Experimental Tectonics: Measurements and Error Propagation in Physics

Introduction to Experimental Science

EXPERIMENTAL
TECTONICS
MEASUREMENTS
AND
UNCERTAINTIES>>INTRODUCTION
Two ingredients are the key of experimental science:
measurements and their uncertainities.
Scientists aim towards designing experiments that can give a
"true value" from their measurements.
However, the limited precision in measuring requires to quote
their results with some form of uncertainty

Methods of Measurement

METHODS OF
MEASUREMENT
The measurement of a property may be categorized defining:
type, magnitude, unit, and uncertainty.

Direct Measurement

DIRECT:
- Compare an unknown quantity (measurand) against a
known quantity (standard). e.g., length, time
- Not always possible or practical (e.g., measuring sound
levels)

Indirect Measurement

INDIRECT: - The quantity can be linked to other quantities that can be
measured in a direct way. (e.g., density, pressure)
data = measurement + unit + error

Uncertainty in Scientific Knowledge

INTRODUCTION
"All scientific knowledge is
uncertain. When the scientist tells
you he does not know the answer,
he is an ignorant man. When he
tells you he has a hunch about how
it is going to work, he is uncertain
about it. When he is pretty sure of
how it is going to work, he still is in
some doubt. And it is of paramount
importance, in order to make
progress, that we recognize this
ignorance and this doubt. Because
we have the doubt, we then
propose looking in new directions
for new ideas."
Feynman, Richard P. 1998. The Meaning of It All:
Thoughts of a Citizen-Scientist. Reading,
Massachusetts, USA. Perseus. P 13
Nobel in 1965

Error Definition and Types

ERROR: DEFINITION
Error - The deviation of a measured result from the correct
or accepted value of the quantity being measured.
Error
There are two basic types of errors: random and systematic.

Systematic Error

SYSTEMATIC ERROR
Systematic Errors - cause the measured result to deviate by
a fixed amount in one direction from the correct value. The
distribution of multiple measurements with systematic error
contributions will be centered some fixed value away from
the correct value.
Some Examples:
- Mis-calibrated instrument
- Readability of an instrument
(T, parallax)
Systematic Errors
Careful design of an experiment will allow us to
eliminate or to correct for systematic errors.

Random Error

RANDOM ERROR
Random Errors - cause the measured result to deviate
randomly from the correct value. The distribution of
multiple measurements with only random error
contributions will be centered around the correct value.
· Some Examples
- Noise (random noise)
- Careless measurements
- Low resolution instruments
- Dropped digits
Random Errors
Data can be dealt with in a statistical manner

Precision vs. Accuracy

PRECISION vs. ACCURACY
Measurements typically contain some combination of random
and systematic errors.
Precision: degree to which an
instrument or process will repeat the
same value (degree of reproducibility)
indication of the level of random
error.
Accuracy: degree of closeness to
true value > indication of the
level of systematic error.
Accurate
Precise
Not Accurate
Precise
Accurate
Not Precise
Not Accurate
Not Precise

Calculating Accuracy and Precision

PRECISION VS. ACCURACY
% error = (True value - experimental value) x 100
How close a value is to the true or accepted value
(an average can be compared to the accepted value)
Only one measurement is necessary for calculating an accuracy but many
numbers is preferred and the accuracy of the average is then taken.
Desired value is zero.
Reference value
Probability
density
Accuracy
Value
Precision
precision is the degree to
which repeated (or reproducible)
measurements under unchanged
conditions show the same results.

Measurement Uncertainty

waffendescomic@aol.com www.comos.com)
" I hear you haven't made any errors yet . "
MEASUREMENT
UNCERTAINTY
offsidescomic@aol.com www.comes.com
" I hear you haven't made any errors yet ."
Measurement uncertainty combines these concepts into a
single quantitative value representing the total expected
deviation of a measurement from the actual value being
measured.

Single Measurement Analysis

SINGLE MEASUREMENT
data = x ± 4x
where Ax is half of the least count, the smallest division that is marked on
the instrument (which is related to the precision of an instrument)
P
Q
A
B
0
1
2
3 4 5 6 7 8 9 10 1
11
12 13
14 15
R
S
X
Y
P
Q
A
B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
O
R
S
X
Y
The error define the number of
significant digits or «figures»

Question for Thought on Single Measurements

SINGLE MEASUREMENT
question for thought
Using two different rulers, I measured the width of my hand to be
4.5 centimeters and 4.54 centimeters. Explain the difference
between these two measurements.
2
3
4
5
cm
4.5 cm
1
2
3
4
5
4.54 cm
CM

Interpreting Single Measurement Precision

SINGLE MEASUREMENT
question for thought
The first measurement implies that my hand is somewhere between 4.5
and 4.9 cm long.
The second measurement implies that my hand is between 4.5 and 4.6
cm long. This measurement is more certain due to its greater precision.
-
2
3
4
5
4.5 cm
cm
1
2
3
4
5
4.54 cm
CM

Significant Figures in Single Measurements

SINGLE MEASUREMENT
question for thought
4.5 cm
2 significant
figures
Uncertain
4.54 cm
3 significant
figures
More certain
due to greater
precision
Significant figures are necessary to reduce uncertainty in our measurements.
Significant figures indicate the precision of the measured value !!

Significant Figures

SIGNIFICANT FIGURES
. Scientists use significant figures to determine how precise a
measurement is
· Significant digits in a measurement include all the known digits plus
one estimated digit
. So when reading an instrument ...
- Read the instrument to the last digit that you know
- Estimate the final digit

Significant Figures Example

SIGNIFICANT FIGURES
example
. Look at the ruler below
8
9
10
11
12
13
14
15
116
117
18
•
. You can read that the arrow is on 13.3 cm
. However, using significant figures, you must estimate the next
digit
· That would give you 13.30 cm

Rules for Counting Significant Figures

SIGNIFICANT FIGURES
rule for counting significant figures
All digits ARE significant except
Zeros preceding a decimal fraction
(ex: 0.0045)
and
Zeros at the end of a number
containing NO decimal point
(ex: 45,000)Repeat,
please

Repeated Measurements

REPEATED MEASUREMENTS
data = x ± 4x
where
X1+ X2+ X3+ X4+ ... +xn
Ax =
Xmax -xmin
X =
n
2
Concentration of standard solution (mg/L)
Repeat number
0.10
0.20
10
2.00
10.00
20.00
1
0.15
0.22
0.93
1.82
9.55
19.07
2
0.14
0.22
0.92
1.86
9.47
19.08
3
0.13
0.22
0.94
1.88
9.70
19.29
Average
0.14
0.22
0.93
1.85
9.58
19.15
Accuracy (%)
141.73
110.25
93.09
92.55
95.76
95.73
Standard deviation (mg/L)
0.005975
0.002222
0.009116
0.030222
0.116727
0.123969

Standard Deviation in Repeated Measurements

REPEATED MEASUREMENTS
standard deviation
Measure of the amount of variation (aka dispersion) of a set of values in
relation to their mean.
low standard deviation -> values tend to be close to the mean (i.e.,
expected value) of the set
high standard deviation -> values are spread out over a wider range.
E- (Xi -X)2
Sx =
n - 1
n
= The number of data points
Xi
= Each of the values of the data
X
= The mean of Xi
Standard Normal Distribution
0.4
0.3
0.2
5
O
3 -2 -1 0 1 2 3

Standard Deviation and Data Distribution

REPEATED MEASUREMENTS
standard deviation
99.7% of the data are within
3 standard deviations of the mean
95% within
2 standard deviations
68% within
+1 standard
deviation
->
μ - 3σ
μ - 20
μ- σ
μ
μ + σ
μ + 2σ
μ + 3σ

Indirect Measurements and Error Propagation

INDIRECT MEASUREMENTS
. Let us assume that some quantity, x, has been measured and that
the corresponding absolute error 8x and relative error 8x/x have
been determined.
11
· Very rarely is the measured quantity itself the end result of the
experiment. Usually the final result is some function of one or more
measured quantities.
· At each step in the calculation, the error in the result is related in
some definite way to the error in the data. If we know the rules for
each kind of operation, we can carry the errors along from step to
step, up to the final result.
· This procedure is called «propagation of error».
72

Error Propagation in Addition and Subtraction

INDIRECT MEASUREMENTS
error propagation
When two or more quantities are added or subtracted, the error in result (E)
is the sum of the errors of the individual quantities
i.e.,
a = a ± Aa
b=b ±△b
E=
A(a + b) = Aa + Ab
E=
A(a - b) = Aa + Ab

Error Propagation in Multiplication and Division

INDIRECT MEASUREMENTS
error propagation
When two or more quantities are multiplied or divided, the relative error in result
(E) is the sum of the relative errors of the individual quantities. Then the error can
be extrapolated.
i.e.,
a = a ± Aa
b=b ±△b
E/measurement=
A (ab)
4b
a b
+
E= A(ab) = a b (Aa + Ab
= bΔα +aAb
E/measurement=
4 (a/b)
a/b
= Aa + Ab
E= A(a/b) = a (Aa + Ab)

Calculations with Significant Figures in Indirect Measurements

INDIRECT MEASUREMENTS
calculation with significant figures
· Adding or subtracting:
- answer can have no more places after the
decimal than the LEAST of the measured
numbers.
. Count # decimal places held
- (nearest .1? . 01? . 001?)
O
· Answer can be no more accurate than the
LEAST accurate number that was used to
calculate it.

Significant Figures Example for Addition and Subtraction

INDIRECT MEASUREMENTS
significant figures
(example)
5.50 g
52.09 ml
+ 8.6 g
- 49.7 ml
2.39 ml -- > 2.4 ml
14.10 g -- > 14.1g

Significant Figures Example for Multiplication and Division

INDIRECT MEASUREMENTS
significant figures
· Multiplying or dividing: round result to least # of sig figs
present in the factors
- Answer can't have more significant figures than the
least reliable measurement.
· COUNT significant figures in the factors

Significant Figures Example for Multiplication

INDIRECT MEASUREMENTS
significant figures
(example)
· 56.78 cm x 2.45cm = 139.111 cm2
Round to 3 significant figs = 139cm2
· 75.8cm x 9.6cm = ?

Study Resources

WHERE TO STUDY?
- These slides.
- Info already collected in other courses
- Material provided by the instructor