Scientific Measurement :2016SPA4103

A. J. MisquittaSchool of Physics and AstronomyQueen Mary University of London

SCHOOL OF PHYSICS AND ASTRONOMY, QUEEN MARY UNIVERSITY OF LONDON

PH.QMUL.AC.UK

Copyright 2016 Alston J. Misquitta

First release, August 2016

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 In-lab assessments 92.1.1 In-lab assessment sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 First formal report 12

2.3 Second formal report 12

2.4 Lab-book assessment 12

3 Lab Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Why maintain a lab book? 13

3.2 Does it have to be neat and tidy? 14

3.3 Do I get any marks for the book? 14

3.4 What if I lose it? 14

3.5 What about the analysis? 14

3.6 What about the error-analysis? 14

3.7 Do I need to write a Conclusions section? 14

3.8 Do I stick graphs in? 15

3.9 What makes a good laboratory book? 15

4

4 Uncertainties & Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Instrument or Measurement uncertainty 17

4.2 Example of how to record data with uncertainties 184.2.1 Significant figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Statistical uncertainties 20

5 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1 Check-list for graphs 27

5.2 Data extracted from graphs 28

5.3 Examples 29

6 Error-propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.1 Outline of error-propagation 33

6.2 Proof of the error propagation formula 35

7 The Formal Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8 Preparation for the lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.1 Work Plan for the lab 40

9 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1. Introduction

Science is the interplay of experiment and theory: we understand the physical world throughtheoretical constructs which often follow experimental breakthroughs. But theoretical ideascan also anticipate experimental verification; indeed, some of the cutting-edge ideas inphysics seem to be guided by theories (consider the Higgs boson). Nevertheless, these twoaspects of scientific discovery are not equal as theoretical ideas remain just that — ideas —unless verified experimentally. Here’s a quote you may have head about:

It doesn’t matter how beautiful your theory is, it doesn’t matter how smart youare. If it doesn’t agree with experiment, it’s wrong. (Richard P. Feynman)

This coming from Feynman may seem rather harsh on some of the most stimulating ideasfrom theoretical physics (string theory is one of these) which still have no experimentalbasis. But here is something from another great theorist:

The measure of greatness in a scientific idea is the extent to which it stimulatesthought and opens up new lines of research. (P. A. M. Dirac)

So we should not always worry about developing ideas that are beautiful but which maynot (yet) have experimental verification. But what if our ideas seem to be disproved byexperiment? Well, here is Dirac again:

If there is no complete agreement between the results of one’s work and theexperiment, one should not allow oneself to be too discouraged.

(P. A. M. Dirac)

Sometimes experiments are wrong. We may have measured something different from whatwe had intended. Or the errors made were just too large. The latter is often the more likely.

6 Chapter 1. Introduction

Anyone remember the faster-than-light neutrino anomaly from the OPERA experiment?Have a look at the Wikipedia article on this anomaly 1 as it makes entertaining reading.

This brings us to one of the reasons for this laboratory:

To understand sources of experimental error, quantify them, and present and analyse theexperimental results within the context of these errors.

You will see more on this in later chapters, but this simply means that unlike your schoollaboratory days when you may have recorded measurements simply as

l = 64.6 cm, (1.1)

here you will be expected to estimate the uncertainty on the measurement and report itaccordingly, perhaps as

l = 64.6±0.5 cm. (1.2)

This uncertainty will then be propagated to other quantities. So in a measurement of g usinga pendulum (something that you will do), you will use the measurement of length (above)and period of oscillation, say T = 1.6±0.2s, to get:

g = 9.7±2.4m/s2. (1.3)

What we have done here is to propagate the uncertainties from l and T to the accelerationdue to gravity g. The final result is much more meaningful than the high-school equivalent:

g = 9.7m/s2. (1.4)

This seemingly suggests that we have done a very good job measuring g as it is quite closeto the expected value of 9.81m/s2, but when we also report the uncertainty, we clearly seethat our measurement is quite imprecise as it could be anywhere between 7.3m/s2 and12.1m/s2.

The second goal of this laboratory is to teach you experimental technique. There are atleast two parts to this: your ability to handle the laboratory equipment, and your ability torecord your measurements and observations logically and systematically in a lab book. Youwill be given a lab book and will be expected to maintain it systematically and neatly. Youwill be assessed based on both of these criteria. This in-lab assessment will be performedby the demonstrators as well as the academics in charge of the lab session.

The third goal of this laboratory is to teach you how to write reports. In a sense, keepinga lab book is a bit like report writing, but it takes some skill to write a coherent formalreport. During the course of this laboratory you will write two short formal reports.

Finally, over the last few years we have been upgrading the experiments to make themmore interesting, more open-ended, and we have been introducing some level of automationusing the Arduino microcontroller. You will first see this microcontroller when you perform

1https://en.wikipedia.org/wiki/Faster-than-light_neutrino_anomaly

7

the pendulum experiment, and some of you will come across it in subsequent experiments.The Arduino is a great platform for your own projects as you can build a lot with it. Weeven have some real experimental kit made using the Arduino. And some of you will learnC++ programming using our BOE BOT robot.

In all, we hope you will enjoy the laboratory and make the most of it.

2. Assessments

Your final mark will be made up of a mixture of in-lab assessments, reports, and marks forthe lab book. Here is how the marks will be allocated:

• 10 marks: In-lab assessment for the first six experiments.• 15 marks: First formal report based on the Pendulum experiment.• 10 marks: Lab-book assessment based on the Oscilloscope experiment.• 20 marks: In-lab assessment based on the long-experiment.• 20 marks: Lab-book assessment based on the long-experiment.• 25 marks: Second formal report based on the Coffee-Cooling experiment.

Notice that the marks are back-loaded: that is, the earlier experiments are not weightedas much as the later ones. This has been done so that you get a chance to learn fromthe first few experiments, improve your techniques, and then, do a better job on the laterexperiments.

2.1 In-lab assessmentsFor most of the semester you will have a demonstrator assigned to you. This demonstratorwill be the one you will need to go to for help (if he or she cannot help you, then they willtry to get someone else to help). Additionally, this demonstrator will question you aboutone or more aspects of the laboratory experiment. The questions will vary from week toweek. The topics covered will include your understanding of the experiment, the way youhave recorded data, maintained your lab-book, recorded and propagated errors, formulatedyour conclusions, and performed the experiment.

This will be a continuous assessment: each category will involve three levels: the firstlevel will be the most basic, the second will be intermediate, and the third an advanced

10 Chapter 2. Assessments

ability. You can progress up these levels. If you are first assessed on your ability to plotgraphs and have achieved only a basic level mark, then you will be told what it is you havedone wrong (or you can use this manual to figure it out). At the next assessment on graphs(this could happen at any time), you will be re-assessed: if you have done better, then youmay progress to the intermediate, or even the advanced level. So if you find that you havenot achieved the best level, take the time to find out what it is you need to do to progress.

For the long experiment — this will span six afternoons — you will be assessed byan academic. The method of the assessment will follow the earlier one. You may expectthe questions to be more difficult at this stage. And we will be looking at your laboratorytechnique in some detail.

A sample in-lab assessment sheet is present on the next page. Each aspect of the lab isassessed in three categories: (A) is a basic category and carries a low nominal mark, (B) isan intermediate category, and (C) is the category you are expected to achieve by the end ofthis module. Sample criteria are indicated for each. These may be altered.

You may use this sheet to keep a record of your assessment level. If you feel you haveprogressed, bring it to the attention of your demonstrator and request a re-assessment. Ifyou take the feedback you receive to heart and work on it, you should be able to achieve thehighest score in each section.

2.1 In-lab assessments 11

2.1.1 In-lab assessment sheet

Student Name:ID no.:Assessor:

Contact tally :1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 20Part: FIRST / SECOND

PreambleA: Date and title of experiment.B: Work-Sheet adequate & pasted in book.C: Can explain what the experiment is about.

DataA: Recorded data in the lab book.B: Recorded with units and uncertainties.C: Caption or text to provide some context to the data (in tables).

GraphsA: Basic graphs present.B: Legend (may be hand-drawn), axis labels, units.C: Error bars present, appropriate fits, some context to the graph (couldbe a caption).

Error-AnalysisA: Some error analysis attempted (need not be correct).B: Basic calculations of standard errors.C: error propagation, extraction of errors from graphs, significant figures.

AnalysisA: Basic attempt at the analysis of the results.B: Important results indicated clearly. Attempt at interpreting results.C: Results reported with uncertainties and units. 3-sigma test performed.

Conclusions/ReflectionA: Basic summary of main results of experiment.B: Good overall presentation of the main results with explana-tions/discussion.C: Reflection on the experiment. Sources of error. Aspects that could beimproved

Nominal marks: A=5, B=10, C=15.

12 Chapter 2. Assessments

2.2 First formal reportThis will be a four page report written based on the pendulum experiment. It will besubmitted online via our QMplus page 1. The deadline for this report will be given to youon that page and in class.

The report must be limited to four pages of 12pt font. Any additional pages will not bemarked. However, you may be asked to supply your raw data for verification. So pleasemake sure you have this data recorded in your lab book. The report must be submitted as aPDF file.

Late submissions will incurr a penalty. Details of this will be posted on our QMpluspage.

Once the report is marked, you will recieve your mark and feedback. Please make noteof this feedback. If there are issues that are not clear to you, ask either the demonstrators, orthe academics about them.

2.3 Second formal reportThis report will be based on the Coffee-Cooling experiment. It will also be a four pagereport with other details the same as with the first report.

2.4 Lab-book assessmentWe will collect your lab books in the week after your long experiment. These books will beassessed based on the long experiment so you will have to complete the entire experiment,analyse the data, formulate your conclusions, and present it all in a reasonably neat mannerin the lab book.

Note that you may need to make a copy of your Coffee-Cooling data before submittingthe lab-book to us as you will need to use it to write up the second report.

Assessment criteria: The lab book entries are not meant to be formal reports: they aremeant to be faithful records of what transpired in the laboratory, with an appropriate analysisof the experiment which would include an analysis of the errors and uncertainties, and areflective conclusion of the experiment.

In the next few chapters we will describe what is meant by this. If you feel that you areunsure of what goes in your lab book, ask your demonstrator for guidance.

1http://qmplus.qmul.ac.uk/course/view.php?id=3067

3. Lab Book

• Why do I need to maintain a lab book?• Does it have to be neat and tidy?• Do I get any marks for the book?• What if I lose it?• What makes a good laboratory book?• What should I record in the lab book?• What about the error-analysis? Does it go here?• Do I need to write a Conclusions section?• Do I stick graphs in? Or can I leave printouts in the book without glueing them down?The goal of this chapter is to shed some light on these and other questions you may have

regarding a lab book.

3.1 Why maintain a lab book?

The short answer is that a good lab book serves as a memory aid: you perform a complextask, one step at a time, and record what you did and your observations at each stage. Thenyou can look it all over and make far reaching conclusions. That is why we all keep somekind of lab book when doing research. But you also need it for reasons of copyright: yourlab book with its details and dates can serve as evidence for discoveries and be useful inpatent disputes. Can you imagine conducting an extended research project — such as youwill conduct in your third and fourth years — without keeping a daily record of what youhave achieved? How would you write it up? How would you make sense of all you haveachieved in the year? It is for these reasons that we place an emphasis on the lab book.Think of it as your research journal. But it is not a random collection of notes, but somethingfar more structured. More on this next.

14 Chapter 3. Lab Book

3.2 Does it have to be neat and tidy?It must be legible and systematic. You will make mistakes. If so, indicate that you havemade one, perhaps with a red pen. But everything you note down is your record. Do nottear out pages or black entries out.

If you feel that your entries are not really readable, then spend time re-writing thingsat home. But in the lab, focus on the experiment and on recording a faithful and completeaccount of your experiment.

3.3 Do I get any marks for the book?Yes. Part of the mark for your long experiment will be based on the contents of your labbook. We will look to see that you have entered your data systematically, performed ananalysis, done the required error propagation, graphs, and formulated your conclusions.

More indirectly, as you will rely on the contents of your lab book to write up the formalreport for the coffee-cooling experiment, part of the marks for that report will be reliant onhow you recorded the information in the lab book.

3.4 What if I lose it?In short: don’t lose it. If you do, and lose the data and analysis for your coffee-cooling andlong experiment, then you will indeed lose a lot of marks. We will not accept any excuses.

3.5 What about the analysis?Once you have recorded your data, if you have time, proceed to the analysis. Hopefully youwill do this as the analysis — even if incomplete — is often needed to give you a chance toassess your data. Maybe you made a mistake. If so, the analysis will tell you so, and if youhave time, you may be able to repeat a part of the experiment. So try to ensure that you useup your time in the lab to perform the analysis. Don’t leave it to the end, or you may be infor an unpleasant surprise.

Record all your analysis in the lab book. Part of your in-lab assessment will be based onthis analysis.

3.6 What about the error-analysis?Record all the error-analysis in the lab book. You may not have time to do this duringthe lab session. That is OK. You will may leave space in the lab book and complete theerror-analysis at home. As with the analysis, part of your in-lab assessment will be basedon the error-analysis.

3.7 Do I need to write a Conclusions section?Yes. This will be needed for every experiment.

3.8 Do I stick graphs in? 15

3.8 Do I stick graphs in?Are lose pages OK to have in a lab book? Surely not. They can fall out. All printouts, extrapages need to be stuck or stapled into the lab book.

3.9 What makes a good laboratory book?The lab book should contain a faithful record of what you did in the laboratory, and alsoany additional analysis you conducted out of the laboratory. You should be able to read anentry in the book and develop a fairly complete idea of what was done, how it was done, theresults, analysis conducted, and the conclusions made.

Broadly, an entry for a particular experiment will contain the following information:• A title and date.• A brief aim of the experiment.• A description of each step along with any important experimental details. This would

include equipment, the setup used, circuit diagrams. Placing all this into the lab bookcan take time so you may want to either refer to figures in the lab script, or simplypaste a printout of the figures in the lab book. It is important that you do not spendtime on this during the laboratory session!• The data recorded neatly. Record as much information as you see fit. Use a simple

rule of thumb: You should be able to perform a complete analysis of the experimentbased on the recorded data.• The analysis of the data.• An analysis of the errors. This may be done after the experiment is complete.• Your conclusions. This would include a comparison with reference experiments or

theory and a reflection on the laboratory experiment. The reflection would cover whatworked and what did not work. How things could be improved. And any other issuesyou may consider important.

We will focus on some of these issues in detail in other chapters in this manual as wellas in the lectures.

4. Uncertainties & Data

Every measurement has an associated uncertainty. The uncertainty is simply a reflectionof the limits of the instruments used, or the limits of the experimental setup. In a goodexperiment we will seek to repeat measurements to find out the statistical uncertainty. In avery good experiment, we will repeat the whole exercise, perhaps with a different team ofexperimenters, and different equipment. This is sometimes needed as every measurementis also susceptible to what is known as a systematic error. This error is often unknown,and takes some ingenuity to understand and reduce. Here we will focus on the first fewuncertainties.

4.1 Instrument or Measurement uncertaintyHave a look at the image used at the start of this chapter 1. There are three rulers used tomake three measurements. What do you think the measurements are? The first ruler hasonly 1cm markings, so its least count is 1cm. But this does not imply that we can measureonly in units of 1cm for surely we can estimate to at least 0.5cm. This would then be ourestimate of the instrument error. Our estimate for the first measurement might then be9.5(5)cm or 9.5±0.5cm.

Likewise, for the second ruler, the least-count is 0.5cm, but we may use a measurementerror of 0.2cm or 0.3cm. So the second measurement would be 8.5(3)cm or 8.5±0.3cm.

Finally, for the third ruler we have a least count of 0.2cm and a measurement error thatcould be the same, or perhaps we could use 0.1cm. So the third measurement would be11.9(1)cm or 11.9±0.1cm..

Notice that in all cases we have used a measurement error that is around half the least-count. This is usually appropriate. But you need to consider carefully what you choose, and

1This figure has been taken from http://www.rit.edu/~w-uphysi/uncertainties/Uncertaintiespart1.html.

18 Chapter 4. Uncertainties & Data

note it down in your lab book.

4.2 Example of how to record data with uncertainties

Here is an example taken from a report writte by one of your seniors Notice the following:

• The report has begun with a title and date.• It contains a brief aim. This is useful as it serves as a quick descriptor of the

experiment.• The data is recorded in a table. Each column is labeled and units are indicated.• Where applicable, the data are recorded with estimated uncertainties.• The number of significant digits is adjusted to be consistent with the estimated error.

More on this below.• The table has a caption immediately below it. The caption describes what is recorded

in the table.

This is an example of a good data table. All points listed above are important, and we willlook for these when we assess your work.

4.2 Example of how to record data with uncertainties 19

20 Chapter 4. Uncertainties & Data

4.2.1 Significant figuresWhat are significant figures? They are the number of digits with leading zeros not included.Here are some examples:• 2.123: four significant figures.• 2.1230: five significant figures. The last 0 is a significant figure.• 0.0123: three significant figures.• 010.0123000: How many here?• 0.000122349900001: and here?

Do you think that the uncertainties recorded in this table are reasonable? A digitaldevice like a multimeter and an oscilloscope will often display a large number of digits.If, say, the oscilloscope displays 6 significant figures, does it imply that the uncertaintyis only on the last digit? Let’s take an example from the report: The third frequency inthe table is reported as 2.60477±0.00001kHz. The uncertainty seems to be very low.Can this be correct?

It may be correct. But are the devices that we have in this lab calibrated to be thisaccurate? They often have tolerances written at the back. If the tolerance was listed as being1% (say!), then the uncertainty on the frequencies would be much larger than reported inthe table.

Additionally, if the readings in the last few digits fluctuated, say the oscilloscope read2.60477±0.00001kHz, 2.60632±0.00001kHz, 2.60210±0.00001kHz, etc. then the realuncertainty would be something like 0.002kHz, so the reading would be 2.604±0.002kHz.

Always report as many significant digits as are consistent with the uncertainty.

In the last example, when we adjusted the uncertainty to be 0.002kHz we also truncatedthe number of significant digits in the reading from 6 to 4 and reported it as 2.604±0.002kHz.

It would be inappropriate to report it as 2.60477±0.002kHz, for then, if we are uncertainabout the third decimal place, surely the digits in the fourth and fifth decimal place aremeaningless. Likewise, it would not be sensible to report the measurement as 2.60±0.002kHz: you must always use the same number of decimal places in the value and it’suncertainty.

4.3 Statistical uncertaintiesEvery measurement is prone to uncertainties that originate from the experimenters, theenvironment (wind, temperature changes, lighting changes, vibrations, ...), the experimentalequipment, and, in very accurate measurements, quantum fluctuations.

If you used a sufficiently accurate measuring device, then there is a very good chancethat each repeated measurement will be different. You can try it out by using a meter stick

4.3 Statistical uncertainties 21

to measure someone’s height: if the meter stick is accurate to 1mm then you will find thatyour measurements will vary. Get another person to perform the measurement and youwill get yet another set of results. It does not mean that the person you are measuring isexhibiting macroscopic thermal or quantum effects (!), but rather that due to variations inyour experimental technique (where did you measure from? How did you judge the topof the head?), and perhaps variations in the persons’ posture, you obtain a whole range ofmeasurements.

Experimental technique: It is possible to reduce the statistical variations in ourobservations by developing good experimental technique. Usually this can be done byunderstanding the sources of error: perhaps the air conditioning causes your equipmentto cool down. If so, perhaps using a baffle can reduce this effect. Timings can beimproved by focusing on when you start and stop the timer.

To some experimenters, most statistical errors are not really statistical, but aresystematic errors that we have not bothered to understand.

Let us call these measurements {xi}. Each xi is a measurement, and let us say we have Nmeasurements in all. If N is sufficiently large — in this course we will consider sufficientlylarge to mean N ≥ 5 — then we may define the average as

x̄≡ 〈x〉= 1N

N

∑i=1

xi. (4.1)

The uncertainty of the measurements {xi} is described by the standard deviation definedas

σx =

√1

N−1

N

∑i=1

(xi−〈x〉)2. (4.2)

Notice that for N = 1 the standard deviation is formally infinite (it is not defined!).The standard deviation expresses the width of the distribution of the measurements {xi}.

That is, if we perform a sufficiently large number of measurements and plot the frequencyhistogram for the measurements, then, if the histogram is normally distributed (i.e., it is aGaussian in shape) then σx is the half-width of the distribution. So the standard deviationexpresses the uncertainty of the measurements: as we will see in the lecture, on average, weexpect 68% of the measurements to be within σx from 〈x〉.

Once we have performed a sufficient number of measurements, the average and standarddeviation may be converged and will not change (by an appreciable amount) as more dataare added. It often takes a fairly large number of measurements for this to happen, but dueto time constraints in this lab we will usually not be able to accumulate enough of data toachieve this. Instead we will try to develop a sense of the uncertainty by taking at least 5 to10 measurements.

22 Chapter 4. Uncertainties & Data

Combining the errors: For a variable X you may have the average 〈x〉, the standarddeviation σx as well as the measurement or instrument uncertainty δx. How do we reportthe final result?

Formally, we combine the uncertainties using the relation:

∆x =√

σ2x +δ 2

x , (4.3)

and report the final measurement as

X = 〈x〉±∆x SI units. (4.4)

Questions:• Can we have δx� σx? If yes, what does it mean to have the measurement error

much larger than the statistical error? Under what conditions might you see thisoccur?• What about the opposite case (which is more usual)? When would σx � δx?

What does this say about your estimate for the measurement error δx?• The standard deviation never gets smaller as we add more data. Does this means

that we can never estimate the mean beyond a certain precision?

5. Graphs

Have a look at the following three graphs:

24 Chapter 5. Graphs

Graph (A) is a very basic (and incomplete) graph: the data is present, as are the axes, but

25

other than suggest that the data are distributed linearly, graph (A) communicates nothingelse.

In graph (B) we see axis labels but still no units. There are also error bars. Notice howthe y-axis starts from 5 and not 0: this allows the data displayed in the graph to span theentire graph sheet.

Finally, in graph (C) we have axis labels and units so we finally know what is plotted!There is a legend explaining (in case it was not clear) the contents of the graph. And wehave a figure number and caption. This is the kind of graph you need to produce.

In this lab we will not normally display error bars on both axes. But you will almostalways have uncertainties along one of the axes. If the uncertainties are too small youshould say that they are too small to see (place this bit of information in the caption).Alternatively, you may magnify the uncertainties by, say, multiplying them by 10 or 100(choose a power of 10). If you do this, say as much in the caption. Bear in mind that anystatistical information calculated from the graph should use the original uncertainties.

In the next figure you see one that was part of a report:

26 Chapter 5. Graphs

Here we see almost all the elements of graph (C). The axes are labeled and units arespecified; the graph has a short but informative caption (just below it); and there is also a

5.1 Check-list for graphs 27

title present. The latter is not really needed as the information in the title should be presentin the caption. The graph does not use a legend, but as this particular graph contains a singledata set, a legend is not essential.

This is a semi-log graph: the y-axis uses a log-scale and the x-axis uses a linear scale.Such a graph is used when we have an exponential relationship between the data. Theanalysis shown makes it very clear how the slope of this particular straight-line fit is relatedto the mathematical form the data are expected to satisfy.

What about the error bars? They appear to be missing in this example. Sometimes theerror bars are simply too small to display. If this is the case, then you should state that thisis so. Here they appear to have been forgotten, but this is otherwise an excellent graph.

5.1 Check-list for graphs

Here are a few points to keep in mind when drawing a graph:

• Decide what you intend to plot. For a pendulum do you plot period T versus length l,or would it be better to plot T 2 versus l? The former results in a curve (a parabola)and the latter results in a straight line (as long as we use the small-amplitude approxi-mation).• Is this to be a linear, semi-log, or log-log graph?• What is a suitable range for the two axes? Use the largest range possible so as to use

up the entire graph sheet.• Do not cut off points or place them on the axes.• Include uncertainties on the data points.• If the uncertainties are too small to see, indicate as much in the caption.• Fit a line only if you have enough of data.• Do not attempt to fit a curve through two points!• Extract fitted quantities like the slope and intercept along with their uncertainties.• If plotting more than one data set, include a legend.• Use different point types and line types for different data sets.• Include axis labels and units.• Include a caption.• Keep the graph neat. Do not clutter it with too much of information.• UNITS: Always make sure you have used the correct units in the graphs.

Here is an example of a good graph:

28 Chapter 5. Graphs

Error-bars are included but they are too small to see. It is the presence of the ‘Chisq’ thatindicates that the error-bars are present. The axes are clearly labeled and units are indicated.The axis ranges are well chosen so as to use the entire graph sheet. The caption is numbered(‘Fig 9’) and is concise and informative. Finally, all the data points are within the graphsheet.

5.2 Data extracted from graphs

First and foremost, a graph is a visual indicator of a relationship — or the lack of one! Butwe typically also extract numerical data from graphs. Here is a short list of quantities youwill look for in the graphs drawn in this lab:• Straight-line graphs will provide a slope and intercept.• These come with uncertainties which will be needed in subsequent error-analysis

steps.• The χ2: as will be explained in the lectures, this quantity is a numerical indicator of

the quality of the fit. It takes into account the uncertainties on the data.• The number of degrees of freedom Nn.d.f.: the n.d.f./ is used together with the χ2 to

do a post-analysis of the fit and the uncertainties assumed on the data points. For afit that is consistent with the uncertainties we will have χ2/Nn.d.f. ≈ 1. If it is muchlarger than 1 then it is possible that the uncertainties are too small (given the qualityof the data), and if it is much smaller than 1 is is possible that the uncertainties aretoo large (given the quality of the data). Look for examples of the former case below.• In the graph example shown above we have χ2/Nn.d.f. = 43.0/6 = 7.2 > 1 so the

error bars on the data points are probably a little too small.

5.3 Examples 29

5.3 Examples

If you think you know how to graph data, then figure out what is wrong (if at all!) with thefollowing graphs:

(A)

(B)

(C)

30 Chapter 5. Graphs

(D)

(E)

(F)

5.3 Examples 31

All of the above have been taken from past reports. Match the each of the examplegraphs with the one of following:• (1) Graph is excellent. χ2/Nn.d.f. indicates that uncertainties are chosen well. Caption

is excellent.• (2) Excellent graph except that there are points that are clipped off as they pass

thought the axes.• (3) Curve through two points!• (4) Uncertainties much too small! Points on axes!• (5) Has probably used the wrong units on the y-axis. And definitely no error bars!• (6) No error bars. No caption. Same point type for two data sets. In other words: do

not use Excel unless you know how to use it!

6. Error-propagation

6.1 Outline of error-propagation

The problem: In the simple pendulum experiment, you have measured the pendulum lengthto be l = 〈l〉±∆l and its period to be T = 〈T 〉±∆T . How do you evaluate g = 〈g〉±∆g?

Just to remind you:

T = 2π

√Lg, (6.1)

so

g =4π2L

T 2 = F(T,L). (6.2)

This is a classic problem that you will encounter in almost every experiment in thiscourse. What we need to do is propagate the errors from the length and period to theacceleration due to gravity. It is easy to evaluate 〈g〉: all we do is use the average values ofthe length and period:

〈g〉= 4π2〈L〉〈T 〉2

. (6.3)

But what about ∆g?

34 Chapter 6. Error-propagation

Error propagation: Consider a function F(X ,Y,Z, ...) where X ,Y ,Z, etc. are inde-pendent random variables.

The random variables could be quantities like length, mass, time. By independentall we mean is that they are not correlated. If the random variables were blood alcohollevel and accident rate then these would not be independent: there is a clear correlationbetween the two.

Assume that we are interested only in the statistical errors of a random variables,that is, ∆X = σX , ∆Y = σY , etc. The question is, given these standard deviations in theindependent random variables, what is the uncertainty in the function F?

Under the assumptions that X ,Y ,Z, etc. are independent random variables, that arenormally distributed, and given that the statistical errors on these variables are smallenough (more on this later), then F will also be normally distributed and will have astandard deviation σF that may be obtained from the formula:

σ2F = σ

2X

(∂F∂X

)2

+σ2Y

(∂F∂Y

)2

+σ2Z

(∂F∂Z

)2

+ · · · , (6.4)

where all derivatives and random variables are evaluated at the expectation values of therandom variables. That is, at their average values.

The assumptions needed to use eq. (6.4) are important. In this lab we will mostlyencounter independent random variables, or we may make the approximation that they areindependent. What about the other assumptions? What if the instrument uncertainty is largerthan the standard deviation? What if all we have access to is the instrument uncertainty? Forexample, when we use the bands of a resistor to determine its resistance, the fourth (or fifth)band will tell us the uncertainty of the resistance. Is this to be taken to be an uncertainty inthe random sense? The short answer is that unless we have good reason to believe that theuncertainty is not random (i.e., not normally distributed), we will assume it to be so.

You will see at lease one example of apparent errors that are not normally distributed.When you encounter this, remember that has been discussed here.

But now let us return to the pendulum example: We will assume that the uncertaintieswe have measured and combined in the manner described in Chapter 4 may be treated asstatistical errors: that is, the uncertainties ∆T and ∆L are statistical uncertainties.

From eq. (6.4) we know than the uncertainty in g is given by

∆2g ≡ σ

2g = σ

2T

(∂F∂T

)2

+σ2L

(∂F∂L

)2

≡ ∆2T

(∂F∂T

)2

+∆2L

(∂F∂L

)2

, (6.5)

where g = F(T,L) = 4π2L/T 2.

6.2 Proof of the error propagation formula 35

Inserting the partial derivatives

∂F∂T

=−2×4π2L/T 3

∂F∂L

= 4π2/T 2,

we get

∆2g = ∆

2T(−2×4π

2L/T 3)2+∆

2L(4π

2/T 2)2. (6.6)

We could use this expression, but let us simplify it by dividing both sides by g2 =(4π2L/T 2)2: we will divide the L.H.S. by g2 and the R.H.S. by (4π2L/T 2)2. This gives usthe simpler expression(

∆g

g

)2

=

(2∆T

T

)2

+

(∆L

L

)2

. (6.7)

Finally, we replace all the random variables in the above expression with their expectationvalues, i.e., with their average values. This gives us the final expression:(

∆g

〈g〉

)2

=

(2∆T

〈T 〉

)2

+

(∆L

〈L〉

)2

. (6.8)

Now we know everything to evaluate the propagated error from L and T to g.

6.2 Proof of the error propagation formulaIn progress!

7. The Formal Report

In this chapter we will see how a formal report should be written.

8. Preparation for the lab

You will have only three hours in a laboratory session. Some experiments span two or moresessions, but in general, you will have a lot to do in a session and only just enough time tocomplete it all. So you need to plan well.

Here is the check list of things to do before and during the laboratory session:• Plan to arrive on time. The labs start at 2pm but you may be able to come in as early

as 1:30pm. By all means do so. But do not come in later than 2pm!• Find out in advance which experiment you will be conducting.• Read the lab script for the experiment and fill in the Work Plan for the lab session.

Such a list is provided on the next page. Paste the completed Work Plan in your labbook.• You will not be allowed to start without a completed Work Plan!• Make sure you have eaten — we do not allow eating in the laboratory.• Make sure you get assessed during the session.• If you know you need to leave early, ask your demonstrator to assess you earlier.• Make sure you fill in the attendance sheet.• If you finish early, do not leave. Instead complete as much as of the analysis and

writing as you can. Your demonstrator may arrive later in the session to see what it isyou have accomplished during the lab.• The lab finishes at 5pm and you must be out by 5:15pm.• Should you find you need extra time, arrange with the lab technicians to come on a

morning or on a Wednesday.

The Work Plan follows on the next page. Before being allowed to start a lab youwill need such a list filled up in a reasonably good manner. You need a new work planfor each lab session so make enough of copies!

40 Chapter 8. Preparation for the lab

8.1 Work Plan for the lab

NAME:Expt:

Date:Part*:

AimsWhat are thegoalsof thisexperiment?StepsList as many stepsas you see fit

DataList data youthink you needto record.

GraphsList the graphsyou may need tomake.State the kindof graph: linear,log–log, semi-log

*: Which part of the experiment? Applicable to multi-day experiments.

9. Experiments