Section 2.3 Uncertainty in Data
• Define and compare accuracy and precision.
• Describe the accuracy of experimental data using
error and percent error.
• Apply rules for significant figures to express
uncertainty in measured and calculated values.
experiment: a set of controlled observations that
test a hypothesis
Section 2.3 Uncertainty in Data (cont.)
accuracy
percent error
precision
significant figures
error
Measurements contain uncertainties
that affect how a result is presented.
Accuracy and Precision
• Accuracy refers to how close a measured
value is to an accepted value.
• Precision refers to how close a series of
measurements are to one another.
Example: The acceleration
due to gravity on Earth is
9.8 m/s2. Look at the lab
data for two students.
Whose data was most
accurate?
Whose data was most
precise?
Sue’s
Trials
1
9.79
2
3
9.81
9.82
Fred’s
Trials
1
How would you describe
Fred’s data?
Calculated
acceleration
(m/s2)
2
3
Calculated
acceleration
(m/s2)
8.85
8.54
9.86
Accuracy and Precision (cont.)
• Error is defined as the difference between
and experimental value and an accepted
value.
•The error equation is
error = experimental value – accepted value.
Example: The acceleration
due to gravity on Earth is
9.81 m/s2. Look at the
lab data for two students.
What is the error for Sue’s
trial 3?
Sue’s
Trials
1
9.79
2
3
9.81
9.82
Fred’s
Trials
What is the error for Fred’s
trial 1?
Calculated
acceleration
(m/s2)
1
2
3
Calculated
acceleration
(m/s2)
8.85
8.54
9.86
Accuracy and Precision (cont.)
• Percent error expresses error as a
percentage of the accepted value. The
percent error is always expressed as an
absolute value.
Example: The acceleration
due to gravity on Earth is
9.81 m/s2. Look at the
lab data for two students.
What is the % error for
Sue’s trial 3?
Sue’s
Trials
1
9.79
2
3
9.81
9.82
Fred’s
Trials
What is the % error for
Fred’s trial 1?
Calculated
acceleration
(m/s2)
1
2
3
Calculated
acceleration
(m/s2)
8.85
8.54
9.86
• In a lab, students were to calculate the density of
aluminum. The accepted value is 2.7 g/cm3.
• In George’s experiment, they found the value to be
2.5 g/cm3. Find:
A). George’s error:
B). George’s percent error:
Section 2.3 Assessment
A substance has an accepted density of
2.00 g/L. You measured the density as
1.80 g/L. What is the percent error?
A. 0.20 g/L
A
0%
D
D. 0.90 g/L
C
C. 0.10 g/L
A. A
B. B
C. C
0%
0%
0%
D. D
B
B. –0.20 g/L
• 3.3 Significant Figures
–(Sig Figs)
Significant Figures
• Often, precision is limited by the tools
available.
• Significant figures include all known digits
plus one estimated digit.
Significant Figures (cont.)
• Rules for significant figures
– Rule 1: Nonzero numbers are always significant.
– How many significant figures in each of the
following:
1234
22
7.7654
Rule 2: Zeros between nonzero numbers
are always significant. (sandwich rule)
How many significant figures?
101
.103
3
3
5500.002
7
Rule 3: All final zeros to the right of the
decimal are significant.
(You shouldn’t tack a zero on after the decimal if you don’t
actually measure it).
How many significant digits?
123.40
103.440
.1230
Rule 4: Placeholder zeros are not
significant. To remove placeholder zeros,
rewrite the number in scientific notation.
In 11 000, the 3 zeros are just place-holders…they are just
showing that the number is eleven-thousand, not eleven.
How many significant figures?
10300
0.0012
0.11030
Rule 5: Counting numbers and defined constants have
an infinite number of significant figures.
What this means….if you say there are 8 people in
the room, that has infinite significance. You can’t have
8.2 people…or 7.9 people.
If you were to average 3 masses, 55.5 grams, 54.6 grams
and 53.5 grams….how would you do that?
The 3 that you divide by has infinite significance since it is
a counting number.
Rounding Numbers
• Calculators are not aware of significant
figures.
• When we find the answers to problems, the
significant figures in the answer will be
calculated based on the significant figures in
the original data. Answers will usually be
rounded.
Rounding Numbers (cont.)
• Rules for rounding
– Rule 1: If the digit to the right of the last significant
figure is less than 5, do not change the last
significant figure.
– Rule 2: If the digit to the right of the last significant
figure is greater than 5, round up to the last
significant figure.
– Rule 3: If the digits to the right of the last significant
figure are a 5 followed by a nonzero digit, round up
to the last significant figure.
Rounding Numbers (cont.)
• Rules for rounding (cont.)
– Rule 4: If the digits to the right of the last significant
figure are a 5 followed by a 0 or no other number at
all, look at the last significant figure. If it is odd,
round it up; if it is even, do not round up. (round to
the nearest even number).
– There is a reason for this rule. If you don’t do this,
you are constantly skewing your lab results to the
high end. Let’s look at an example.
Trial
#
Time
(sec)
1
5.5
2
3
4
5
6
6.5
7
7.5
6
7
8
8.5
What is the
average?
Trial # Time
(sec) old
rounding
1
5.5
Trial # Time (sec)
New
rounding
1
5.5
2
3
4
6
6.5
7
2
3
4
6
6.5
7
5
6
7
7.5
8
8.5
5
6
7
7.5
8
8.5
What is the
average of the
rounded values?
What is the
average of the
rounded values?
• If we consistently round up in a lab setting
(like you have probably done your whole
life), you are constantly skewing the
results upward.
• By rounding with the even rules, the round
“ups” balance the round “downs” and the
data is more true to what it should be.
Rounding Numbers (cont.)
• Rule for Addition and subtraction with
significant figures
– Round numbers so the answer is significant
to the same column as the least significant
number.
– Huh????
43.24 m
+ 2.2 m
• Let’s look at an example:
– Joe, Sally, and Fred count the money in their piggy
banks.
– Joe counts only the dollars. He says he has $43. in
his bank.
– Sally counts only the dollars and dimes. She says
she has $53.4 in her bank.
– Fred counts all the money in his bank. He ends up
with $22.56.
– If we add their totals together, we get
43
53.4
+ 22.56
118.96
Is this really the total for the three banks?
– If we add their totals together, we get
43
53.4
+ 22.56
118.96
Is this really the total for the three banks?
• Of course not….because we don’t know how many
pennies Sally had…or dimes or pennies that Joe had.
• The best we can do is to say that the three had a total of
about $119.
• In the same way, the results in an addition
or subtraction can only be good to the
least accurate place of any one of the
numbers.
• Examples:
5.514
+ 22.33
22.34
+ 195.63
• Rule for Multiplication and division with
significant figures
– Round the answer to the same number of significant
figures as the original measurement with the fewest
significant figures.
– In other words, your answer can’t have more
significant figures than the number with the
fewest significant figures.
• Examples: Multiply or divide the following,
maintaining significance.
43.224 x 22.3 =
13.1 x 102 =
22.0 x 0.0015 =
Section 2.3 Assessment
Determine the number of significant
figures in the following:
8,200, 723.0, and 0.01.
A. 4, 4, and 3
A
0%
D
D. 2, 4, and 1
C
C. 2, 3, and 1
A. A
B. B
C. C
0%
0%
0%
D. D
B
B. 4, 3, and 3