Lab Manual
PHYS 130
University of Alberta, Dept. of Physics
Fall 2022
PHYS 130 – Page 2
PHYS 130 – Page i
Copyright: This Lab Manual is Copyright ©2021 by the University of Alberta, Department of
Physics. Reproduction is permitted exclusively for material included in Lab Report Submission to
our program.
PHYS 130 – Page ii
Contents
1
Introduction
2
Physics Lab Policies
2.1 Laboratory Administrative Staff . . .
2.1.1 eClass . . . . . . . . . . . . .
2.2 Assignments and Marking . . . . . .
2.2.1 Tutorials . . . . . . . . . . .
2.2.2 Lab Reports . . . . . . . . . .
2.2.3 Assignments . . . . . . . . .
2.3 Attendance . . . . . . . . . . . . . .
2.4 Supplies . . . . . . . . . . . . . . . .
2.5 Safety and Student Behaviour . . . . .
2.6 Collaboration and Academic Integrity
2.7 Reassessment of Graded Work . . . .
3
Assessment and Marking
4
Lab Reports
4.1 Introduction . . . . . . . . . .
4.2 Formatting and Requirements .
4.3 Components of the Lab Report
4.3.1 Introduction . . . . . .
4.3.2 Methods . . . . . . .
4.3.3 Results . . . . . . . .
4.3.4 Discussion . . . . . .
4.3.5 Conclusion . . . . . .
4.3.6 References . . . . . .
4.3.7 Acknowledgments . .
4.3.8 Appendix . . . . . . .
4.4 Voice and Style . . . . . . . .
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Assignments
5.1 Introduction . . . . . . . . . . . . .
5.2 Formatting . . . . . . . . . . . . . .
5.2.1 Documents vs. Spreadsheets
5.2.2 Calculations . . . . . . . .
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1
CONTENTS
5.3
5.4
5.5
PHYS 130 – Page 2
5.2.3 Page Limits . . . . . . . .
Style . . . . . . . . . . . . . . . .
References and Acknowledgements
Conclusion . . . . . . . . . . . .
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6
Measurements and Uncertainties
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6.1 Measurement and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Significant Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.3 Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7
Graphical Analysis
7.1 Graph Construction . . . . .
7.1.1 Analysis of graphs .
7.2 Using LINEST . . . . . . . .
7.2.1 Microsoft Excel . . .
7.2.2 Google Sheets . . .
7.3 Computers and spreadsheets
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Lab 1: Graphing and Spreadsheets
8.1 Pre-Lab . . . . . . . . . . . . .
8.2 Purpose . . . . . . . . . . . . .
8.3 Introduction . . . . . . . . . . .
8.4 Lab setup . . . . . . . . . . . .
8.5 Procedure . . . . . . . . . . . .
8.5.1 Making a Table . . . . .
8.5.2 Making a Graph . . . .
8.5.3 Linearizing Data . . . .
8.5.4 Significant Digits . . . .
8.5.5 Uncertainty Propagation
8.6 Assignment . . . . . . . . . . .
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Lab 2: Oscillations of a Spring
9.1 Pre-Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.2 Oscillating spring . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.1 Static measurement of the spring constant . . . . . . . . . . . . . .
9.5.2 Spring oscillations . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7 Using a DIG/Sonic Sensor . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.1 Sensor initial set up . . . . . . . . . . . . . . . . . . . . . . . . . .
9.7.2 Measuring a constant height . . . . . . . . . . . . . . . . . . . . .
9.7.3 Measuring position, velocity, and acceleration for oscillating spring
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CONTENTS
9.8
PHYS 130 – Page 3
Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
10 Lab 3: Polarization
10.1 Materials . . . . . . . . . .
10.2 Purpose . . . . . . . . . . .
10.3 Introduction . . . . . . . . .
10.3.1 Polarization of Light
10.3.2 Arduino . . . . . . .
10.4 Procedure . . . . . . . . . .
10.5 Analyzing Your Data . . . .
10.6 Appendix . . . . . . . . . .
10.7 Assignment . . . . . . . . .
11 Lab 4: Speed of Sound
11.1 Pre-Lab . . . . . . . .
11.2 Purpose . . . . . . . .
11.3 Introduction . . . . . .
11.4 Theory . . . . . . . . .
11.5 Procedure . . . . . . .
11.6 Analysis Questions . .
11.7 Standing Sound Waves
11.8 Assignment . . . . . .
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12 Lab 5: Diffraction
12.1 Purpose . . . . . . . . . .
12.2 Theory . . . . . . . . . . .
12.2.1 Diffraction Grating
12.2.2 Spectrometers . . .
12.2.3 Spectra . . . . . .
12.3 Procedure . . . . . . . . .
12.4 Extra Information . . . . .
12.5 Assignment . . . . . . . .
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Chapter 1
Introduction
Welcome to Physics Labs. In this class, we ask you to spend some of your time outside of the
lecture hall (or online lecture), working with the actual systems that explore the topics that are being
discussed in the lecture component of the class. In your lecture, you are mostly learning about the
theory of physics where we deal with the mathematical model for how the universe behaves. In the
lab, you will work with the experimental aspects of physics, testing how well the mathematical model
actually represents the world that we see. The dialogue between theory and experiment drives the
progress of science as a whole.
While this purpose for physics lab seems lofty with the whole “progress of science” theme, this
course is an introductory course and many of the physics ideas are well established. For example, the
physics of a ball rolling down a ramp seems rather simple, so the experiment itself may seem rather
mundane. In other introductory sciences, like biology and chemistry, the topics are also fundamental
but you are also learning important techniques like “don’t squirt the acid in your eyeball.” The
techniques you learn in this lab course appear more subtle since they focus more on the practice and
presentation of scientific knowledge.
Physics labs have a few major goals. By the end of the lab sequence, you should be able to:
• Test physical theories with simple experiments and use numerical data to evaluate a physical
model.
• Quantify your results using numerical measurements and analyze these results with a computer.
• Quantify the uncertainty in your measurements to understand the limitations of your experiment.
• Use basic equipment for measuring physical quantities in the lab.
• Present your findings in a clear, concise manner through a report or answering assignment
questions.
4
CHAPTER 1. INTRODUCTION
PHYS 130 – Page 5
Since many of the things we work with in introductory labs are not dangerous, we also leave many
of the instructions open-ended. The object of these labs is not to teach you how to follow cookbook
recipes.
Remember that in real research labs, we are never following any cookbook recipe toward a known
result. We are usually faced with some problem or issue that we want to learn about and solve. We
may want to design a device or obtain data to test a theory. We are “given” equipment to do this: the
equipment in our lab or those of our colleagues with whom we collaborate. We must then figure out
how to conduct our experiment in order to properly test the theory or create the device, just like we
proposed we would do in our research grant application. We have to design the experiment, think of
our sources of error, determine whether we are properly answering the questions we set out to answer,
and look for ways to improve our approach. We should also keep our eyes open for unexpected or
surprising results. In science, we cannot just “follow recipes.”
You will see the same theme throughout these labs. You will often be given a problem to solve,
but somewhat general instructions on exactly how to do it. That will often be up to you. We will
supply only the tools and perhaps some good advice. Then you are a bit like a real experimental
physicist where given a set of tools and a problem to solve. There may be more than one correct way
to solve the problem. It is very important that you keep your mind open to the wonder of what you
are seeing, even if it is something we normally take for granted.
A vitally important part of all physical experimentation is the ability to communicate the results
to our colleagues and, often, the general public. Communication is a key skill in science. If you
communicate poorly, you can get misquoted and have embarrassing credibility issues. Essentially,
good science can easily be ruined by poor communication. In fact this happens all too often.
Your primary tool for communication of your results is your lab report, though some courses forgo
the lab report in favour of question-based assignments. Our reports are modelled on the research
papers that physicists write up to describe their science. Research papers have a standard format: Abstract, Introduction, Methods, Results, Discussion, Conclusion, References and Acknowledgments.
One reason for this format is so that other researchers can find what they want in your paper as easily
as possible. In other words, sticking to the format helps people who are going to read your paper.
Your lab report will not have the exact same style as a research paper, but it will follow a required
format. This will greatly assist your lab TA, who has to read and grade every one of your reports.
In conclusion, our goal is not just to test physics theories, but also to develop your skills in
describing scientific results both quantitatively (through measurements, uncertainties, and computer
analysis) and verbally (through your lab reports).
Acknowledgments
The physics experiments and laboratory manual have been the result of contributions by the
faculty, staff, and students of the Department of Physics at the University of Alberta. The persons
who have primarily worked on developing the experiments, writing the laboratory instructions, and
PHYS 130 – Page 6
editing the manuals are:
Kyle Foster
Prof. Mark Freeman
Dr. Isaac Isaac
Prof. Al Meldrum
Dr. William Morrish
Prof. Erik Rosolowsky
Wladek Rudzinski
We are grateful for the extensive comments and suggestions been made by the Teaching Assistants
who run the laboratories. The technical staff has also contributed enormously in the design and
building of laboratory equipment.
Chapter 2
Physics Lab Policies
This chapter acts as the syllabus for labs. It outlines the policies that govern how the lab operates.
Any amendments to this document will be displayed on eClass at the beginning of the term.
2.1
Laboratory Administrative Staff
Your laboratory section will be taught by a teaching assistant from the Department of Physics.
The TA has several different roles in the lab setting: (1) The TA oversees the safety of the laboratory
and ensures that everyone is aware of how to operate the lab equipment safely. (2) The TA provides
instruction on how to complete the individual labs as well as laboratory skills. (3) The TA answers
your questions about the lab and assists you with operating the equipment. (4) The TA assesses your
laboratory work, including your attendance at your registered lab section, your completion of pre-lab
assignments (if applicable), and any lab reports or assignments you submit.
The Undergraduate Laboratory (UGL) program is supervised by the academic staff who ensure
that the labs are running smoothly. If you have any concerns with your lab, you should contact the
UGL staff.
Lab Coordinator Lab Coordinator
Name Kyle Foster
William Morrish
Office CCIS L1-183
CCIS L1-187
Email: physugl@ualberta.ca
Academic Director
Prof. Al Meldrum
CCIS 3-201
Important: In all email to the UGL staff include your course and lab section in the subject line.
Example: PHYS 124 D11.
7
2.2. ASSIGNMENTS AND MARKING
2.1.1
PHYS 130 – Page 8
eClass
Students will be expected to access content on the labs eClass page such as assignment submission
pages, worksheets, due dates, and pre-lab quizzes. All information related to the labs will be posted
on eClass.
2.2
Assignments and Marking
Your lab and tutorial marks count for 15% ± 5% of your total grade with the weight being set by
your professor for the lecture component of the course. You must achieve at least 50% in the labs
in order to pass and obtain credit for the whole course. There are a variety of different assessment
methods used through the course and the assignments have different weighting as described in the
Assessment and Marking Chapter.
Most lab grading will be completed ONLINE through eClass and must be submitted as PDF files
only. After submission you should verify that the file can be downloaded intact. Files that fail to
open will receive a grade of zero. Make sure you upload the correct file to eClass, as whatever is
uploaded will be marked. Also double-check that the formatting has not changed after converting
your document to a PDF as page limits are very strict.
The lab reports will be due 48 hours after the end of your class. For example if you have a
Fall/Winter class Monday at 8 a.m. until 11 a.m., your assignment will be due Wednesday at 11 a.m.
If you have a lab on Friday from 2 p.m. to 5 p.m., the assignment will be due on Sunday at 5 p.m.
Any assignments handed in late without prior approval of the TA or PhysUGL will receive 0%.
The TA will mark your lab report according to a marking rubric set by the Department of Physics.
They will provide brief written comments on the report. Some labs will follow more specific grading
rubrics. These expanded rubrics will be posted ahead of time for each assignment.
If you have concerns about the grade you received or about missing work, you must raise the
concerns first with your TA within 5 days of the release of grade or at the next lab session, whichever
comes first. If you do not receive a satisfactory response, email physugl@ualberta.ca with your
concerns. Any requests received after this time will be processed on a best-effort effort and we may
not be able to resolve your issue.
2.2.1
Tutorials
Tutorials will be sets of theoretical problems related to class, to be solved in the weekly tutorial
session. Many times, they will require a computer to graph some data or an equation, or for various
other purposes. So, it is highly recommended to bring your laptop if you have one–otherwise you
can use one of the older lab computers.
CHAPTER 2. PHYSICS LAB POLICIES
PHYS 130 – Page 9
Some tutorials will be posted on eClass before your scheduled tutorial sessions. Helping you
with the tutorials and answering questions is the job of your lab TA. If you have any problems, your
TA is there to help. But once the lab is over, you are on your own. So, it is best to finish the tutorial
before leaving. Your professors and TAs will not be answering any office-hour questions about the
tutorials.
The tutorial report does not have a strict format. Think of it like an assignment you will be
submitting. Make sure you answered all the questions clearly, and your work can be followed and
understood by the TA. Any graphs required should also be included. Having it fairly freeform like
this is nice and convenient for the student, but it means one thing: if your TA cannot understand what
you are doing, you may get marks deducted.
2.2.2
Lab Reports
Lab reports are sufficiently important that they get a chapter all of their own (Chapter 4). However,
there are a few basic expectations. Lab reports have a maximum of 4 pages, not including the cover
page, and must be submitted as a Portable Document Format (PDF) document to eClass within a
specified window of the end of your lab period. During the Fall and Winter terms, this period is
normally 48 hours after the end of the lab. Announcements made on eClass about the due date take
precedence.
2.2.3
Assignments
Depending on your course, you may have a mixture of lab reports and assignments or just
assignments. These assignments are discussed in Ch. 5.
2.3
Attendance
Your attendance in the lab and tutorial sessions is mandatory. Your TA will be taking attendance
at the beginning of each lab and when you leave. If you fail to sign in or sign out, you will be given a
zero for that lab report or tutorial. You cannot hand in lab reports or tutorials without attending lab
sessions. It is your responsibility to ensure that your name is on the attendance sheet and that you
have been signed out before leaving the lab. Your TA will not sign you out unless you have gathered
all of your raw data and shown it to them. Not getting signed out will result in a zero for that lab.
The lab period is the primary time for completing the lab and asking your TA questions. If you
leave before the end of the 3 hour lab period, the TA will not be expected to help you outside of class
(e.g., through email or office hours). You should use the time in the lab effectively, and ask your TA
questions during the lab period. If you come to the lab session more than 15 min past the start time,
2.4. SUPPLIES
PHYS 130 – Page 10
without a valid reason that adheres to the University’s policy on attendance, your attendance may not
be counted.
The attendance policy regarding missed sessions is based off the University Calendar1. If you
miss a session, fill out the missed session form on eClass and contact your TA. You should make
note of the following from the University Calendar:
• To apply for an excused absence where the cause is incapacitating mental and/or physical illness
and most other cases including severe domestic affliction, you must submit the excused absence
form found on eClass within two working days following the scheduled date of the Lab, or as
soon as the student is able, having regard to the circumstances underlying the absence.
• For an excused absence where the cause is religious belief, you must contact the lab coordinator
within two weeks of the start of Fall or Winter classes; and within three days of the start of
Spring or Summer classes.
• All other accommodation requests covered by the Duty to Accommodate Procedure should be
discussed with the coordinator as soon as the you are able, having regard to the underlying
circumstance.
The UGL Staff will try to arrange a make-up lab for your absence. This is done on a “best-effort”
basis and you are expected to respond within one business day to any communications regarding
rescheduling. The need for prompt communications is required to ensure that we can effectively
coordinate attendance in the different labs. Thus, if you fail to follow-up on communication, you will
not be able to make up the lab and will receive a 0% on the lab. If rescheduling is not possible, the
UGL staff may exempt you from completing the lab, but this is only for short absences. Your average
will be calculated without exempted labs.
2.4
Supplies
You will need some personal equipment for the laboratories and tutorials. Always bring a
calculator and writing materials. You will need access to a computer. It is highly recommended to
bring a laptop and a smartphone if you own one. The laboratory classrooms have a limited number
of computers with the appropriate software. All software used in the labs is available through eClass
and the laboratory computers. Some labs require a portable USB drive (also called a thumb drive or
flash drive) but you do not need to own one.
Safety equipment: You do not need to bring any personal protective equipment such as lab
goggles or a lab coat. Any specialized safety equipment will be provided.
1https://calendar.ualberta.ca/content.php?catoid=6&navoid=806#Attendance
CHAPTER 2. PHYSICS LAB POLICIES
2.5
PHYS 130 – Page 11
Safety and Student Behaviour
Students who engage in unsafe or disorderly conduct will be asked to leave the lab by the Instructor
or Undergraduate Laboratory Staff. Students asked to leave will be unable to complete their labs,
thus earning a grade of 0% for that lab. You are liable for the costs of any equipment that you break.
2.6
Collaboration and Academic Integrity
Students work in the lab in pairs. If there is an odd number of students in the lab section, one
group of three is allowed. Once the experiment is completed and the raw data are collected, each
student should make an original lab report or tutorial submission independently.
This section provides supplemental material regarding plagiarism in lab reports highlighting
some additional principles:
• Unless cited or otherwise attributed, all content in your lab report is assumed to be your
own content. This includes figures, even those abstracted from the lab manual. Note that
paraphrased content is NOT yours and needs to be cited.
• While you have a lab partner, your lab report submission is entirely your own. The only
material you should share is the raw data collected from the experiment. We recommend
that you “work together but write separately.” You cannot share derived quantities created by
editing the spreadsheet or plots.
• Students may not share their lab report, directly or indirectly, in draft or completed form, with
other students. If a shared lab report is used in whole or in part in another student’s submission,
the student who shared the report will also be reported to the Faculty of Science Discipline
Office, for alleged breach of the Code of Student Behavior2 for Cheating3, and by Participating
in an Offence4.
• Never copy-paste. We fully recognize that it is standard practice to copy-paste material and
then tweak words and phrases to avoid plagiarism concerns. With clear citation of the source
material, this can become acceptable, however it can also frequently remain plagiarism. When
you copy-paste, you are starting by plagiarising and then editing to avoid it. It is far better to
directly type all the entire content of the lab report yourself.
• Physics Lab report submissions are now electronic submissions which will be retained as
an ongoing record of content submitted in the lab program. Be aware that this creates a
comprehensive library of lab submissions and your report will be compared to all other
2https://www.ualberta.ca/governance/media-library/documents/resources/policies-standards-and-codes-of-conduct/
cosb-updated-july-1-2020.pdf
330.3.2(2) in the Code of Student Behaviour
430.3.6(5) in the Code of Student Behaviour
2.7. REASSESSMENT OF GRADED WORK
PHYS 130 – Page 12
submissions, past and present, to identify instances of Plagiarism, Cheating, and Participation
in an Offence.
2.7
Reassessment of Graded Work
Lab reports are usually graded by the lab instructor for your lab session. If you have concerns
about how part of your lab was evaluated, you have several potential courses of action. In all cases,
requests for reassessment are holistic, so reassessment may raise or lower your grade.
1. Informal Reassessment – Your first step should always been to approach your lab instructor
and explain your concerns. You should be prepared to demonstrate why you think the lab was
incorrectly evaluated. All requests for informal reassessment must be made within 5 business
days of the lab being returned, your next lab session, or by the final exam date of your class,
whichever comes first. A business day refers to a day when the University is open, regardless
of whether classes are in session.
2. Formal Reassessment by Lab Staff – Within 5 days of receiving a response from your lab
instructor, you may request appeal of your work under the guidelines of the Faculty of Science
Academic Appeal Policies5.
Requests for formal reassessment are initiated with the Undergraduate Lab Coordinator
(physugl@ualberta.ca) with a written request explaining why you think there has been an error
in evaluation plus your original work as evaluated. In your request, include as much detail as
you can to justify your concerns. If you want several pieces of work re-evaluated, you need to
make a separate request for each piece of work individually. You should retain copies of the
papers you hand in. The Physics UGL staff (Lab Coordinators and the Academic Director) will
consult as to whether a grade change is warranted. You may be requested to attend a meeting
to discuss your concerns.
3. Formal Reassessment by Faculty – If you think reassessment by the Lab Staff is in error,
you can submit a request for formal reassessment by the faculty. Contact the Physics Office
(CCIS 4-181) and make your request to the Associate Chair, Undergraduate Studies for the
Department of Physics.
5https://cloudfront.ualberta.ca/-/media/science/studentservices/studentforms/forms-cabinet-2018/academic_
appeals-policy-2016.pdf
Chapter 3
Assessment and Marking
Your lab grade will determined by your performance on a series of assignments. The weighting of
these are given on eClass under grades and the due dates can be found on the assignment. Typically
each assignment will be due 48 hours after the end of the lab session. You can find the schedule of
labs on eClass under course information. Due dates are those currently planned but may be updated
on eClass. In all cases, the eClass due dates are the ones you should respect.
Assessments will consist of submissions to eClass. These will consist of completing some
analysis, creating graphs, carrying out some calculations, and answering some questions. We will
be asking you to submit lab reports or worksheets depending on the experiment. Lab reports are
equally weighted throughout the semester. The in-depth rubric for full lab reports can be found under
the course information page of eClass. All labs will be equally weighted and we will be allow for a
make-up lab for a lab of your choice! You must arrange with physugl to obtain a make-up lab (See
make-up lab form). See the schedule for more details.
13
Chapter 4
Lab Reports
Brevity is the soul of wit.
– William Shakespeare’s Hamlet
4.1
Introduction
All of science, but physics in particular, is a dialogue between theory and experimentation. The
lecture part of the course focuses on the theory of physics. The Laboratory portion of your course
is a counterpoint to the lecture, exploring the experimental side of the science. In the lab, we have
two principal objectives. First, we want to develop your skills as a scientist conducting experiments.
Second, we want to develop your technical writing skills, which is the focus of this document.
Technical writing has several key features that we summarize as Clear, Complete, Concise, and
Correct. In the context of a lab report, clear writing means that you are effectively communicating
your understanding of the experiment you performed. Complete writing means that all of the
necessary information to support your work is present in the document. However, the writing is still
concise, being as brief as possible while still communicating effectively. Finally, technical writing
should be correct. In contrast with other types of writing like opinion pieces, technical writing can
be wrong.
In science, the main medium of technical writing is the scientific journal article. These journal
articles are how scientists report their results to each other. In Physics Laboratories, we evaluate your
work based on the Lab report, which mimics the standard structure of a journal article: Introduction,
Methods, Results, and Discussion (IMRaD). This structure organizes the information in the lab report
so that the reader can readily assess the results of your experiment. A standard lab report includes
these four main parts plus a few other components that have become standard in journal articles as
well.
14
CHAPTER 4. LAB REPORTS
PHYS 130 – Page 15
• Introduction: The introduction connects your lab report to the broader context of science. When
a reader finishes the introduction, they should know what the objective of the experiment is
and what the theory that motivates this experiment is.
• Methods: The methods section explains how you conducted the experiment. The standard for
the methods section is that the reader should be able to replicate your experiment after reading
this section.
• Results: The results section summarizes the data you collected in the experiment. Data can take
many forms, but in physics, it is usually well described by tables and graphs of measurements.
This section is designed to be a presentation of results without judgment or conclusion.
• Discussion: The Discussion section uses the content of the results section to form conclusions.
It is in this place that the results are evaluated in comparison with the theory outlined in the
introduction.
• Conclusions: The conclusions serve the same purpose as an abstract in other forms of technical
writing. It is a short summary of the core of the lab report. No new information is presented.
• References and Acknowledgements: Citations are an essential part of any piece of academic
writing and you should cite all works that informed your lab report either directly (quote or
figures) or indirectly.
• Appendix: The appendix represents material that supports but is not essential to the main part
of the lab report. This provides evidence and motivation for your interpretation. No new ideas
are presented in the appendix.
As you grow accustomed to writing in this standard structure, you will develop your skills in both
reading and writing journal articles as you move through your scientific career.
4.2
Formatting and Requirements
There are several “business rules” for writing Physics Lab reports. Your submission should
address all these points.
• The entire lab report must be typed including equations. The only exception to this are diagrams
or situations where the Lab Manual expressly asks for a drawing.
• The lab report has a maximum length of four (4) pages, not including References, Acknowledgments, or the Appendix. Your assignments should not have a cover page. If you have a lab
partner, mention them in the Acknowledgements.
• The lab report may contain an Appendix that exceeds the four page limit. The Appendix can
only contain (1) answers to questions specifically asked in the lab manual that are not included
in the main body of the report and (2) extra raw data to demonstrate that you completed part
4.3. COMPONENTS OF THE LAB REPORT
PHYS 130 – Page 16
of the experiment that cannot fit within the confines of the main report (for example, figures or
extended data tables).
• Labs with pre-lab quizzes will have a shorter page limit that will be specified in the lab and
announced on eClass.
• The document should be formatted to have dimensions of a standard ANSI Letter size. This
is the standard North American paper size that and measures 215.9 by 279.4 mm (8.5 × 11
inches).
• You must use a size 12-point font or larger.
• You must include at least 1.9 cm (3/4 inch) margins on all sides of the text: top, bottom, left,
right.
• For clarity, all tables and pictures must be labeled (Table 1, Graph 3, Diagram 2...) and
described with a caption. Table captions are typically above the table, while figure/graph
captions are typically below.
• If the TA has a hard time reading your report or following your ideas, marks will be deducted.
• The lab report is considered to be entirely your own work, unless you give proper credit to
other sources. This includes the lab manual, your lab partner, and sources on the internet.
These rules will help guide you with the clarity and conciseness of your report.
4.3
Components of the Lab Report
This section describes the components of the lab report in more detail. Overall, a good model
for the lab report is that you are telling a story. It is the gripping tale of the lab that you completed,
even though it may not be an instant bestseller.
4.3.1
Introduction
The introduction orients the reader to what is going to be presented in the lab report. It makes the
connection between all of physics down to what happened in your lab. In making this connection,
the introduction also motivates the lab and explains why the results of the lab will be important or
interesting.
In terms of content, the introduction is 2-3 paragraphs that (1) state the objectives of the labs, (2)
give a background of the theory, and (3) present the key equations that will be used.
We do not want a drawn out introduction; it should be short and to the point. You do not need
to list every equation in the lab manual, but instead you should focus on the equations most relevant
CHAPTER 4. LAB REPORTS
PHYS 130 – Page 17
to the lab. The final form of your equations does not need to be here, only the basic theory. The
analysis section will show the more detailed calculations necessary for the lab.
Most plagiarism cases in Physics Labs come from material in the introduction. You should never
copy-paste from the lab manual or any other sources. Even when you are paraphrasing another
source, remember that it must be cited and that you should type your paraphrasing into the lab report
rather than editing pasted text. If there is a direct quote from the lab manual it must be quoted and
directly referenced after the quote. If are unsure if something needs to referenced, you probably need
to reference it anyway. This includes any figures you did not create.
A few tips can help with writing the introduction. First, you can outline the introduction before
the lab and even develop some content; however, you should revise it after the lab period to ensure
that it is relevant to the rest of the report. For example, you won’t know how important an equation
is until you write up your report. Second, if you’re having a hard time getting started, just start with
“The purpose of this laboratory is...” and finish the sentence. Explain briefly what you will do and
what theory you are exploring.
4.3.2
Methods
The experimental method should be a short description on how to complete the lab. In journal
articles, the standard for this section is to provide the reader with the necessary information so that,
if they have the same equipment, they could complete the lab the same way you did it and get the
same results. In practice, that can contain too much information. Instead, you should make sure that
your Methods section gives the key steps that are relevant to the experiment and outlines the things
that you did to get your particular results. This section will be read by your instructor to understand
anything that might explain why your data are messed up1.
In terms of format, this should not be a cookbook recipe. It should be a description of what you
did. Your statements should be statements that start with “We...” “I...” or “My lab helper...” and not
be instructions to the reader telling them what to do.
• List the equipment used in the setup and describe its purpose. This does not have to be a bullet
point list but you should identify every key piece of equipment in the setup and what it was
used for.
• Include a description of your setup. A photo or diagram with caption is best but a detailed
description could work.
• You can use a list form to make the procedure easier to follow. This should include the steps
that are essential to getting your data: How did you calibrate the device? What kind of tests
did you do? How did you do the tests?
1One datum, many data, but we recognize that “data is” can also be used.
4.3. COMPONENTS OF THE LAB REPORT
PHYS 130 – Page 18
The methods section is not a justification of why you did the lab the way you did, but if you made
changes to the procedure, you can explain why you did that here or in the Discussion. Remember
that all pictures and diagrams should be labeled and cited.
4.3.3
Results
The results section presents your data but it does not make any conclusions or interpretation. The
Discussion section that follows is the place for such conclusions. In scientific journals, the material
in the Results section should never be wrong. The content simply is the results of the experiment as
described in the Methods section.
In your lab report, this section should contain the data from your experiment and the results of any
calculations. A good lab report will have a clear presentation of the data, both in raw and summary
forms.
The raw data are usually presented as a table, which can be pasted in from your spreadsheet. The
summary data should be graphs of the raw data and the calculations used to develop derived data
(e.g., a linearization of the relationship you are fitting). If your data table is long, you can give a
sample of your table and a short description of the range of the columns and include the full data in
the Appendix.
You should have a sample calculation of any row/column in which you are manipulating data.
This is not only to show if you did it correctly, but also helps the TA retrace your steps to see where
you went wrong. Extensive calculations can be shown in the Appendix, but the lab report should
contain a clear reference to that part of the Appendix and a description of what that analysis is used
for.
All graphs that are required by the lab manual and all graphs for your linear fits must be shown
and labeled. If you are linearizing an equation, you need to show your equation and identify the
corresponding values, explaining which variables are 𝑦, 𝑥,𝑚 and 𝑏. For more information on plotting,
see the graphing chapter.
The Results section should clearly show your final results. For example: if you are measuring
the gravitational acceleration, you should clearly state the value of 𝑔 ± 𝛿𝑔 that you measure. This
result should also show the error analysis. Your results should be rounded to have one significant
digit in the error. For example, if your linear fit gives that 𝑔 = (3.14659843 ± 0.0323119) m/s2 , the
error should be rounded to ±0.03 m/s2 , one significant digit. Your measured value cannot be more
precise than the error so your final value would be 𝑔 = (3.15 ± 0.03) m/s2 . All values should have
appropriate units; answers without units are wrong.
You are submitting an individual lab report so all work in this section should be done independently. All graphs and calculations must be done independently. However, you should cite what data
you received help collecting.
CHAPTER 4. LAB REPORTS
4.3.4
PHYS 130 – Page 19
Discussion
The Discussion is where you interpret the results and it is the most important section of the lab
report. The Discussion should not include any calculations or error analysis, which appear in the
preceding Results section. The point of the Discussion is to evaluate your data in the context of the
purpose of the experiment, i.e., what you outlined in the Introduction.
The Discussion section should address several points.
• You should start by identifying and restating the main result(s) from the Results section: “We
measured 𝑔 = (3.15 ± 0.03) m/s2 .”
• You should then make a conclusion regarding whether your results are consistent with the theory
presented in the Introduction. Assessing consistency involves asking whether the theoretical
value 𝑔 = 9.81 m/s2 falls with the error range of the experimental value. We express the
agreement based on how far the theoretical and observational values are from each other in
terms of the error interval:
– If the experimental values are within one error interval, i.e., 𝑔 = (3.15 ± 0.03) m/s2 , i.e.,
3.12 m/s2 - 3.18 m/s2 there is “good agreement” between the theory and the experimental
results.
– If the experimental values are within two error intervals, i.e., 𝑔 = [3.15 ± 2(0.03)] m/s2 ,
i.e., 3.09 m/s2 - 3.21 m/s2 there is “modest” agreement with the between the theory and
the experimental results.
– If the experimental values are within three error intervals, i.e., 𝑔 = [3.15 ± 3(0.03)] m/s2 ,
i.e., 3.06 m/s2 - 3.24 m/s2 there is “poor” agreement with the between the theory and the
experimental results.
– Experimental values that are more than three error intervals from the theoretical value
are “not consistent” or “disagree.”
In the example we’ve been working with,
𝑔theory − 𝑔observed 9.81 m/s2 − 3.15 m/s2
=
= 222.
𝛿𝑔
0.03 m/s2
(4.1)
The experimental and theoretical values disagree, and the lab report should now turn to why.
Relative errors with percentages (i.e., “This result is 30% of the theoretical value.”) provide
useful context but the only analysis that matters is the comparison to the error range. Reporting
relative errors without examining the absolute error range will be penalized.
• You should then discuss the quality of your results, addressing whether the linear fit represents
your data well and whether there is individual data that are clearly in disagreement with the
overall trend. You can also speculate why these data might disagree with the others and
potentially re-analyze your results without these data included. Note that you should ALWAYS
report the data, even if you describe why they shouldn’t be included in the results. You should
highlight anything that may have affected your results.
4.3. COMPONENTS OF THE LAB REPORT
PHYS 130 – Page 20
• You should then address the dominant uncertainties in your analysis and discuss ways to
improve the results. These points can be pure speculation like “use an ultraviolet laser instead”
or they can be readily implementable strategies like “repeat the experiment 10 times.”
• You can answer any questions that are asked in the lab manual provided the answers easily fit
into your narrative. Otherwise, these can be included in the Appendix.
Given everything that goes into the discussion, it should not be a single paragraph. Split your
ideas up and create good flow between the sections. For example, the first paragraph you could talk
about your value for 𝑔 and how it compares to your theoretical results. Your second paragraph could
justify why your answer did or did not make sense. Your third paragraph could be about any errors
you encountered or troubles with data. The last paragraph could be any specific questions asked in
the lab manual. This allows the TA to easily make sense of your report as a whole.
4.3.5
Conclusion
The conclusion section should be a short one paragraph summary of the experiment. There
should be no new information in the Conclusion. The conclusion should be a brief description of the
lab, including the physics principle and what experiment is being done. It should also include your
main results 𝑔 = (3.15 ± 0.03) m/s2 and whether your result is consistent with the theory. The reader
should be able to read the conclusion and get a good idea of what the experiment is about and the
main findings. Do not put all the questions, issues, or steps that were in the lab, only key findings.
4.3.6
References
This section just includes your citations. There is no preferred citation style but you should be
clear about what sources you cite. If you are at a loss for citation style, try the Chicago style2. The
References are not counted in the 4-page limit on the report. Remember that everything in the lab
report is assumed to be your original work unless you declare it otherwise. An example citation for
this lab manual would look like this:
Isaac, et al. 2020. Lab Manual PHYS 124. Edmonton: University of Alberta, Department of Physics.
Change the date, course number and authorship as appropriate.
2https://www.chicagomanualofstyle.org/tools_citationguide/citation-guide-2.html
CHAPTER 4. LAB REPORTS
4.3.7
PHYS 130 – Page 21
Acknowledgments
Use an Acknowledgments section to attribute ideas to other people. This is a good place to
recognize contributions from your lab TA, other students in the class, or lecture information that
was helpful in developing your lab report. If someone shares a good idea with you that you end up
including in your report, use this section to describe it. As you go through your career, you will
engage in collaborative and group work. In all such cases, giving and receiving credit for ideas
prevents a lot of trouble and upset. Acknowledgments do not count toward the 4-page limit on the
report.
The Acknowledgments section is free-form. You may use a narrative description. For example,
you can write “I received help from the online class forum on calculating my errors,” “I received
advice from a TA in the Physics Help Centre,” or “My friend Marie Heisenberg gave me advice on
graphing.”
In general, the Reference section gives credit to resources that anyone would be able to access,
i.e., it is restricted to citable sources. The Acknowledgments section gives credit to those resources
that only you or members of your class would be able to access.
4.3.8
Appendix
The Appendix is supplementary information to the main report and is not counted against your
four-page limit. All key points must be made in the body of the report. You should think of the
Appendix as an Evidence section that provides backing for the main points present in the body of the
report. Material you may include:
• Long data tables that cannot fit in the four-page main report. A short summary of the table or
an excerpt of a long table should appear in the main body.
• Extra figures to illustrate the experimental set-up.
• Answers to questions present in the lab manual that you are required to answer but do not easily
integrate into your lab report.
You are evaluated on the content in the body of your report not the Appendix.
4.4
Voice and Style
Writing should not be boring or awkward, and technical writing is no exception. To make your
lab report sound natural, you should write it using a first-person active voice. Sentences subjects
like “I” or “We” are encouraged since you are reporting on your experiment. A sentence like “We
4.4. VOICE AND STYLE
PHYS 130 – Page 22
then broke into the chemistry lab to dispose of the plutonium in their sinks.” would be a good
component of Methods section, but maybe not so great from a safety perspective. You should avoid
using sentences in passive voice since the passive voice is not clear about what is doing the action.
If a sentence makes sense when you add “by zombies” to the end of it, it is in the passive voice3. For
example, “The laser was directed into the two slits.” is in the passive voice because it would make
sense to say “The laser was directed into the two slits by zombies.” You should instead say “We
directed the laser into the two slits.”
Another key component of technical writing is mathematics. You are expected to type up
mathematics in your writing and this mathematics is considered part of the writing. Equations are
sentences and can be integrated directly into your writing. You can include things like Newton’s
Second law, which is
® = 𝑚®a,
F
(4.2)
as part of the sentence. You should explain the reasons for equations appearing in your text and
® is the net force, 𝑚 is the inertial mass, and a® is the acceleration. Note
define the variables, such as F
that equations can be numbered, as above, for future reference in the lab report.
3This tip courtesy of Dr. Rebecca Johnson, Professor of Ethics with the US Marines.
Chapter 5
Assignments
5.1
Introduction
Depending on which physics course you’re taking, labs may involve a combinations of lab reports
and assignments, or just assignments. These assignments will typically involve answering a series
of questions that have been posted to eClass and submitting a PDF containing your answers along
with any graphs, tables, or other materials to eClass. This chapter is meant to help familiarize you
with what is expected when submitting these assignments.
5.2
Formatting
While we do not expect the same level of formality for assignments that we might expect from
a full lab report, it’s important to submit a nicely formatted, professional document. This helps the
TAs follow your work more easily and, more generally, give you a chance to practice skills that will
be useful throughout your University career and beyond; many, perhaps even most, jobs will require
you to be able to work with spreadsheets or produce professional-looking documents at some point.
Communication is a skill and requires practice and instruction.
There are a number of basic formatting requirements:
• All documents should be typed; this includes all text, captions, calculations, and labels
• All graphs, diagrams, and other figures should be produced electronically unless otherwise
indicated
• Use 12-pt font or larger and select an appropriate font; good choices include: Times New
Roman, Garamond, Calibri, Helvetica, or any other font that looks clean and professional
• Use at least 2.54 cm (1 inch) margins for the body of the text
23
5.2. FORMATTING
PHYS 130 – Page 24
• Adhere to the page limit posted for each assignment on eClass
• Answer all questions in complete sentences
• Submit only a single PDF
• You should create your assignment in a word processing program like Microsoft Word or
Google Docs; do not submit an Excel or Google Sheets spreadsheet with the answers typed
into the cells
Some of these points will be discussed in more detail below.
5.2.1
Documents vs. Spreadsheets
Spreadsheets are very useful for performing calculations on large tables of data and for producing
graphs, but they are not suitable for writing out the answers to an assignment or for producing lab
reports. Instead, you should write your assignment in a word processing program such as Microsoft
Word or Google Docs (both are free through the University). These programs offer many more
options for properly formatting your assignments.
If you create your tables and graphs in a spreadsheet, they can easily be copied over to your
document by selecting the relevant material in the spreadsheet, right-clicking, selecting “copy”, and
then pasting it into your document. As long as you’re using programs from the same company (e.g.
Word + Excel or Docs + Sheets) this process should be essentially seamless and is just as easy as
taking a screenshot but produces a much higher quality figure in the final document. This also allows
you to continue to edit the graph or table directly in your document instead of having to take a new
screenshot. Avoid taking a screenshot of graphs or tables whenever possible.
Once your document is complete, you can then export to PDF or print to PDF. These are essentially
the same process, but you can use whichever you prefer. In Google Docs, select File > Download >
PDF document (.pdf). In Microsoft Word you can use File > Save As... and then select PDF from
the “File Format” drop-down box. Whenever you create a PDF, you should double-check that the
formatting is the same as it was in your original document. While Word is pretty good at recreating
the exact formatting, Google Docs sometimes ends up shifting things slightly which can cause you
to go over the page limit.
You should also check when uploading your assignment that you are uploading the correct
assignment. It is a relatively common occurrence that students submit a previous assignment or the
lab manual itself. If you catch this immediately, it’s relatively easy to fix. If it isn’t caught until the
assignment is graded and handed back, there’s very little that can be done.
CHAPTER 5. ASSIGNMENTS
5.2.2
PHYS 130 – Page 25
Calculations
When you’re typing out equations in your document, you should use the available equation editor
whenever possible. Most word processing programs have one available and the equation editors in
Microsoft Word, Google Docs, and Pages are very easy to use. In most programs, you can find the
equation editor under the ’Insert’ menu. After clicking on this, you’ll have the option to begin typing
your equation or using the visual editor by selection options such as fractions and exponents from
the toolbar.
There are some shortcuts that can make typing equations easier. For example, rather than hunting
for the appropriate Greek symbol from the menu, you can usually just type a backslash followed by
the character. Typing \delta would produce 𝛿 and \Delta would produce Δ. This does require
knowing how to spell the name, but can be much faster than using the menu. You can also use
underscore (_) to create a subscript while a caret (^) produces a superscript. You can look up all the
available shortcuts, but we’ll provide a short table of them below. If you find yourself producing a
lot of documents that are heavy in mathematical notation, you may want to consider using LATEX, a
typesetting system for preparing documents. There is a certain amount of overhead in learning an
entire typesetting system but may be worth it for you and it is sometimes required or preferred in
some physics lab courses.
Table 5.1: A short listing of some shortcut commands that can be used in
the equation editors of common word processing programs including Microsoft
Word and Google Docs.
Command
x^2
\int_a^b
\sum
\omega_1
\Omega_i^2
5.2.3
Result
𝑥2
∫𝑏
𝑎
Í
𝜔1
Ω𝑖2
Page Limits
Unless otherwise noted on the assignment itself on eClass, there is a strict 4-page page limit in
place for all assignments. A title page (if you have one) does not count against the limit and neither
does the References and Acknowledgements section. Any material exceeding 4 pages will simply not
be marked. TAs are instructed to ignore anything on page 5 and beyond (unless they’re References
and Acknowledgements). The only time the page limit might be an issue is if you have a very long
table of data. Usually we will only ask for a sample table consisting of 5 - 10 rows of data. If the
entire table is necessary, you can put a sample table in the main part of your assignment and place
the full table at the very end of the report in a section entitled “Appendix”. Material in the Appendix
is not marked, but it can be useful for TAs to have the full data set to refer to while marking the rest
of the assignment. If you do this, make sure to mention the full data set in the caption of the sample
table.
5.3. STYLE
5.3
PHYS 130 – Page 26
Style
Your answers should be in complete sentences. While some questions may be very direct, most
questions will contain something along the lines of “Why?” or “Explain.” Here we are looking for
you to provide some kind of evidence or line of reasoning to explain how you got to your answer.
Sometimes this may involve proving a concept using mathematical reasoning, showing a calculation,
using outside, credible sources to back up your answer, or using data from the experiment to show
the validity of your answer. We’re usually looking for somewhere between 3-6 sentences but it can
be more or less depending on how well you word your answer and how many separate points you
have to make to back up your answer.
Try to think about the best way to support your answers. If you’re asked whether the result of an
experiment seems reasonable, you might be surprised how many different ways you can support your
answer. You can look at the size of the uncertainty compared and how many intervals your answer is
from the generally accepted answer. You can also discuss whether there were many outliers in your
dataset, whether the linear fit seems to match your data well, and whether the slope and y-intercept
of the fit are within uncertainty of their expected values. If we ask for a theoretical or experimental
number that requires calculation, you should include the calculation of the value and its uncertainty
in your answer unless otherwise specified.
Make sure to at least read the sections on Formatting (4.2), Results (4.3.3), and Discussion (4.3.4)
from Ch. 4. These sections should be helpful in understanding the formatting requirements and how
to thoroughly explore your answers and provide evidence to support them.
5.4
References and Acknowledgements
Every assignment should end with a section of References and Acknowledgements. References
are used to cite sources of information or ideas such as textbooks, internet sites, and research articles.
Acknowledgements are used to acknowledge anyone that you collaborated with; you should include
their names as well as a very brief explanation of how they helped.
When you are citing sources, the best practice is to put a number at the end of any sentence where
the idea is not your own. We assume that everything in your assignment is your own idea and written
in your own words unless you indicate otherwise. Properly citing sources and acknowledging the
people you worked with is key to avoiding issues with plagiarism. Even if you reference a particular
source, you should always put the ideas in your own words. If you wish to use the exact wording of
the source, you must also put quotation marks around the parts that are identical. It’s still best to try
to put everything in your own words, however. If your entire assignment is simply quotations from
another (properly referenced) source, you won’t get in trouble for plagiarism, but we can hardly grant
a high grade if you don’t appear to understand the material well enough to reword it.
Here is an example of how you might put references into an answer and then what your References
and Acknowledgements section might look like at the end:
CHAPTER 5. ASSIGNMENTS
PHYS 130 – Page 27
Q1: What is the lifetime of a free neutron?
The neutron decay lifetime is (888.3 ± 2.5) s [1].
References and Acknowledgements
[1] Werth, G. “Fundamental particle properties using particle traps.” Journal of Physics
G: Nuclear and Particle Physics, vol. 20, no. 12, pp. 1865-1883, 1994.
John Smith - my lab partner for helping set up the equipment, helping with data collection,
and for discussing assignment formatting
Of course, depending on the question, your answers may be entire paragraphs with multiple
references. All of the references should be placed into the References section at the end of the
report. Your references don’t need to be quite as complete as the one shown above, but it should be
sufficiently precise that someone would be able to find the original source. For example, simply citing
http://hyperphysics.phy-astr.gsu.edu/hbase/index.html would be insufficient, but http://hyperphysics.
phy-astr.gsu.edu/hbase/conser.html#coneng should allow someone to find the original text.
5.5
Conclusion
As always, if you’re unsure about assignment requirements you should contact your TA. Make
sure to ask them specific questions. Questions like “Is my report okay?” are too broad and it would
be unfair for your TA to go through your entire report unless they do the same for every student.
However, they should be able to promptly answer more specific questions. You can also discuss with
your fellow students. If you do, make sure to acknowledge them and try to avoid sharing the actual
report itself or showing your report through screen sharing as there can be a temptation to plagiarize.
Chapter 6
Measurements and Uncertainties
6.1
Measurement and Uncertainty
In physics lab, we will be spending most of our time making measurements of different physical
experiments. Since these experiments are usually simple (“ball rolling down ramp”), one of the
main objectives of physics lab is to improve the sophistication with which you are viewing these
measurements. These ideas span all of science, but in first-year physics, they are especially clear and
we can spend some time thinking about them. Many of the topics you read about here may sound
familiar if you are in or have ever taken STAT 151, but we will bend these general concepts to suit
our specific tasks.
When we make a measurement, we will obtain a value (“2.3 metres”), which can frequently be
compared to a theoretical expectation (“the theory predicts 2.1 metres”). We need some way of
describing whether the physics we have used to describe the experiment is complete or whether there
are factors in the system that are not explained by the theory. The classic “unexplained factor” in
first-year physics is friction or air resistance, but other phenomena can also lead to disagreement
between your results and expectations. To answer whether these factors are important, we need to
quantify how good our answer is. We do this with what we usually call an error but is more precisely
described as an uncertainty. We will use these two names interchangeably but remember that an
“error” does not necessarily mean that something was done incorrectly.
A key part of all science is reproducibility but no experimental setup can ever be perfectly
replicated. Nor can any values in the experiment be perfectly well known. We use errors to quantify
how well the experimental values can be known. We denote the value of an error with a 𝛿, so the
error in the quantity 𝑥 is denoted 𝛿𝑥. The uncertainty can be thought of as a “warranty” where you
are promising that if someone else made this measurement, they would likely get a result that is
within 𝛿𝑥 of the value 𝑥. In lab, you are not trying to get the theoretical answer, but you are trying to
make sure that your warranty is good.
Statistics and Uncertainties: In the above paragraph, we were cagey about the exact meaning of
28
CHAPTER 6. MEASUREMENTS AND UNCERTAINTIES
PHYS 130 – Page 29
the error range and that results were “likely” to be within 𝛿𝑥 of the measured value 𝑥. This hesitation
comes from the nature of uncertainties, specifically in how they are governed by probability and
statistics. Often small random factors (e.g., air currents) will affect our results and we cannot account
for their presence exactly but we must describe the results statistically with a probability density
function1. An example of this statistical measurement is shown in Figure 6.1.
Probability Density Function
1.0
Data
0.8
0.6
0.4
0.2
0.0
0.2
x
1
2
x x x+ x
3
Measurement
4
5
Figure 6.1: Example of the results of the experiment and the resulting uncertainty.
In this Figure, we show the results of an experiment conducted 20 times as vertical lines in the
top of the plot. Note that on the measurement axis, these values range from 2.1 to 4.2. These values
are scattered through this range, but most of the values are near the centre. The mean of these data
is 𝑥 = 3.2. If a 21st experiment were conducted, it would likely not get 3.2 as an answer but some
value in the range shown. Below the data we show in gray a Gaussian distribution or a “bell-curve”
that describes these data well. We assume that measurements are usually well described by this error
distribution but this is not always the case in physics. However, for first year physics labs, this is a
good place to start.
For a Gaussian distribution, we show the error ranges 𝛿𝑥 for the measurement, indicating the
range from 𝑥 − 𝛿𝑥 to 𝑥 + 𝛿𝑥. Most of the data are in this range, but some lie outside of it. With the
assumed Gaussian shape, statistics makes clear predictions that
• 68% of the data are between 𝑥 − 𝛿𝑥 and 𝑥 + 𝛿𝑥,
• 95% of the data are between 𝑥 − 2𝛿𝑥 and 𝑥 + 2𝛿𝑥, and
• 99.5% of the data are between 𝑥 − 3𝛿𝑥 and 𝑥 + 3𝛿𝑥.
1You don’t need to know what this is, but if you have a statistics background, this is what we’re describing.
6.1. MEASUREMENT AND UNCERTAINTY
PHYS 130 – Page 30
When you give an error in a measurement, you are making the claim that the true value of the
measurement is extremely likely to within 3𝛿𝑥 of your measured value. Remember: it is perfectly
fine if the theoretical value is outside this range. There are factors beyond our control. However, you
are making claims about the quality of your measurement and someone else ought to be able to set
up your apparatus and get the same quality of measurement.
Making Judgments: In making measurements, you are frequently called upon to estimate the error
in your measurement. In making such an estimate, there may not be a completely correct answer.
Unfortunately, there are clearly wrong values. If you are using a metre stick to measure the length
of a pencil, an error estimate of 1 m is clearly incorrect since it points to the inability to correctly
use the metre stick. A typical metre stick is typically marked off with tick marks every 1 mm, so the
uncertainty is probably around this magnitude, though it does not matter if the value of the error is
1 mm or 0.5 mm. In particularly tricky measurements, say for a vertical distance without knowing
exactly what direction is straight down, the error might even be 3 to 5 mm. However, asserting that
you can measure the distance with a metre stick down to a scale of 10 𝜇m implies absurd skill with
the metre stick that is also incorrect. Throughout lab, you will be called upon to justify your error
values for measurements. Think about what limits the precision to which you can measure your
data. A general rule of thumb is to take half the smallest division for analog measurements, and the
smallest division for digital measurements.
Repeating Experiments: The golden standard for characterizing errors is through repeating an
experiment. If the experiments are independent, the answers you get should vary within the range
of errors. To use this method effectively, you need several to repeat many times, but even going
from one to two experiments gives you a sense of how big the uncertainties should be. In particular,
repeating experiments will help you get the correct uncertainties out of using linear fits which will
be described in the graphing chapter.
When you make repeated measurements of the same quantity, we get a new way of measuring
the true value. Since the measurements scatter around the true value by the error, we can get a better
estimate of the true value by calculating the mean (average) of the data, usually denoted 𝑥:
¯
𝑁
1 Õ
𝑥𝑖 .
𝑥¯ =
𝑁 𝑖=1
(6.1)
Here 𝑁 is the number of data and 𝑥𝑖 is the 𝑖th measurement. Add up all the data and divide by the
number of measurements.
We can then calculate the error on an individual measurements since this is the typical separation
from the mean value. This is measured by the standard deviation, which we denote as 𝜎𝑥 since that’s
the variable they use in STAT courses.
v
u
t
𝑁
1 Õ
𝜎𝑥 =
(𝑥𝑖 − 𝑥)
¯ 2
(6.2)
𝑁 − 1 𝑖=1
Staring at the formula, you should see that this is the average of the squared separation between a
measurement and the mean. For the case of measurement errors, we take 𝛿𝑥 = 𝜎𝑥 .
CHAPTER 6. MEASUREMENTS AND UNCERTAINTIES
PHYS 130 – Page 31
This value represents the error you can determine for each measurement, so each 𝑥𝑖 has an error
of 𝜎𝑥 . The other advantage of making repeated measurements is that you can average down the error
of each individual measurement. The error in your measured mean value is:
𝜎𝑥
𝛿𝑥¯ = √ ,
𝑁
(6.3)
which is called the standard error in the mean. This is the uncertainty you can use for your
measurement of the mean.
6.2
Significant Digits
We now take brief diversion in discussing uncertainties to describe how we record the actual
numbers used in the analysis. Thanks to computers and calculators, it is very easy to keep 10 to
12 digits in every number, but in science we rely on the use of significant digits to communicate
how precisely we know a measurement. Significant digits have a reason to be there and you should
consider these reasons. For example, if you report your height as 1.754968403 metres, it implies
that you were able to measure it with an error of one part in a billion, down to the last nanometre.
It shows that you certainly did not use a metre stick, because we cannot read a metre stick with that
kind of precision.
In physics, we deal with significant digits in one of two ways. The first way is simple and actually
not very rigorous: that way just says “use common sense.” This is what we happens in online
homework assignments (mainly because the computer grading cannot propagate sig-digs very well).
Thus, we usually ask for 3 significant digits unless a different number is specified. In doing this, we
are admitting that we do not care much about the uncertainties in the numbers you are using in the
homework. As such, we should not read anything special into the number of sig-digs, we are just
skipping it in the name of convenience. The second way is to do the rigorous treatment of significant
digits, which we will deal with more frequently in lab because the error in our measurements matters.
Generally, it is always better and more complete to maintain significant digits in all your calculations. But sometimes it can end up just being a real time-consuming pain and take your mind off
what we really wanted to teach you. So, we will tell you when to worry about it and when not to.
This is for reasons of consistency with the homework assignments, and is also because we want to
stress that while sig-digs have an important meaning that you should be well-aware of, we do not
want every lab to get bogged down with error propagation and significant digit calculations. Hint:
whenever you have to do a real error calculation, it will define your significant digits for you. If you
do not have to do any error calculations or significant digits, then use 3 significant digits.
The rules for significant digits are as follows:
1. If there are no errors, the number of significant digits in the answer is equal to the smallest
number of significant digits in any of the variables.
6.2. SIGNIFICANT DIGITS
PHYS 130 – Page 32
2. If there are errors, the error analysis will define the number of significant digits for you. In
most cases, you should round your error to one significant digit and round the value to the
same place value (i.e., number of decimal places).
3. For counting the number of significant digits a number has, a simple rule to follow is that all
numbers are significant digits except the leading zeros. Here are some examples:
1205
four significant digits.
1205.00
six significant digits.
0.041
two significant digits.
100.0001
seven significant digits.
0.000000109 three significant digits.
1,000,000
one significant digit.
six significant digits.
1,000,000
1,000,000.
seven significant digits.
Only the 1,000,000 is a bit tricky, because it is not obvious whether the measurement was exactly one
million by chance, or whether it was rounded, in which case the number of actual significant digits
would be smaller than 7. For example, when we say the sun is 150 million km away, do we really
mean 150,000,000 exactly? Actually, no. None of the last seven zeros are significant. We can deal
with the ambiguity in two ways. The first way is to put a line over or under the last significant digit as
is shown in the next-to-last case. The other way is to use scientific notation; all digits in the number
are significant. Writing 1,250,000 as 1.2500 × 106 specifies which of the zeros are significant.
There are a set of rules for writing a value and its uncertainty:
• In introductory labs, uncertainties have only one significant figure, such as 0.2 or 0.001. Not
1.2374. (The uncertainty in the uncertainty is an advanced topic.)
• A value should have the same number of decimal places as the uncertainty: 62.4 ± 0.1 cm.
Obviously, if the measurement is reported as 62.4 ± 0.01 cm, it would be pretty weird because
it implies that we are more certain about our error than we are about the actual value. Bizarre.
Similarly, 62.4 ± 1 cm does not make sense, because it says that our measurement is more
precise than its uncertainty.
• The value and its uncertainty should have the same units. Measurements are reported in the
format 𝑥 ± 𝛿𝑥, followed by the unit. Here 𝑥 is the value and 𝛿𝑥 is the uncertainty.
• A value and its uncertainty should use the same power of 10. These last two rules are there to
avoid confusion – if you use different units and different powers of 10, you will be likely to get
into a nasty mess.
CHAPTER 6. MEASUREMENTS AND UNCERTAINTIES
6.3
PHYS 130 – Page 33
Error Propagation
The last subtlety in the dealing with uncertainties is how to determine a calculated value from
other values that have uncertainties. For example, let’s say we want to obtain the mass of an object
by applying a force and measuring its acceleration. Then we can get the mass from Newton’s second
law: 𝑚 = 𝐹/𝑎. However, we cannot measure our applied force perfectly, nor can we measure the
acceleration perfectly. In fact, we have a force uncertainty 𝛿𝐹 and an acceleration uncertainty 𝛿𝑎 to
worry about. So what is the error in our measurement of the mass?
For Gaussian errors, the error is determined by the partial derivatives of the function. Specifically,
the error is given by the chain rule in partial differentiation, that is, if 𝑓 (𝑥, 𝑦, 𝑧, ...), then
s
2
2
2
𝜕𝑓
𝜕
𝑓
𝜕𝑓
𝛿𝑓 =
(𝛿𝑥) 2 +
(𝛿𝑦) 2 +
(𝛿𝑧) 2 + · · ·
(6.4)
𝜕𝑥
𝜕𝑦
𝜕𝑧
The process of computing the implied error in a calculated value based on the function used to
calculate the results is called error propagation. Using this approach, we can work out several
formulas that are given in Table 6.1.
Table 6.1: Error formulas
given 𝑓 =
𝑥
𝑥𝑛
𝑥±𝑦
𝑥𝑦
Then 𝛿 𝑓 =
𝛿𝑥
|𝑛𝑥 𝑛−1 𝛿𝑥|
p
(𝛿𝑥) 2 + (𝛿𝑦) 2
r
2 𝛿𝑦 2
+ 𝑦
|𝑥𝑦| 𝛿𝑥
𝑥
r
𝑥/𝑦
𝑥
𝑦
𝑥𝑎 𝑦𝑏
𝑥𝑎 𝑦𝑏
ln 𝑥
𝛿𝑥
|𝑥|
𝛿𝑥 2
𝑥
r
𝑎2
+
2
𝛿𝑦
𝑦
𝛿𝑥 2
𝑥
+ 𝑏2
2
𝛿𝑦
𝑦
Table 6.2 shows some specific examples. In these examples, we want to find the error 𝛿 𝑓 in the
function 𝑓 , by propagating the errors in all the other variables 𝑥, 𝑚, 𝑔, 𝑅, 𝑦, 𝑡 etc.
Example: Using our above example of measuring the mass by using Newton’s second law, let us
say the applied force was 10.44 ± 0.02 N and the acceleration was measured to be 0.53 ± 0.01 m/s2 .
What is the mass?
6.3. ERROR PROPAGATION
PHYS 130 – Page 34
Table 6.2: Using error formulas: examples.
Given 𝑓 =
6𝑥 3
𝑚𝑔
𝑥
𝑅2
𝑥𝑦 2
𝑡6
Then 𝛿 𝑓 =
18𝑥 2 𝛿𝑥
q
𝑚𝑔
𝛿𝑔 2
𝛿𝑥 2
2
( 𝛿𝑚
𝑥
𝑚 ) +( 𝑔 ) +( 𝑥 )
2𝑅𝛿𝑅
r
𝑥𝑦 2
𝑡6
𝛿𝑥 2
𝑥
2
𝛿𝑡 2
+ 2 𝛿𝑦
+
6
𝑦
𝑡
We actually have to consider two things here. If we blindly use the formula 𝑚 = 𝐹/𝑎, we get
10.44 kg · m/s2
= 19.69811321 kg
0.53 m/s2
on our calculator screen. So we have a significant digit problem. Now let us look at the error. Our
formula is 𝑚 = 𝐹/𝑎. According to the above tables, the error formula we will use is the one for
𝑓 = 𝑥/𝑦. This gives the error 𝛿𝑚 as:
s
2 2
𝛿𝐹
𝛿𝑎
𝐹
+
𝛿𝑚 =
𝑎
𝐹
𝑎
Hint: You can always check the units to help see if your error formula is correct. Here, they better
come out in kilograms. Putting numbers in, we get:
s
2 2
10.44 kg m/s2
0.02 N
0.01 m/s2
𝛿𝑚 =
+
= 0.3735...kg
10.44 N
0.53 m/s2
0.53 m/s2
Now that we have the error, we apply the rule to round the error to one significant digit:
𝛿𝑚 = 0.4 kg. Then, we know the answer should have the same number of decimal places as the
uncertainty. Our final answer is then
𝑚 = 19.7 ± 0.4 kg.
This is not the same number of significant digits that we would have gotten if we did not have
errors. Without errors, the final result has the same number of significant digits as the original value
with the smallest number of significant digits. Since the acceleration had only 2 significant digits,
the correct answer without error would have been 𝑚 = 20. kg.
Finally, if we were ignoring the sig-dig calculations and just reporting three significant digits, as
in the homework assignments, the answer would be 𝑚 = 19.7 kg.
Chapter 7
Graphical Analysis
Graphing is used as a method of aiding the analysis of data obtained in a laboratory experiment.
Almost all of your experimental work will involve one or more graphs. Therefore, it is important to
learn how to construct and use graphs and to learn how to interpret them.
A graph is a convenient method of presenting data. The analysis of data graphs can help your
understanding. By drawing a smooth curve through the points, one can obtain a fairly accurate
reading for any unmeasured points on the curve. The shape of the curve can give some idea about
the functional relationship between the plotted variables. When a certain relationship between two
variables is anticipated from theory, a graph will clearly show any regions where the experimental
data deviates. Finally graphs are an excellent way to communicate your results to readers, so we
follow specific conventions to make your data easy to interpret.
Even so, not every set of numbers requires a graph for plotting. Instead, tables can be good
summaries of small numbers of data. If you are interested in communicating the actual values of
data, you should use a table. If you want to illustrate the relationship between two or more sets of
data, then a graph is ideal. A good rule of thumb is that if one of your graph axes is not a physical
variable, the data should probably be presented as a table (I’m looking at you, bar graphs).
7.1
Graph Construction
Creating an effective plot is as much an art as it is a skill. Entire texts have been dedicated
to describing the best way to present quantitative information in graphical format and effectively
communicating with the viewer1. In Physics labs, we ask that you take some time to improve the
way you communicate your results. An example graph is given in Figure 7.1. There are a few basic
rules are required for any graph of data, which are important for each component of the graph.
1One of the great works in the field is The Visual Display of Quantitative Information by Edward R. Tufte.
35
7.1. GRAPH CONSTRUCTION
PHYS 130 – Page 36
Axes: All axes must be labelled with a title and units (if appropriate). In most cases, a legible
numeric scale is required.
Captions: All graphs require a caption explaining what the reader is looking at. The caption
typically begins with a phrase summarizing the figure (like a title). This starting phrase is then
followed by any additional information including where the data come from. Finally, a caption can
include a summary statement of what the reader should take away from the figure.
US Science and Space Budget (US, M$)
Legends: If figures contain multiple plotting symbols, these can be explained in a legend or in
the caption. Legends should not obscure any of the data visible in the plot.
30000
27500
25000
22500
20000
17500
6000 7000 8000 9000
Deaths by Strangulation (US, Annual)
Figure 7.1: Correlation between annual deaths by strangulation and US science
budget over the years 1999 to 2009. These data are selected from the Spurious
Correlations website of Tyler Vigen (tylervigen.com). While the correlation
appears to be tight, both axes are correlated with population growth so the
underlying trend is meaningless.
These are the rules that govern the creation of graphs. There are several more guidelines that are
used to create good graphs. In making your graphs, please bear these in mind:
• The axis ranges should chosen to highlight the full range of data being shown and should not
show large amounts of blank space. Axis ranges should be labelled clearly to indicate the
numerical values of the data.
• Minimize non-data ink. Graphs quickly become cluttered with lines that do not communicate
any information. In particular, grid lines should be used sparingly and only lightly drawn, if at
all, to avoid detracting from the presentation of the data.
• When you are plotting multiple sets of data on the same set of axes, you should not only colour
CHAPTER 7. GRAPHICAL ANALYSIS
PHYS 130 – Page 37
to distinguish your trends—change both the colour and the plotting symbol to highlight the
differences.
• Your data should always be plotted using plotting symbols (dots, squares, triangles, etc.). Any
lines connecting the data should be summary trend lines and NOT a point-to-point connection.
• Text should be legible and approximately the same size as your lab report (12 points). If
you are capturing a graph from another program such as LoggerPro, one trick to coax good
proportions out of those programs is to make your plotting window smaller so that the text and
symbols will expand to be a good size when inserted into your lab report.
7.1.1
Analysis of graphs
1. The straight-line (linear) graph
The straight-line graph is the easiest graph to analyze because the functional relationship between
the plotted variables is easily determined by comparing to the standard format of the equation:
𝑦 = 𝑚𝑥 + 𝑏
where
𝑦 represents a variable quantity plotted on the vertical axis,
𝑥 represents a variable quantity plotted on the horizontal axis,
𝑚 represents a constant quantity whose value equals the slope,
𝑏 represents another constant quantity whose value equals the intercept of the line with the 𝑦-axis
when 𝑥 = 0.
Example:
If you were asked to experimentally determine how the current 𝐼 through a resistor depends on voltage
𝑉, you would perform an experiment by varying the voltage and noting the corresponding current
readings. Suppose you were to obtain the data shown in Table 7.1. You would then plot a graph of
the dependent variable 𝐼 on the 𝑦-axis against the independent variable 𝑉 on the 𝑥-axis. The graph
obtained would look like Fig. 7.2.
This graph is linear and, therefore, introducing constants 𝑚 and 𝑏, we write the equation:
𝐼 = 𝑚𝑉 + 𝑏
Looking at the graph we see that 𝑏 is equal to zero. Therefore, the functional relationship between
the variables is 𝐼 ∝ 𝑉 and the equation reduces to
𝐼 = 𝑚𝑉
7.1. GRAPH CONSTRUCTION
PHYS 130 – Page 38
Table 7.1: Current and voltage sample data. The voltages were chosen to be very
5 V, and the currents were somewhat randomly chosen to give an approximate
slope of 0.1 A/V.
Voltage 𝑉
(𝑉)
5
10
15
20
25
Current 𝐼
(𝐴)
0.51
1.02
1.48
2.02
2.48
Figure 7.2: Current is plotted against voltage for the sample data from Table 7.1.
The solid line is a linear trendline for the sample data. The value given by the
slope is (0.099 ± 0.001 A/V and the initial current given by the y-intercept is
0.02 ± 0.02 A.
The next step is to use the LINEST function is Excel, which will be covered in Lab 1, to calculate the
Δ𝐼
and has the value of 0.101 A/V. Now,
slope and its uncertainty. In this case, the slope is defined as Δ𝑉
V/A is simply the unit ohm for resistance 𝑅. Knowing that the slope represents 1/𝑅, we compare
(1/slope) = 9.90 ohms to the known resistance of 10.0 ohms used in the experiment. Since the two
values of resistance agree within 1%, we can write the equation of our line as
𝐼 = (1/𝑅)𝑉
Here we have used a graph as a method for analyzing data to investigate the mathematical relationship
between voltage 𝑉 and current 𝐼.
CHAPTER 7. GRAPHICAL ANALYSIS
PHYS 130 – Page 39
2. Non-linear graphs
When two variables are plotted and the resulting graph is non-linear, it is difficult to determine
the functional relationship between the two variables from the shape of the curve. However, there
are several techniques, which can be used to turn a non-linear graph into a linear one, which is then
analyzed as in the previous section.
a) Change of variables method:
Suppose we are being asked to experimentally verify the following equation for the motion of a
freely-falling body:
𝑣 2 = 2𝑔ℎ + 𝑣 20
(7.1)
Since 𝑣 is a function of ℎ, we collect data for 𝑣 for varying values of ℎ. Now a plot of 𝑣 against
ℎ would yield a non-linear graph (Figure 2), which is not very useful in verifying the relationship.
However, if we were to plot a graph of 𝑣 2 against ℎ, we would obtain a linear graph with slope equal
Figure 7.3: Velocity vs Height for a falling body.
to 2𝑔 and with a vertical intercept equal to 𝑣 20 . We have employed a change of varyable where the
variable 𝑣 2 represents the 𝑦-axis. Since 2𝑔 and 𝑣 20 are simply some constant values, we have
𝑣 2 = (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) × ℎ + (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
which agrees with the format prescribed by
𝑦 = 𝑚𝑥 + 𝑏
This linear graph would then be experimental verification of Equation 7.1 and could be used to
determine 𝑔 and the initial velocity 𝑣 0 of the falling body.
A second application of the method of change of variables is shown in the following example.
Suppose the equation 𝐹 = 𝑚𝑎 is to be tested, where 𝐹 is minted constant for the duration of the
7.1. GRAPH CONSTRUCTION
PHYS 130 – Page 40
Figure 7.4: Velocity Squared vs Height for a falling body.
experiment. The relation between 𝑎 and 𝑚 of 𝑎 = 𝐹 (1/𝑚) is linear with a slope of 𝐹.
b) Log-Log method for power relations:
An equation of the following form:
𝑦 = 𝑚𝑥 𝐴 + 𝑏
(7.2)
is called a power relation. This type of relation occurs frequently in physics, and graphically, yields
a curve when 𝑦 is plotted against 𝑥. However, it is difficult to determine the power 𝐴, simply by
looking at the shape of the curve. A simple technique, called the log-log method, solves the problem
by showing whether a power relation actually holds, and also gives the numerical value of 𝐴. As the
name implies, the method involves taking logs of both sides of Equation 7.2 as follows:
log(𝑦 − 𝑏) = 𝐴 log 𝑥 + log 𝑚
(7.3)
where the quantity 𝑏 must first be subtracted from 𝑦. The value of 𝑏 is obtained from the “𝑦 vs.
𝑥” graph by noting where the curve intercepts the 𝑦-axis for 𝑥 = 0. By comparing equation 7.3 to
𝑦 = 𝑚𝑥 + 𝑏, we see that a graph of log(𝑦 − 𝑏) on the vertical axis versus log𝑥 on the horizontal axis
yields a straight line of slope 𝐴. Once 𝐴 is known, we can immediately plot a graph of 𝑦 vs 𝑥 𝐴 which
yields a straight line whose slope is the value of the constant 𝑚.
If the log-log graph is not linear, obviously the relation between 𝑦 and 𝑥 is not a power relation, and
we must try different methods to linearize the curve.
c) Semilog method for exponential relations:
A very common function is one where the change in the function (population increase, radioactive
decay, etc.) is proportional to the function itself. This is an exponential function given by the general
expression of the form:
𝑦 = 𝐴𝑒 𝑘𝑥
(7.4)
CHAPTER 7. GRAPHICAL ANALYSIS
PHYS 130 – Page 41
Figure 7.5: Power relation.
Taking the logarithms of both sides of Equation 7.4 gives
ln 𝑦 = 𝑘𝑥 + ln 𝐴
(7.5)
Figure 7.6: Exponential relation. Figure 7.7: Semi-log graph for exponential relation.
Note that, unlike the log-log method, you would plot ln 𝑦 versus 𝑥.
3. Error in slope and intercept
Linear regression, also called “Least Squares”, is a mathematical method for calculating the slope
and intercept of straight lines as well as their corresponding error estimates from the data points.
This is done via a built-in function in Excel called LINEST function. You will learn how to use this
function in Lab 1. The justification of this method requires the use of statistics and is beyond the
scope of this manual. The method is only practical when a computer can be used to perform the
tedious calculations required.
The “LINEST” function does not determine if a relationship is is linear. Regression calculations
will blindly generate slopes and intercepts for any set of data with at least 2 distinct x values and
generate error estimates for any set of data with 3 point regardless of whether or not the underlying
relationship is linear. It is essential that you visually check out the fit between the data points and
the linear regression line to look for systematic departures from a linear relationship. If you find any
clear blunders in the data, remove them from the data table. Otherwise, the statistical calculations of
slope and intercept are meaningless.
7.2. USING LINEST
PHYS 130 – Page 42
7.2
Using LINEST
7.2.1
Microsoft Excel
To use this function, hit the little 𝑓𝑥 button in the formula bar. Select Statistical, and go to LINEST.
Enter your 𝑥 and 𝑦 data, and under Constant and Stats, enter 1 (logical TRUE). This will just give
you the slope in the cell you chose for LINEST. To get the rest of the stats, select a 5 × 2 array of cells
with the first one being the one you entered the LINEST formula in (now containing the slope), and
press F2, and then press CONTROL + SHIFT + ENTER. On a Mac, press CONTROL + U, and then
press CONTROL + SHIFT + RETURN instead. Google sheets will do this without having to press all
the buttons; just highlight the 5 × 2 area and type in the LINEST command. The 5 × 2 array will fill
with numbers whose meaning is shown in Table 1.
Table 7.2: Meaning of the numbers in the 5×2 array of the LINEST function.
slope
error in slope
correlation coefficient
F-test overall
Regression sum of squares
intercept
error in intercept
error of the regression
degrees of freedom
Residual sum of squares
You are only worried about the slope, intercept, and their errors. The size of the errors will set
the number of significant figures to use when you report the slope and intercept (see introductory
material).
7.2.2
Google Sheets
For Google Sheets, the linest command is used entering a cell and typing =LINEST(YRANGE,
XRANGE,TRUE,TRUE). Here, the YRANGE is a spreadsheet range of values representing the 𝑦-data in
a graph, specified using a colon (:), e.g., A2:A7 would represent the data in column A from rows 2 to
7. Similarly XRANGE is the spreadsheet range representing the 𝑥-values in the graph. The first TRUE
value indicates that we want LINEST to fit the 𝑦-intercept of the line and the second TRUE shows that
we want the program to complete a full statistical analysis and report the values in the same pattern
as Table 7.2.
7.3
Computers and spreadsheets
At this stage of your career, you must be able to use a computer and know how to use a word
processor and a spreadsheet. Your lab report will of course be a professionally presented document;
CHAPTER 7. GRAPHICAL ANALYSIS
PHYS 130 – Page 43
therefore, you need to use a word processor to type it up and a spreadsheet to make tables and
graphs. It does not matter which spreadsheet or which word processor you use, but most people
are experienced in Microsoft Office. However, many of the experiments involve electronic data
acquisition using sensors which gather data into a special software called “LoggerPro”, which also
has many useful graphing and data analysis functions. We will be using this software extensively,
therefore, it is installed on all lab computers. At the beginning of the year, we will give you an
installation code to install LoggerPro on your laptop and we encourage you to use it for data analysis.
While it is not an object of the physics lab to teach you a basic level of computer literacy, there are
many online tutorials out the web.
We strongly encourage all students to write their lab report on their own laptops, which may be
brought to every lab and tutorial session. While we have not made laptops mandatory yet, this is a
strong possibility for the future. We do have a few computers in the labs, which students can use as
well. However, you will not be able to take them home, so it is better to use your own machine.
A digital camera or a USB stick may be needed for some the labs. You’ll often be asked to
take a picture of your apparatus or some part of your experiment, or copy some electronic data from
somewhere. Or, you may want to voluntarily include photographs in your lab report to help illustrate
some point you’re trying to make. All students should therefore have access to a digital camera. Any
camera will do, even the ones on your cell-phone.