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. 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 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

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. 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).

Number of significant figures How many significant figures ? 1) 654,000 ; 2) ; 3) ; 4) 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. There is a spread of the measurements. The histogram of data. After initial data-analyses, you found that one datum having value = 57 mm was in the list. What should you do? Spread of data

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. 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. Detail discussions will be done in a future lecture.

Whenever we present a value for a physical quantity we convey information about precision. Explicit: “Best” ValueAbsolute uncertainty Relative Uncertainty (Precision) 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 = ± cm; y = ± cm

The uncertainties from the measurements of x and y are independent or uncorrelated. These two independent errors combine in quadrature. Math

For the silicon wafer (note) The technique is general. The “combine fractional errors in quadrature” applies to cases of multiplication and division and can be extended for more factors. x = ± cm; y = ± cm; A = cm 2 (from Excel) 1)A = cm 2 ; or 2) A = cm 2  A = A = (2.285 ± 0.026) x 10 1 cm 2.

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.