.6
D. 5
E. none of these 96.6  43.90909090 = 44
2.20
Ans. 2 sig. fig.
17
Scientific Notation
Let’s pinpoint exactly what is meant by
“scientific notation.”
It is to express a number so that the decimal
is behind the first nonzero digit.
4.271 x 106
e.g. 427.1 x 104 =
5.1 x 10-3
0.0051 =
1302 x 10-5
= 1.302 x 10-2
Expressing a number in sci. notation should not
change the # of sig. fig.
18
Do NOT use scientific notation indiscriminately!
Scientific notation is needed ONLY....
1) ...if the number has ambiguous zeroes.
e.g. 150 x 2.0 = 300 = 3.0 x 102
2) ...if the number is very small (smaller than
0.01)
e.g. 0.003 = 3 x 10-3
3) ...if the number is already in exponential
notation (but not scientific notation)
e.g. 24.831 x 1056 = 2.4831 x 1057
19
What is wrong with using scientific
notation when it is not necessary?
It shows your ignorance on the use of
scientific notation.
Would it make sense for someone to tell
you to measure out 3.5x101 mL, when all
he/she needs to say is 35 mL ?
Would it make sense for someone to tell
you to weigh out 4.5 x 10-1 g when all
he/she needs to say is 0.45 g ?
20
Significant Figures When Calculating Averages
What is the average of 35.7 and 35.8?
You probably can come up with an answer
without a calculator or doing mental arithmetic.
In math, it would be the midpoint, 35.75.
In science, however, if these were measured
numbers you might consider the operation:
35.7 + 35.8
71.5
=
= 35.75
2
2
What is wrong with this answer?
The average cannot have more precision than
the numbers from which it is derived! Ans. 35.8
One last question on sig. fig......
Think before you blurt out the answer!
What is the answer to the correct sig. fig.
for this calculation?
3.42 x 10-32 + 2.1 x 10-38
Ans. 3.42 x 10-32 not 3.4 x 10-32
3.42 x 10-32
+ 2.1 x 10-38
5.52 x 10-??
NO!!
3.42 x 10-32
+ 0.0000021 x 10-32
3.42 x 10-32
3 sig. fig.!
More practice problems are available on my HomePage: Click here.
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