# 7. Writing a Lab Report¶

The lab report is the classroom version of an experimental research article. It isn't enought to perform a good expereiment and to carry out the data analysis carefully. You also need to communicate your results to other scientists in a way that convinces them that your work is interesting and should be taken seriously. In a lab report, you need to practice writing in a way that convinces your readers that you did good work and reached a correct conclusion.

Most experimental physics articles follow the following fairly standard format:

ABSTRACT
I. INTRODUCTION
A. Motivation
B. Summary of the experiment
II. THEORY
III. PROCEDURE
A. Description of the apparatus
B. Description of the experimental procedure
IV. ANALYSIS
A. Method of analysis
B. Presentation of results
C. Discussion of results
V. CONCLUSIONS
A. Summary of the results
B. Pertinence of the results to the questions raised in the introduction

What did you do? (Procedure)
Why did you do it? (Introduction, Theory)
How did you do it? (Procedure, Analysis)
What happened? (Analysis, Conclusions)

The format also provides some shortcuts for busy readers. Most scientific prose tends to be fairly dense, so readers like to find out in a hurry if a paper is actually of interest or importance to them. The abstract section provides a concise summary of the article and its most important results, which allows the reader to determine after a few sentences if the entire article is relevant. The introduction and conclusions briefly give a little more information.

Each section of a lab report is discussed in a separate section of this chapter. You will probably find it helpful to read over the entire chapter the first time you are asked to write a section of a lab report to get some sense of how the pieces together. When you write a full lab report, you should go back and read the entire chapter again.

## 7.1 Abstract¶

An abstract is a very short summary of the entire paper, which is about three to six sentences long. In a journal, the abstract appears just below the article’s title and list of authors. The purpose of an abstract is to give readers with a brief glimpse into the subject of the article, which helps them decide whether or not to read the whole thing. Researchers often begin a project by searching for recent articles that might be helpful. Good abstracts make it possible to determine quickly which articles are relevant. The abstract should make sense on its own without the rest of the paper.

The structure of the abstract is essentially a miniature version of the structure of the article, except that each of the five major sections (introduction, theory, experimental design, analysis, and conclusions) might be represented in the abstract by only a sentence or even a phrase. Often the theory section is omitted completely from the abstract unless the paper is theoretical, which will not be the case for your lab reports. In fact, you should avoid using equations in the abstract. The outline for the whole article is a pretty good starting point for the outline of the abstract as well. The abstract should always summarize the introduction and conclusion sections, which means that it will always include a short summary of what question you were seeking to answer, what your results were and what they imply. In particular, an experimental physics abstract should include a summary of any quantitative results you report in your conclusions. Remember, the abstract is the "hook" you use to get people to read the rest of the paper, and you can best capture their attention with a nice quantitative result with a promisingly small experimental uncertainty. Although the abstract is the first section of a lab report, you may want to write it last because it is a summary.

Checklist for the abstract:
◻ Summarize the question you were seeking to answer
◻ Very briefly describe the method used to answer the question
◻ Present your quantitative results with uncertainties
◻ Mention what your results imply
◻ 3-6 sentences long

## 7.2 Introduction¶

The introduction section is meant to provide the reader with the answers to two very important questions:

What is the question that this experiment is intended to answer?
Why is answering this question interesting (and/or important)?

In a published journal article, this section often begins with a brief summary of previous related research, a statement of a problem that this research has raised, and a brief description of the experiment in question and how it addresses the problem. Detailed descriptions are not appropriate in this section. This section should capture the interest of your readers, provide them with some general orientation, and convince them that what you are doing is interesting and worth reading about.

After giving the motivation for the experiment, you should give a brief summary of the experimental method you will use. This need not be extensive since the detailed description goes in the procedure section.

Checklist for the introduction:
◻ Explain why answering this question interesting and/or important
◻ Briefly summarize the experimental method used

## 7.3 Theory¶

The theory section is meant to provide the reader with the background to understand how the experiment works, what assumptions have been made, and how the experiment is related to the physics being studied. This section may be very short if the theory is well-understood and the connections between the theory and the measurements are straightforward and obvious. It can be quite extensive, however, if the experiment is complex or the measurements being made are related in a complicated way to the results being compared to the theory.

If, for example, you were measuring the average velocity over some interval for your experiment, your theory section would be very short: you measure a distance and a time, divide the first by the second, and there’s your average velocity. Suppose, on the other hand, that your experiment was the determination of an acceleration in a situation where you couldn’t be sure the object was starting from rest. It is still possible to find the acceleration, but you have to measure two time intervals over two distances, and the connection between those measurements and the final result involves a fair amount of algebra. In that case, you would be expected to derive the connection for your theory section, which you could expect to be one or two pages long. You don't need to show each algebraic step, but you should show some intermediate results, especially if they involve complicated algebra, a substitution, or some trick of manipulation.

The amount of theoretical background that you provide also depends on the expertise of your intended audience. For the purposes of this course, you should imagine your reader to be a typical classmate (not a professor), who for some reason has not done the lab in question and knows nothing about it. This is similar to a researcher whose audience has quite a bit of general knowledge about physical principles and experimental techniques, but no experience with the specific experiment.

For General Physics III (PHYS 233), at least, you should start your theory section with first principles or the equation that defines the phenomenon you'll be studying. In a journal article, you wouldn't go this far back, because starting from first principles to get to the result would take up too much space. However, this will be a good idea in the introductory lab when you're learning how to write a theory section. Since you're doing experiments that are usually close to the basic principles, starting with those principles helps you to examine your assumptions carefully.

Mathematics is used as a language in physics to express ideas concisely and precisely, so theory sections tend to involve equations. The rules for including equations in your writing follow from the idea that equations should be treated as words and phrases in your writing, even though they are not English ones. The following is a good example of some mathematics properly integrated into text:

Since the aluminum block does not undergo a phase change and its temperature change, $\Delta T$, is relatively small, we can approximate the change in the block’s internal energy by $$\Delta U = m c_{\rm Al} \Delta T, \tag{1}$$ where $m$ is its mass and $c_{\rm Al}$ is the specific heat for aluminum around room temperature.

The rules that the example above follows are:

(1) Integrate mathematics into your text. Your writing should flow smoothly, even in sections that include mathematics. Therefore, each equation should be part of a sentence, rather than just being dropped in between sentences.

(2) Punctuate and capitalize properly. If an equation is integrated in a sentence, then the proper punctuation should surround it. In case you’re not certain about the punctuation required, try reading the sentence with the equation replaced by a noun. In the example, it should be clear that you don’t need any punctuation before the equation because no punctuation would be used if the word "dogs" replaced the equation. A comma is needed after the equation to seperate it from the phrase, "where $m$ is ...". Note that the word “where” just after the equation is not capitalized because it’s a part of the same sentence.

(3) Italicize mathematical symbols. In general, whenever you switch from writing in English to writing in another language, you’re supposed to italicize the other language to cue the reader that it's not English. For example, you might write “The attempted coup d’état was unsuccessful.” Since coup d’état is French, you italicize it. In the example above, the same rule holds for the language of mathematics. Whenever you use a mathematical symbol, whether it’s alone or in an equation, you should italicize.

(4) Define mathematical symbols. Unless a foreign word is commonly understood by English speakers (such as coup d’état), you also need to provide your readers with an English translation the first time that you use it. This rule also holds for mathematics, so the symbols $\Delta T$, $\Delta U$, $m$, and $c_{\rm Al}$ are defined in English. If a symbol is reused after it has been defined, there is no need to define it a second time. You wouldn't insult your readers by defining a foreign word each time you use it.

(5) Separate and center equations. Although an equation should be part of a sentence, that doesn’t mean that it should be on the same line as the English words. It takes extra focus to read equations, so they should be placed on a separate lines and centered. You should also leave sufficient blank space above and below them to make them easy to read. Note that the line below the equation is not indented because it isn't the start of a new paragraph.

(6) Number each equation. You will often refer to an equation again after it is introduced. Even if you don't, a reader may want to refer to an equation when communicating with someone else about your work. That’s why you should number each of your equations, not just the ones that you reference later. To make it particularly easy for your reader to find the equations, you should place the number on the far right of the page.

Checklist for theory section:
◻ Show all algebraic steps that aren’t obvious
◻ Describe any assumptions or approximations made
◻ Integrate equations into sentences with appropriate punctuation
◻ Italicize all mathematical symbols
◻ Define all mathematical symbols
◻ Center equations on separate lines with equation numbers to the right

## 7.4 Procedure¶

In the procedure section, you should convince the reader that you carried out an experiment carefully enough that your experimental results should be taken seriously. You should think of the reader as someone who is unfamiliar with the particular experiment you are doing. In addition, you should assume that the reader is somewhat skeptical about your results, so picky about your procedure. This may not sound very friendly, but professional scientists act just this way reading other authors’ papers, especially if they've done similar experiments.

Suppose that you measured the period of a pendulum in a lab. Simply writing, "We measured the pendulum period as a function of mass hanging from the end" isn't a sufficient description of your procedure. Performing the experiment carefully to reduce the uncertainty in your results requires multiple measurements, timing several periods for each measurement, and choosing a particular starting and stopping point in the swing. You should give enough detail so that the reader could reproduce what you did. You should also explain why you went to all that trouble because that enhances your credibility with the reader by providing evidence that you thought carefully about the experiment.

Typically, the procedure starts with a description of the equipment and how it was set up. You should not make a list of the equipment used in your report. Instead, the equipment should be described in text and diagrams. Well-labled diagrams are very helpful because they reduce the number of words needed. However, all diagrams should be refered to in the text (by their figure numbers). Note that a photograph of the equipment is not a good substitute for a diagram because photos don't emphasize the important details. Each diagram should have a figure number and caption. Also, you should refer to each diagram by its figure number in the text.

You also need to explain what you did with the equipment. You should do this in a logical order, but not be too "step-by-step" about it. Specifically, avoid a numbered list of steps. At the other extreme, you should avoid narratives such as, "First we did (whatever), but that didn't work, so then we tried (something else) to fix the problem with the first measurements." Refine your procedure to remove these false steps, and present it in enough detail so that the reader can clearly understand what you did without being overwhelmed by irrelevant details. One detail you should definitely include at this stage is the number of times you repeated any given measurement. Students often forget the importance of repeated measurements. Repeating repeatable measurements is the typical way to determine the uncertainty of the measurement. Finding the appropriate level of detail is difficult. You don’t need to tell the reader everything, but you do have to say enough. For example, you don't have to explain how to use standard equipment like a stopwatch or a vernier caliper. The ideal procedure section is one that provides just enough detail so that the reader could repeat the experiment and get results consistent with yours.

Articles in professional journals sometimes refer to other papers for details regarding equipment or procedure. In order to practice writing procedures, your lab report should be self contained, rather than refering to other sources, such as the lab handout. You should also be careful to explain what you did, rather that giving instructions, which is what lab handouts tend to do.

The procedure should be written in the past tense since it is descibing work that you've already done. Historically, scientific writing used the passive voice (“The ball was hit by the batter.”), instead of the active voice (“The batter hit the ball.”). In the natural sciences, we hope that the identity of the experimenter does not affect the result of the experiment, and writing in the passive voice deemphasizes the role of the individual experimenter. Unfortunately, the passive voice can be boring to read, partly because it is wordier and partly because it dilutes the sense of action. The place where the historical convention really requires the passive voice is the procedure section. In other parts of the report, "the spring exerts a force," or "a results suggest an inverse-square law, so you're out of the picture anyway. However, the procedure section is the place where you describe what you did, except that your identity isn’t supposed to be important. Times are changing, and it is becoming more acceptable to write the procedure in the active voice. For example, authors of a scientific paper are writing things like, “We observed NGC 253 on seven consecutive nights looking for supernovae.” You may use the active voice in the procedure section.

Checklist for the procedure section:
◻ Describe the equipment in sentences, not a list
◻ Include a well-labeled diagram (with figure number and caption) of the setup
◻ Refer to each diagram by its figure number in the text
◻ Describe all measurements, roughly in order
◻ Describe how uncertainties were determined
◻ Do not write the procdure as instructions

## 7.5 Analysis¶

There are three main goals to accomplish in the analysis section. First, you should present the data in an origanized way. More often than not, this will involvle making graphs. Second, you need to explain how you went from a bunch of measurements to few numbers (maybe only one) that characterize those measurements. You should give enough details that the reader can decide if you used the appropriate methods and carried them out correctly. You should present your case for the implications of your numerical results. Finally, you should draw some conclusions about your results.

If there isn't too much data, it can be presented in a table. (A table is often a good way to present a smaller set of results extracted from the raw data.) However, it is usually easier to see trends in a graph, rather than a table. Often, you will not simply make be a graph of your unprocessed data. You will more likely plot means of sets of data with appropriate uncertainty bars. Simply including a graph isn't enough. In the text, you must also reference the graph and explain it's significance. For example, you might write, “The dependence of distance the ball has fallen versus time is shown in figure 2.” Notice that the graph is identified both by the data being displayed and its figure number. Identifying the graph by the data tells the reader why this graph is part of your logical argument about the meaning of your data and results. Identifying the graph by the figure number makes it easy to find. Review Chapter 4 for details about making good graphs. You should include error bars on your graphs, unless they’re too small to be visible. If this is the case, say so explicitly so that your reader does not assume that you have simply forgotten about them.

The details of your analysis from here depend on exactly what question you are trying to answer with your data. You should have worked out an expected relationship between the variables that you are measuring in the theory section. Refer to equations in the theory seciton by number, rather than repeating them. For example, you might write, “From equation 3, we expect a graph of $d$ vs. $t^2$ to be linear with a slope of $g/2$.” If the expected relationship is linear or can be linearized (see Ch. 6), you can check that the data that you have graphed are consistent with that prediction. The result you are after in an experiment is often related to the slope and/or intercept of a linear graph. In that case, you would use linear regression to find the slope, the intercept, and their uncertainties (see Chapter 5). If you did some calculations to extract the value you want from the slope or the intercept of a graph, give enough details of the calculations so that the reader could duplicate your work.

An essential part of any analysis is the propagation of uncertainty to find the uncertainty of your results (see Ch. 3). Careful treatment of uncertainty is essential if you are to draw meaningful conclusions from your data. If you have to estimate the uncertainty of any measured quantities, describe how you did your estimate, unless you already did this in your procedure section. If you computed the uncertainty of a value, describe how you did that calculation and show an example calculation. Report uncertainties with units and in the same form and to the same precision as your results. For example, use 3.981 ± 0.068 N, not 3.981 ± 0.06832 N. If you are using scientific notation, report both the result and the uncertainty with the same exponent. For example, use (1.10 ± 0.06) × 10−6 m, rather than 1.10 × 10−6 ± 6.2 × 10−8 m. Comparing the size of the uncertainty relateive to the result is much simpler with the preferred format.

Checklist for the analysis section:
◻ Present the data in tabular and/or graphical form
◻ Include linearized graphs of data, if appropriate
◻ Refer to each graph in the text by figure number
◻ Refer to equations from the theory section by their equation numbers
◻ Explain how you calculated the slope and intercept of any linear graphs
◻ Show the calculation of any quantities derived from slope or intercept
◻ Show calculations of uncertainties
◻ Discuss consistency with theoretical predictions or accepted values
◻ Discuss the results and their implications

## 7.6 Conclusions¶

A conclusions section should review the purpose of the lab and summarize the implications of your experimental results in one or two paragraphs. You should remind the reader of the basic question that the experiment was to address (as presented in the introduction), and then briefly explain how your results bear on that question. This section should be a summary of information presented elsewhere rather than a place to present new information. The purpose of this section is to close the report with a review that highlights the most important results. As with the abstract, you should report quantitative results and their experimental uncertainties.

You might wonder what's the difference between the conclusions and the abstract. Both are summaries which both contain quantitative results. The main differences have to do with location. The abstract is the hook at the beginning to get the reader interested, and should contain hints of what's to come. It also summarizes the entire report. The conclusion comes at the end, and should give some sense of finality or closure. It will emphasize your deductions from your data analysis, describing them in more detail than is given in the abstract. Both the abstract and the conclusions should report comparisons between predictions (made in the theory section) and your measurements or their consequences.

Checklist for conclusions:
◻ Review the basic question addressed
◻ Report quantitative results with uncertainties
◻ Summarize the implications of your results