Revised schedule of my weekly lecture
Week 1 (1/19): Syllabus
Week 2 (1/26): “good” and “bad” examples of lab books
Week 3 (2/2): concept of significant figures
In the past two years, after the first three lectures, the topics of “fundamental constants”,
“basic physical concepts”, “random and system errors”, “error propagations”, “distribution
functions”, “least square fitting”, “chi-square” etc. were presented in systematic and logical
ways. As a result, the last two topics were discussed toward the end of the semester. Students’
responses were somewhat negative.
This year, I plan to first discuss some of the key ideas directly related to what you need for the
labs. Unavoidably there will be some gaps which will be discussed later with some repetitions.
Your feedbacks and suggestions at any time will be extremely helpful and valuable.
Week 4 (2/9): random errors, error propagations through simple math manipulations
Week 5 (2/16): least square fitting, basic concept of chi-square
2/19 evening, starting to pick your topic of paper #1.
Week 6 (2/23): key components of writing a “good” scientific paper
Week 7 (3/1): plotting using excel program, “good” and “bad” figures
Week 8 (3/8): in-class peer-reviewing paper #1
Reminders
It would be a good idea to obtain the UMNx500 ID's from your lab partners of all assigned
labs. While you pick a lab to write your paper #1 and #2, you must fill in the x500 ID’s of your
lab partner(s).
The first HW assignment has been posted on class web-page.
The assignments for this class will not be collected and graded. Your incentive to do it
(apart from intellectual curiosity) is that most of quiz problems will be based on the HW
assignments.
Number of significant figures
How many significant figures ?
1) 654,000 ; 2) 654000 ; 3)654000. ;
4)654.000
Please write down the answers in your paper. Four students
have been picked to write their answers on the white board.
Random vs. Systematic errors
Mainly discuss random errors now
The experiments:
• Things that we measure (with a
ruler, micrometer, scale, ohmmeter,
voltmeter, photo-diode,
photomultiplier, interferometer,
etc.) always have uncertainty.
• Things that we count (the number of
students in this university, the number of
students in this class room (now), the
amount of money in your pocket, fish in a
pond, etc.) may or may not have
uncertainty.
In carrying out experiments, we use apparatus of appropriate precision, develop and
apply consistent procedures to maximize accuracy, and carefully assess and record
all uncertainties.
One important goal of the first half of this semester, you should learn how to assess
all errors or uncertainties and keep a good record.
Errors ≡ Measurement Uncertainties
Not (necessarily) mistakes!
• Random Errors
– Variations from trial to trial
even with consistent and
reliable technique.
– Fluctuations in counting with
limited samples.
– Can be minimized, but not
eliminated: better instruments
and/or procedures, larger
samples, careful calibration, ...
• Systematic Errors
– Individual sources, once
recognized, can be
eliminated or corrected for.
– Biases built into the
procedure or analysis that
affect every trial the same
way.
– Scale offsets, bad calibration.
– Non-random or
nonrepresentative samples.
L. Y. Zhao et al., Nature
Phys 12, 32 (2015).
Random Errors:
• In five lab sessions of Physics 2605,
a clear plastic ruler and a
rectangular stainless sample plate
are passed around. Each student
measures the length of the sample
plate to the nearest millimeter.
• The data (total 75) are collected and
entered into an Excel spreadsheet.
A subset of data is shown.
Random Errors:
•
Can be treated statistically.
• The histogram of data.
•
How do we estimate the “best” value for a set
(distribution) of independent measurements?
• Usually, the mean:
Random Errors:
•
Can be treated statistically.
• The histogram of data.
There is a spread of the measurements.
Spread of data
After initial data-analyses, you found that one datum having
value = 57 mm was in the list. What should you do?
•
How do we describe the “dispersion” (“spread” or
“width”) of a set of independent measurements?
Deviation with respect to the mean:
(Not very useful – sum is zero)
Magnitude of deviation w.r.t. the mean:
(More useful, but inconvenient)
Mean-squared deviation (variance)
and standard deviation:
A subtlety…
• There is a bias because the mean is determined from the same set of data.
• A better (or unbiased) estimate of the standard deviation
N→ N−1 reflects the number of “degrees of freedom” after the data are
used to determine the mean. The difference is unimportant unless the
number of measurements is small.
Detail discussions will be
done in a future lecture.
This applies to cases where the errors are small and random, so that the distribution
of measurements (deviations) follows a normal (Gaussian) distribution.
Detail discussions of various distribution functions will be done in near future.
•
Whenever we present a value for a physical
quantity we convey information about precision.
Explicit:
“Best” Value
Relative Uncertainty
(Precision)
Absolute uncertainty
Implicit: Significant Figures (the # of digits
in a value in which we are confident)
Error Analysis: Propagation of Errors
Suppose we measure the length x and
width y of a rectangular silicon wafer and
understand our measurement uncertainties
well. The information we really want is the
area A of the wafer and its error?
x = 3.670 ± 0.003 cm; y = 6.226 ± 0.005 cm
•
Math
The uncertainties from the measurements of x and y are independent or
uncorrelated. These two independent errors combine in quadrature.
For the silicon wafer (note)
•
x = 3.670 ± 0.003 cm; y = 6.226 ± 0.005 cm;
A = 22.84942 cm2 (from Excel)
1) A = 22.85 cm2 ; or 2) A = 22.849 cm2
DA = 0.037
A = (2.285 ± 0.026) x 101 cm2.
The technique is general. The “combine fractional errors in
quadrature” applies to cases of multiplication and division
and can be extended for more factors.
•
Sums and differences:
The absolute errors combine in
quadrature. (Note the case of the
simple sum/difference, α = 1, β = ±1.)
•
Exponential:
We could keep doing this all day long.
More examples for practice are given
in HW#1 posted on the web.
Plot a function
What are the best ways to present the data (x, y) which you know that
they should be described by the following functions (y =f(x), for x > 0)?
Please write down the answers in your paper.
Four students have been picked to write their
answers on the white board.