Error in Measurement
Precision
Accuracy
Error Types
Significant Digits
Error Propagation
"Accuracy" means truth.
"Precision" means detail.
Accuracy
• Accurate: measured value very close to actual
value.
– What affects accuracy?
– How can you improve accuracy?
– Systematic (correctable) errors
– Calibration of instrument, e.g. zero and gain
Precision
Precise: Multiple measurements close together
-What affects precision?
- How can you improve precision?
- Random errors (not correctable, need trials)
- Instrument capabilities
Significant Digits in Measurement
• Conveys precision
• Rule of thumb: estimate 1 digit
• Digital: half of the last digit
Sig Fig Rules
•
Examples
– 7000.00 kg has six: there is a decimal point, so all the digits are significant.
– 0.00040 kg has two: there is a decimal point, so the 4 and last 0 count.
The other zeros are all "leading".
– 208,000 g has three: the 2, the leftmost 0, and the 8. The trailing zeros do
not count, since there is no decimal point.
– 4440700 km has five: the 4s, the sandwiched 0, and the 7. The trailing
zeros do not count, since there is no decimal point.
• In scientific notation, significant digits count in the first part of the
number.
Examples
– 5.8 x 10^3 has two significant digits.
– 2.00 x 10^-8 has three significant figures.
Sig Fig in calculations
For multiplication and division,
your answer should match the
smallest number of significant digits
in the question.
For addition and subtraction,
your answer should match the
smallest number of decimal places
in the question.
1. If I ran 10.0 m in 2.0 s, how should my speed be rounded off?
2. What if I ran it in 2.01 s? What about 5 sec?
3. What value of force causes a 25.0 kg ball to have an acceleration of 4 m/s^2?
4. If I raise my 36 m flagpole by 20.0 cm, how tall is it?
5.0, 4.98, 2
100, 36
Pendulum Lab Measurements
Which were accurate?
Which were precise?
How do you know?
How do you compensate?
What kind of errors were
there?
How many sig fig did you
use?
Reporting error
• Error: disagreement between measurement and
true/accepted value. We don’t always know this.
• Uncertainty: refers to measured value, it is the
interval within which repeated measurements are
expected to lie.
• Examples:
5.0 +/- 0.2 cm (absolute)
Mean +/- 1 std dev (68%, or about 2/3)
9.8 +/- 2% (relative)
Histograms
Practice
A metal rod about 4 inches long has been passed around to several groups of students. Each group is asked
to measure the length of the rod. Each group has five students and each student independently measures
the rod and records his or her result.
Student
Student 1 Student 2 Student 3 Student 4 Student 5
Group
Group A
10.1
10.4
9.6
9.9
10.8
Group B
10.135
10.227
10.201
10.011
10.155
Group C
12.14
12.17
12.15
12.14
12.18
Group D
10.05
10.82
8.01
11.5
10.77
Group E
10
11
10
10
10
1) Which group has the most accurate measurement?
2) Which group has the most precise measurement?
3) Which group has the greatest error?
4) Which group has the greatest uncertainty?
We now receive a report from the machine shop where the rod was manufactured. This very reputable
firm certifies the rod to be 4 inches long to the nearest thousandths of an inch. Answer the questions below
given this new information. Note that the questions are slightly different. (4.000 inches = 10.160 cm)
5) Which group has the least accurate measurement?
6) Which group has the least precise measurement?
7) Which group has the smallest error?
8) Which group has the smallest uncertainty?
Answers
•
•
•
•
•
•
•
•
•
•
Answers:
1) Unknown. Accuracy can only be decided when the true or accepted value is known. Every value
above is as likely as any other value.
2) Group C: The measurements of this group are very close to each other (repeatable). Individual
measurements in Group B have greater precision but they do not agree with each other as closely
as Group C.
3) Unknown. Error can only be decided when the true/accepted value is know. We may suspect
error for measurements such as 8.01 cm or 12.18 cm but unless we observe a clear mistake in
measurement method we must accept all values here on the same basis.
4) Group D: The average variation in measurements from this group is nearly 1 cm, almost 3 times
more than any other group. (read about average deviation)
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5) Group C: While they are the most consistent set presented, they are consistently far away from
the accepted value.
6) Group D: Precision refers to the data set itself, not to the comparison between the data and the
accepted value. The values for Group D do not agree with each other very well.
7) Group A: This data shows a pretty wide scatter of values but the average is closer to the
accepted value than for any other group.
8) Group C: Uncertainty is not changed by knowledge of the accepted value. This group has the
least variation so the least uncertainty. Here you see a clear difference between uncertainty and
error.
Practice
In a ballistic pendulum experiment, suppose the digital timer shows 0.02 s for the time of flight of the projectile. The
manufacturer information about the precision of the timer is nowhere to be found. What error would you quote on
your measurement?
0.002 s
0.005 s
0.01 s
In a projectile motion experiment suppose you have the following series of measurements of the distance x
traveled by the projectile: 30 cm, 32 cm, 29 cm, 28 cm, 31 cm. The mean value is therefore 30 cm. What is the
error (standard deviation) of the result?
0.8 cm
1 cm
1.4 cm
2 cm
Suppose you counted the number of people in line each time you came to your favorite coffee shop last month. Here are the results of your
investigation.
Number of people
0
1
2
3
4
5
6
7
8
9
10
Occurrences
1
3
3
4
5
8
5
4
3
2
1
What is the error determined by the 2/3 rule?
1.7
2
2.5
3
3.5
Answers:
In a ballistic pendulum experiment suppose the digital timer shows 0.02 s for the time of flight of the projectile. The manufacturer information about the
precision of the timer is nowhere to be found. What error would you quote on your measurement? .005 s
In a projectile motion experiment suppose you have the following series of measurements of the distance x traveled by the projectile: 30 cm, 32 cm, 29
cm, 28 cm, 31 cm. The mean value is therefore 30 cm. What is the error (standard deviation) of the result? 1.4 cm
Suppose you counted the number of people in line each time you came to your favorite coffee shop last month. Here are the results of your
investigation. 2.5
Error Propagation
Spirit Travel Example:
Length = 4825 +/- 5 m
Stop at 3260 +/- 10 m
How far to go? 1565 m, +/- ??
Can calculate extremes: 1580 and 1550 m
Or add uncertainty
Either way, get 1565 +/- 15 m
General rules
For sum and diff:
add absolute errors
For Product/Quotient:
add relative errors
For Powers:
relative error of the
original quantity times
the power.
Practice
You measure the following quantities:
A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m, C = 2.5 m/s ± 0.5 m/s, D = 0.10 s ± 0.01 s.
Choose the correct answers for the following expressions:
A+B=
3.0 m ± 0.0 m
3.0 m ± 0.2 m
3.0 m ± 0.4 m
B−A=
1.0 m ± 0.0 m
1.0 m ± 0.2 m
1.0 m ± 0.4 m
C×D=
0.25 m ± 0.05 m
0.25 m ± 0.08 m
0.25 m ± 0.51 m
B/D=
20. m/s ± 4 m/s
20. m/s ± 10 m/s
20. m/s ± 20 m/s
3×A=
3.0 m ± 0.2 m
3.0 m ± 0.3 m
3.0 m ± 0.6 m
The square root of (A × B) =
1.4 m ± 0.1 m
1.4 m ± 0.2 m
1.4 m ± 0.4 m
Answers:
________________________________________
A + B = 3.0 m ± 0.4 m
B − A = 1.0 m ± 0.4 m
3 × A = 3.0 m ± 0.6 m
C × D =0.25 m ± 0.08 m
The square root of (A × B) = 1.4 m ± 0.2 m
B / D = 20. m/s ± 4 m/s