MSc Laboratory Handbook 2012-13

7/31/2019 MSc Laboratory Handbook 2012-13

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Department of Physics

MSc Laboratory Handbook

Academic Year 2012-13

Abridged from the original by Professor Ben Murdin


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Fundamental Constants

Electron rest mass Me 9.11x10-31 kgProton rest mass M p 1.67x10-27 kgElectronic charge e 1.60x10-19 CSpeed of light in vacuum c 3.00x108 m s-1Planck's constant h 6.63x10-34 J s

h/2 1.05x10-34 J s

c 197 MeV fmBoltzmann's constant k 1.38x10-23 J K -1

8.85x10-5 eV K -1 Molar gas constant R 8.31 J mol-1 K -1 Avogadro's number NA 6.02x1023 mol-1 Standard molar volume 22.4x10-3 m3 mol-1 Bohr magneton B 9.27x10-24 J T-1 Nuclear magneton N 5.05x10-27 J T-1 Bohr radius a0 5.29x10-11 mFine structure constant = e2/(4 0c) (137)-1 Compton wavelength of electron c =h / (mec) 2.43x10-12 mRydberg's constant R 1.1x107 m

R hc 13.6 eVStefan-Boltzmann constant 5.67x10-8 W m-2 K -4 Gravitational constant G 6.67x10-11 N m2 kg-2 Proton magnetic moment p 2.79 N Neutron magnetic moment n -1.91 N

Other Data and Conversion Factors1 angstrom 10-10 m1 fermi fm 10-15 m1 barn b 10-28 m2 1 pascal Pa 1 N m-2 1 standard atmosphere 1.01x105 Pastandard acceleration due to gravity g 9.81 m s-2 permeability of free space 0 4x10-7 H m-1 permittivity of free space 0 8.85x10-12 F m-1 1 electron volt eV 1.60x10-19 J

eV/hc 8.07x105 m-1 eV/k 1.16x104 K

1 unified atomic mass unit (12C scale) u 931 MeV/c2 1.66x10-27 kg

wavelength of 1eV photon 1.24x10-6 m base of natural logarithms e 2.718ln 10 = loge 10 2.303

Abbreviations

A ampere H henry mol gramme mole T teslaC coulomb Hz hertz N newton V volteV electron volt J joule Pa pascal Wb weber F farad K kelvin S siemens W wattg gramme m metre s second ohm

Prefixesf femto 10-15 p pico 10-12 n nano 10-9 micro 10-6 m milli 10-3 c centi 10-2 k Kilo 103 M Mega 106 G Giga 109 T Tera 1012


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ContentsCONTENTS ................................................................ .................................................................. ........................ 3

1. INTRODUCTION .......................................................... ..................................................................... ............. 5

2. WORKING IN THE LABORATORY ....................................................................... .................................... 6

2.1 SAFETY IN THELABORATORY -----------------------------------------------------------------------------------------6 3. THE LABORATORY NOTEBOOK............................................................ .................................................. 7

4. UNCERTAINTIES IN MEASUREMENTS ............................................................. ..................................... 9 4.1 I NTRODUCTION----------------------------------------------------------------------------------------------------------9 4.2 SYSTEMATICERRORS------------------------------------------------------------------------------------------------- 10 4.3 R ANDOMERRORS----------------------------------------------------------------------------------------------------- 10 4.4 NUMBERS ANDU NITS ------------------------------------------------------------------------------------------------ 11 4.5 PARENT ANDSAMPLEDISTRIBUTIONS------------------------------------------------------------------------------ 11

4.5.1 Introduction ......................................................... ................................................................... ........... 11 4.5.2 Sample Distribution ................................................................. .......................................................... 12 4.5.3 Parent Distribution ............................................................. ............................................................... 14 4.5.4 The error in the mean .................................................................. ...................................................... 15

5. PROBABILITY DISTRIBUTIONS ............................................................. ................................................ 17 5.1 I NTRODUCTION-------------------------------------------------------------------------------------------------------- 17 5.2 PROBABILITY DENSITY FUNCTIONS--------------------------------------------------------------------------------- 17 5.3 GAUSSIAN OR NORMALDISTRIBUTION----------------------------------------------------------------------------- 18 5.4 DISCRETE PROBABILITY FUNCTIONS-------------------------------------------------------------------------------- 20 5.5 BINOMIALDISTRIBUTION-------------------------------------------------------------------------------------------- 20 5.6 POISSONDISTRIBUTION---------------------------------------------------------------------------------------------- 21

6. PROPAGATION OF ERRORS............................................................ ........................................................ 23 6.1 MATHEMATICALBACKGROUND------------------------------------------------------------------------------------- 23 6.2 SPECIFICEXAMPLES-------------------------------------------------------------------------------------------------- 25

6.2.1 Addition and Subtraction ....................................................... ............................................................ 25 6.2.2 Multiplication and Division ...................................................................... ......................................... 26 6.2.3 Power laws.............. ..................................................................... ...................................................... 27 6.2.4 Logarithms and Exponentials ................................................................... ......................................... 28

7. REPRESENTING DATA .................................................................. ............................................................ 29 7.1 DRAWINGS, PHOTOGRAPHS ANDTABLES -------------------------------------------------------------------------- 29

7.1.1 Photographs............................................................ ............................................................... ............ 29 7.1.2 Tables....... ............................................................... ................................................................ ........... 29

7.2 GRAPHS ---------------------------------------------------------------------------------------------------------------- 30 7.2.1 Error Bars......... ................................................................ ................................................................ . 30 7.2.2 Drawing The Line .......................................................... ................................................................... . 30 7.2.3 More than One Line on the Same Graph ........................................................ ................................... 30 7.2.4 Linear Scales................................................ ................................................................ ...................... 31 7.2.5 Logarithmic Scales.......................................................... ................................................................. .. 32

8. FITTING DATA USING THE LEAST SQUARES TECHNIQUE.......................................................... 34 8.1 I NTRODUCTION-------------------------------------------------------------------------------------------------------- 34 8.2 BESTSTRAIGHTLINEFIT: LINEAR R EGRESSION------------------------------------------------------------------ 35 8.3 CORRELATION--------------------------------------------------------------------------------------------------------- 36 8.4 THE 2 DISTRIBUTION: TESTING THEGOODNESS OFFIT---------------------------------------------------------- 36 8.5 FINALR EMARKS------------------------------------------------------------------------------------------------------ 37

9. THE LABORATORY REPORT ......................................................... ......................................................... 38 9.1 General Comments...................................................................................... .......................................... 38 9.2.1 Plagiarism and Copying ........................................................ ............................................................ 38 9.2.2 Title, Authors and Affiliation .......................................................................................................... ... 39


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9.2.3 Abstract ................................................................ ................................................................... ........... 39 9.2.4 Introduction ......................................................... ................................................................... ........... 39 9.2.5 Theory ............................................................. ............................................................. ...................... 39 9.2.6 Experimental Arrangements and Techniques ........................................................... ......................... 40 9.2.7 Procedure..................................................... ................................................................ ...................... 40 9.2.8 Results ..................................................................... .............................................................. ............. 40 9.2.9 Discussion.......................................................... ..................................................................... ........... 40 9.2.10 Conclusions............... ............................................................... ........................................................ 40 9.2.11 Acknowledgements................................................................... ........................................................ 40 9.2.12 References ..................................................................... ................................................................ ... 41

10. BIBLIOGRAPHY........................................................ ..................................................................... ............ 42

APPENDIX A: SUMMARY FORMULAE FOR ERROR ANALYSIS....................................................... . 43

APPENDIX B: KEY UNITS AND CALCULATIONS IN RADIATION PHYSICS ................................... 44

APPENDIX C: GLOSSARY ............................................................ .............................................................. ... 46


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1. Introduction

The MSc laboratory classes provide an opportunity for you to further develop the practical skillsassociated with radiation, medical and nuclear physics. You should gain experience of studying problems and designing experiments to test many scientific theories. To do this successfully youwill need to analyse, critically assess and interpret the data you obtained. Finally, you mustdevelop the skills necessary to describe the work clearly for the benefit of others. These areamongst the most important skills that a physicist possesses and are often the ones most requiredin professional life.

The experimental work that you will carry out in the MSc laboratory will give you specific skillsin the operation and understanding of a variety of complex and specialist nuclear detectionequipment. The experiments will also help you to explore new aspects of nuclear physics, andthe underlying processes related to radioactivity and nuclear measurements. In addition thelaboratory encourages you to think about experimental problems and how to solve them, and youwill extend your general skills and techniques that are fundamental to good experimental work.

There are a number of textbooks and other resource material that you may find useful background reading for your laboratory work. An extensive bibliography is included in chapter 10. For a good general introduction to laboratory practice students are referred to the recent book by Kirkup [1].


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2. Working in the Laboratory

2.1 Safety in the Laboratory

You are required to exercise a sense of responsibility in the laboratory at all times, and towork in ways that do not endanger either yourself or other people. You are required tocomply with the following instructions.

As a new student you must attend a radiation safety induction and a laboratorysafety induction before you can work in the MSc laboratory. Students will beissued with radiation badges which must be used at all times.

Smoking, eating or drinking is not allowed within the laboratories. Thisincludes chewing etc. your pen or pencil.

Always wear a laboratory coat. Coats and any bags should be stored in thecloakroom area, not at your work bench.

Remove your laboratory coat and wash your hands before leaving the lab.

Use tweezers to move radioactive sources. Sources should not be moved awayfrom your work area without the approval of the laboratory supervisor. Youmust never take a source out of the laboratory.

Ensure that you follow the written procedures in the lab script safely and takeheed of any verbal instructions by staff. Risk assessments are available for theexperiments as part of the help folders. If you are in any doubt consult ademonstrator or supervisor.

IMPORTANT:

Never turn the power on a NIM crate ON or OFF - leave the crates as you find them


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3. The Laboratory Notebook

It is said that Rutherford discovered Radon by observing that the results of hisexperiment depended on whether the laboratory door was open or closed!

A laboratory research notebook is one of a physicists most valuable tools. For a practising physicist it contains a permanent written record of his or her mental and physical activitiesfrom experiment and observation through data analysis to the ultimate understanding of the phenomena under investigation. The act of writing in the notebook should cause the physicistto stop and think about what is being done in the laboratory. In this way it is an essential partof doing good science.

It is a requirement of the course that all students keep an account of all experiments in alaboratory notebook. Keeping good notes is an acquired skill that can be of tremendous benefit in any career. Writing good notes requires practice and discipline. It is not a skill thatcomes easily to most people. For research students the notebook will be the prime source of information required to write a PhD thesis. The researcher may even wish to make moneyfrom an invention or result! In the case of important data for a patent, for example, theexperiment and the written record should also be witnessed and signed by a competentobserver. The notebook may play a vital role in obtaining legal rights to the invention!

A bound notebook is provided at the start of your course. If need to purchase another itshould be a hard back book. It must be brought into the laboratory on every laboratory dayand used exclusively for entering notes on the conduct of the experiment. All informationshould be recorded clearly in ink in the notebook on pages used consecutively with no gaps.All entries should be dated. Note that, unless it is a help to you in your learning, it is a wasteof time to copy material from this booklet or the experiment instruction sheets into your notebook. However, any information you do consider important should be photocopied and permanently glued into the notebook.

The written record should include information on:

Preliminary reading, settings, adjustments.

Readings together with uncertainties and settings used in the final datagathering.

Any procedure or precaution not fully described in the printed script or elsewhere.

Any relevant external conditions (room temperature, barometric pressure, etc.)

Precautions taken to optimise performance and/or minimize errors.

Sources of important information, components and advice.


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The written record should NOT be:

Censored;Corrected;Written after the event

Before taking any readings, rough estimates of the expected results should be made andnoted. This is so you will have some idea how to setup instruments initially, how to scaleaxes on graphs so that these can be plotted as readings are taken, and so on. It also helps youspot gross errors in the experimental arrangement or data more easily if you have some ideaof what should happen.

Most importantly, readings should be tabulated with uncertainties as they are taken. Thetables should have headings that identify the measurements taken, the equipment used, over what range and in what units. Normally results should be graphed as the readings are taken.If, for some unusual reason this cannot be done, the graph should be drawn immediatelyafterwards and certainly before leaving the laboratory or proceeding to the next part of the

experiment.

After the data has been recorded the physicist begins to study it. The notebook provides theforum in which the data and observations are analysed, evaluated and interpreted. Thisfurther analysis must also be written in the notebook. This final complete record may then beused as the basis for writing reports, technical papers, patent disclosures and correspondencewith colleagues. The information can also be used to review progress and plan future work.

It should be clear from the above just how important the laboratory notebook is. A well-keptnotebook should be a valuable asset but do not spend time perfecting the appearance of your notes at the expense of working on the scientific problem.


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4. Uncertainties in Measurements

4.1 Introduction

Experimental physics is about studying the world around us. This invariably involves makingmeasurements - and this is where the problems start! The spinning of a coin only has two

possible outcomes but we still cannot tell what it will be. A Geiger Muller tube measuresaccurately 12 disintegrations in one 60s period but 15 in the next! I ask ten people to tell methe length of the same piece of string and get 10 different values. How is a physicist meant tomake sense of this data?

In addition, how should physicists report their findings to others? For example if we measurethe length of a piece of string and get the answer of 1m what can we say? As scientists theone thing we should not say is:

The length of this piece of string is 1m.

However, we might say:

We measured the piece of string and got the result of 1m.

This immediately means that we have to be concerned about confidence. Are other peoplegoing to be confident in our measurements or not! We might instead have written:

We find the length of the piece of string to be 1m with 95% confidence that it lies between 0.90 and 1.10m

This is still our opinion of the measurement. If real confidence is to be justified then enoughinformation must be supplied for the reader to make their own assessment of themeasurement. There have been many instances where work, even by eminent physicists, has been shown later to contain errors much larger than the limits they originally quoted!Gradually as the experimental techniques and methods are improved the results tend toapproach a single value. In a dictionary error is often defined as the difference between theobserved or calculated value and the true value. In general this is not what error means to ascientist. The problem with experimental science is that the true value is not known.

One class of errors can be dealt with easily. Simple mistakes in measurements or calculationare usually easily spotted and eliminated by carefully repeating the experiment or calculation.

Random fluctuations in measurements and systematic errors are more difficult to correct for.In everyday language accuracy and precision mean much the same thing. In science thedifference between accuracy and precision is very important. The accuracy of an experimentis a measure of how close the result of an experiment is to the true value. The precision of anexperiment is a measure of how well the result has been determined without reference to itsagreement with the true value. This distinction is illustrated in Figure 4.1. Generally when wequote uncertainties in an experiment we are talking about the precision with which the resulthas been found.


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X Axis

Y A x

i s

X Axis

Y A x

i s

(a) (b)

Figure 4.1 : (a) Accurate but imprecise data and (b) precise but inaccurate data. The solidline represents the "true" relationship between x and y.

4.2 Systematic Errors

The accuracy of an experiment is usually limited by how well we can control systematicerrors. These are not easily detected but can be estimated from careful study of theexperimental conditions and techniques used. For example, they may be due to problemswith:

Instrument Calibration

Instrument Reproducibility

Biased observation

A significant part of planning of an experiment should be devoted to understanding the origin

and reducing the sources of systematic error.4.3 Random Errors

The precision of an experiment depends on how well random errors can be reduced. As before, we can refine the experimental method and techniques. We also know intuitively thattaking a sufficiently large number of readings may reduce the effect of random errors on the precision of the measurements.

As Figure 4.1 shows there is usually little point in reducing the random error significantly below the systematic error, although of course it is rare that we actually know how much thesystematic error is.


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4.4 Numbers and Units

Quantitative information is expressed in numbers and units - unless the quantity isdimensionless in which case there are no units. You should use the International System of Units (SI units) together with other units that are in use with the International System; theseare summarised in Appendix B and on the inside front cover of this handbook.

The number associated with the unit should contain the minimum number of digits necessaryto adequately express the quantity with due regard to the precision with which it has beendetermined. For example, if a length has been determined to a precision of 0.1m it would bemisleading to express it as 17.1154m, as might happen if you had simply copied the digitsfrom a calculator. Instead, write it as 17.1m, or, more informatively 17.10.1m; theremaining digits do not contain useful information and should be omitted.

It might be that you have a good reason for wishing to express this length in mm. If withoutcomment you simply write 17100mm, it is not clear whether the two zeros are significant or whether, as in this example, they are there simply to get the decimal point in the right place.It is possible of course, to write 17100100mm, but although this is probably less misleadingthere is still some uncertainty as to the significance of the zeros in the100. It would be better to write 1.71x104mm, or, more informatively, (1.710.01)x104mm.

It is often preferable to avoid using zeros that are there only to get the decimal point in theright place; such zeros are referred to here as superfluous zeros because it is possible toavoid the need for them. Superfluous zeros can occur not only when the number is muchlarger than unity, as in the above example of 17100mm, but also when the number is verymuch smaller than unity; for example in writing 0.003920.00001m, again the zeros are thereto show where the decimal point is. One technique of removing superfluous zeros has beenshown in the previous paragraph, namely to use 10 raised to an appropriate power as amultiplier; in this case to write (3.920.01)x10-3m reduces the number of superfluous zeros toone. Another technique is to change the size of the unit by using one of the prefix letterswhich form part of the SI system; for example, 0.003920.00001m can be written3.92 0.01mm. Again the number of superfluous zeros has been reduced to one.

4.5 Parent and Sample Distributions

4.5.1 Introduction

If we consider some quantity e.g. the length of a specific piece of string, it is reasonable toassume that there is a true value for that quantity, which we would like to know. However random fluctuations mean that there is some variation between any measurement and this truevalue. In cases where the quantity is discrete then the number of possible results is limited ina way that the values of a continuous quantity are not. However there is still a truedistribution for the results and hence a true mean value which is likely to be continuous.Imagine that we make repeated measurements of our piece of string, and we plot the resultsobtained on a graph. If this graph represents all the measurements we could have made, i.e.an infinite number of measurements then this is called the parent distribution. See Figure 4.2for example. Such a distribution is also sometimes called a population or universedistribution.


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x

f ( x)

0.0

0.5

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

Figure 4.2 : Example of three parent distributions of data having the same true value, ,which in this example is zero, but different amounts of spread, , as indicated.The distribution depicted is the Gaussian distribution (see below section 5.4)

The spread of measurements around the central peak of the histogram is the result of randomerrors in the measurement process. These arise, in this case, mainly from the limitations of the measuring device that prevents the observer from obtaining exactly repeatable results.

In reality of course the parent distribution described above cannot be measured. However, atrial or sample distribution can be obtained by taking a finite number of measurements,n, of the observed quantity x. Clearly it is assumed that in the limit asn that the sampledistribution tends to the parent distribution. We want to know the following things from thesample distribution:

1) What is the best estimate of the mean of the parent distribution we can obtain?i.e. what is the best number to quote for the measurement made?

2) What is the spread of the parent distribution? This gives us information aboutthe precision of our measurements.

3) What error should we quote for the best estimate of the mean? This is usuallycalled the standard error in the mean. It is not the same as the spread.

4.5.2 Sample Distribution

Clearly, as we never take an infinite number of measurements we can never know the meanor spread of the parent distribution. Instead, the best we can do is to take a finite number of measurements.

For a given number of trialsn the sample mean,m, is defined by:

x xn

m ni

i1

1(4-1)


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This is just another way of saying that the mean of a sample of data x1, x2, x3, xn is the sumof all the data values, divided by the number of data. We use the bar above x to meansimply the mean value of.

The median and mode of a distribution are ways alternative to the mean of talking about thelocation or centre of the distribution, but are less frequently used (see Bevington [2]).

The most commonly used way of describing spread of a distribution or data set is with thevariance or the standard deviation (s.d.). [The reason for this is that it is easy to calculate theshape etc of the Gaussian, Poisson and Binomial distributions from their s.d., and as we shallsee later, and these are the most important distributions in statistics]. The variance and s.d.are given by:

2)var( n s x (4-2)

2221

22 1 m xm xm xn

sn

i

in

(4-3)

In other words the variance is the mean value of the square of the deviation of each data pointfrom the mean, or for short, the mean square deviation. The s.d. is just the square root of the variance. It is also known as the root mean square deviation, or for short, the r.m.s.deviation. It is often convenient to find sn using the last equality in 4-3, but beware, the twoterms are large and the difference is very small. This means that you should keep as much precision as possible until the end.

There are many convenient ways of calculating the standard deviationa) Most calculators, including those sanctioned by the Physics Department, have a

statistics mode. In the stats mode, you should clear the stats memory, then enter eachdata value. Functions will be available to tell you the number of data values, themean, the standard deviation of the sample sn (and the estimated population s.d. sn-1,see next section)

b) Use Excel. Type in the data into a blank spreadsheet and use the functions STDEVPand SQRT. N.B. in Excel STDEV is used to calculate sn-1 (see next section).

c) By hand. Calculate the sum of all the data values and divide byn. This is the meanvalue; squaring it gives the second term in the last equality of equation 4-3. The sumof the squares of all the data values divided byn is the first term. To find sn subtractand square root.

It is often helpful to know that if you have data values which only differ in the last fewdecimal places there is a shortcut to the calculation, which can be especially helpful if youare calculating by hand or with a calculator (though not really necessary if you are using aspreadsheet like Excel). The following formulas say how to calculate the mean and standarddeviation when you subtract off some number A from all your data xi and divide by a scalefactor by B to get a new data set, yi.

B A x

y ii

You can then calculate the mean and s.d. in y and get back to the mean of x using:

A y B A B

A x B x


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which is just B times the scaled mean plus A, and the s.d. in x is:

yn xn Bs B A x

B A x

B x x s ,

2222

,

i.e. just B times the s.d. of the scaled data. E.g. if your x data set is 20.012174, 20.012146,20.012198, 20.012186, then subtract off A=20.0121 and scale by B=0.000001 to give a scaled

y data set of 74, 46, 98, 86. The mean of these y = 76. The mean of the squares of the scaledset is 6148, so using (4-3) the scaled error is

19766148 2, yn s Hence the original unscaled mean of x is 0.000001 76+20.0121 = 20.012176 and the originals.d. is 0.000001 19=0.000019.

4.5.3 Parent Distribution

If the number of possible outcomes of the measurement is small, i.e. the total population issmall, then it may be possible to actually make all possible measurements, e.g. whencounting moons around each planet in the solar system. In this case the mean and standarddeviation of the population are obtained using equations 4-1,2,3.

If there are an infinite number of possible measurements, then the population distribution hasa mean, , defined by:

n

iin

xn 1

1lim (4-4)

Note that it is quite common for the characteristic properties of the population distribution tohave Greek letters (e.g. ) while the sample distribution normally has Latin letters (e.g.m).

Following the logic of 4-1 to 4-4, the standard deviation of the population distribution isdefined by:

n

iin

xn 1

22 1lim (4-5)

Eqns 4-4 and 4-5 are just another way of saying that the population mean and s.d. are themean and s.d. of a sample of data as the sample size tends to infinity. Of course it is not possible to make an infinite number of measurements. In fact it is often not possible (due totime constraints) even to measure a large number of data, so we are almost always required tomake an estimate of and from a small sample. In the case of , life is simple, as the best possible estimate of is just m.

It would seem intuitive that that the best estimator for would be sn, and it certainly has thesame limiting value as equation 4-5 asn . However, there is a flaw in this argument thatis important at small values of n, i.e. just the situation where we probably want to use it [3].The best estimate of obtained from the sample distribution has been derived in a number of texts [2,3]. It is given by:


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n

iin m xn

s1

221 1

1

(4-6)It is now clear why we used the subscriptn for the s.d. of eqn 4-3; because the denominator isn, and this is what distinguishes it from 4-6 where the denominator isn-1.

N.B There is great confusion between various texts about the names of the two quantitiesdefined by equations 4-3 and 4-6. Some texts call the former the sample s.d. and the latter the population (or parent) s.d., but many use exactly the reverse! The important thing is not toremember the name, but what they are for. Equation 4-3 is an exact calculation of thevariance of the sample data (which is most useful if the data include the whole population of possible results hence the confusion with names), and equation 4-6 is for estimating the population s.d. from a small sample (again, you can see why confusion can arise). Theconfusion disappears if you use the symbol sn or sn-1. In physics, it is very rare that you will be considering a measurement where you are able to take the entire population, so you willalmost always be interested in sn-1.

It is sometimes useful to note that sn-1 and sn are related by:

221 1 nn

sn

n s

(4-7)

4.5.4 The error in the mean

Having taken a small sample of data and used it to estimate the mean of the parent population, it is extremely important to give some indication of how uncertain this estimateis, as explained in section 4.1. It is important to note that the standard deviation, sn-1 or ,

determines the width of the histogram or distribution of measurements. It is not a measure of how closem is to , i.e. the precision to which the mean,m, is known. The likely difference between m and is called the standard error in the mean or more simply the standarderror. Here it is given the symbol sm, see figure 4.3.

How do we determine the standard error? Even a small number of measurements will give adistribution that is a rough approximation to the parent distribution. If this has a narrow peak and hence small , as shown in Figure 4.2, the measurements will be close together and themean m of the sample will be close to . Alternatively, if the peak is broad the measurementswill be more widely scattered and the meanm is less likely to be near . So , or its bestestimate sn-1, must be related to the error inm. We intuitively expect that by makingn larger and larger we can make the histogram of the sample look more and more closely like the population, and so the error inm should become smaller and smaller. In fact we should beable to get m as near to as we like, even if our individual data points have poor precisionand hence large s or . In short, the error inm decreases with increasingn. However, sn-1 does not change much withn, it just gets closer and closer to , so the error cannot simply be

sn-1. It can be shown that the likely difference betweenm and , the standard error in themean, is

n

s s nm

1

(4-8)

You can see that if only one measurement is taken the error is the same as the standarddeviation, but that the error goes down the more measurements are taken. [Of course it is


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very difficult to guess the s.d. unless you take more than one measurement]. There is a law of diminishing returns due to the square root: you need to quadruple the number of data to halvethe error, but take a hundred times more data to get the error down by a factor of ten.

Figure 4.3 : Example of a sample histogram (cross-hatched) taken from a parent population (solid line). The population has mean, , while the sample has meanm. The bestestimate for the population standard deviation, , from the sample data is sn-1. The likelydifference betweenm and , i.e. the error inm, is sm, the standard error in the mean.

To understand more precisely what the standard error in the mean is, imagine taking a smallsample of data and calculating the mean. Imagine then taking another sample, the same size,and calculating the mean again. Keep taking samples until you have a large number of valuesfor the mean. The mean of the means will be the same as the true mean of the population (if you take a very large number of samples), but the sample means have a spread, because eachindividual sample is quite small and therefore its mean has some error. The spread of thesemeans about the true mean is the standard error in the mean. To be exact the standard error isthe likely standard deviation of those sample means. The derivation of Eqn 4-8 is not difficultand may be found in most statistics texts.

So finally we can summarise as follows. From a sample distribution of n measurementsequation 4-1 gives the best estimate of the population mean, while equation 4-6 gives the best

estimate of the standard deviation. Finally equation 4-8 gives the best estimate of the error inthe mean. The result of all this is a measurement quoted as:

m sm (4-9)Most calculators, including the standard University approved calculator, have a statisticalmode that easily allows you to enter a set of data and obtainm, sn-1 and n. There is thereforeno excuse for not taking multiple measurements of physical quantities in the laboratory andquoting m sm. The final rule to remember is that the error should normally be quoted onlywith one significant figure, and the mean should have the same number of decimal places asthe error (see section 4.4). Dont forget that the error might be much larger than your original precision on the measuring apparatus you used to take the data (if the data has lots of intrinsicvariation or noise), or the error may even be much smaller than the original precision (if youtook lots of measurements)!

s m

m

sn-1


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5. Probability Distributions

5.1 Introduction

This section will describe the properties of some of the most important probabilitydistributions in physics. The most frequently used, though you may sometimes be unaware of

it, is the Gaussian distribution. It is the familiar bell shaped curve that governs most physical parameters. We also describe several other distributions which have specialised use, theBinomial, Poisson and Lorentzian distributions. We start by looking at general properties of probability distributions.

5.2 Probability density functions

The Gaussian distribution, for example, is for continuous variables. Continuous variables cantake any value with infinite precision (even if the measurement has reduced precision), and ineffect it is therefore impossible to get a measurement result exactly the same as some testvalue. It therefore makes no sense to ask what is the probability of obtaining any particular value? (as this is zero). Only the probability of obtaining a value between two limits has ameaning. We therefore use a probability density function P ( x) which gives the probabilitydensity for any given test value x, but it is the area under the function between two limits thatgives the probability. The probability of the measurement lying between x and x+dx is

P ( x)dx. For a finite sized interval the probability of lying betweena and b is given by:

b

adx x P p (5-1)

The sum of all probabilities must be unity, so the probability density function must be

normalised, i.e. it must satisfy the condition:

1dx x P (5-2)

The most likely value from a p.d.f., i.e. its expectation value, written < x>, is given by:

dx x xP x (5-3)

In fact the expected value of any function f of x is

dx x P x f f )( (5-4)

This is useful if you know, say, the probability distribution for tolerances on a resistor, andyou know how different values affect the output of the circuit, you can work out the expectedoutput. The expected output is not necessarily just the output calculated from the expectedresistance, if the output is non-linear and the probability distribution is asymmetric and. Wecan use equation 5-4 to calculate the likely value of f ( x) = ( x- )2, i.e. the likely square of thedeviation of a measurement from the expectation value, called the expectation variance

dx x P x x 222 (5-5)


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5.3 Gaussian or Normal Distribution

The Central Limit Theorem (which has a quite simple proof, see for example ref [2]) statesthat quantities which have many random contributions are governed by a distribution thattends towards a Gaussian for large numbers of contributions. This is the reason why itgoverns so many measurements in physics and other disciplines almost everything you can

measure is affected by many different random factors and types of noise. Measurement of a piece of string will depend on the exact temperature (which may make the ruler expandslightly), the steadiness of your hand, parallax, the humidity (making the fibres expand) etcetc. Throwing one die gives a distribution that is equal for each result, throwing two gives atriangular distribution peaked at 7, but throwing 10 dice gives a distribution that is verynearly Gaussian, because of the large number (in this case 10) of random contributions to theresult. This is interesting considering that the distribution for results from each individual dieis flat, and nothing like a Gaussian. Similarly the mean value of a sample of data will obey aGaussian distribution due to the random contribution from each data point, even if each datavalue is governed by another distribution. This fact is used in the proof of equation 4-8.

The family of Gaussian or Normal probability density functions with different means andstandard deviations is usually written:

2

21exp

21,;

x x P

(5-6)

It can be shown quite simply by substituting equation 5-6 into 5-3 that the expectation value< x> = , and into 5-5 that the expectation variance is 2. In fact it is in order to be able toarrive at these results that the distribution is written in this way. Most importantly, it is

because features explicitly in this distribution that it is so often used as the measure of spread.

Fig 4.2 illustrated three distributions with mean value of zero and different values of the s.d.Figure 5.1 shows the area under the distribution between is 0.683, which means that thereis a 68.3% probability of any one observation lying within the limits from the expectationvalue. Conversely it also means that if a standard error is quoted then what it really means isthat there is 68.3% confidence that the true value lies with one error of the quoted value. Italso means that for any graph with error bars corresponding to one s.d., only 68.3% of thedata points will overlap the true line. The corresponding percentages for 2 , 3 and 4 are 95.45%, 99.73% and 99.99%, respectively.

It can be seen from equation 5-6 that the width of the Gaussian distribution as described bythe standard deviation , occurs where the height of the curve has dropped from e-1/2 of its peak value. The width of the distribution can also be characterised by its full-width at half maximum (FWHM), . This is also often called the half-width and is defined as the width of the distribution between the values of x where the height of the distribution is half itsmaximum value. From equation 5-6 it can be shown that:

= 2.354 (5-7)

The relation between these parameters is also shown in Figure 5.1.


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x x + x

x

-40 -30 -20 -10 0 10 20 30 40

P ( x ;

)

0.00

0.01

0.02

0.03

0.04

0.05

Figure 5.1 Distribution showing area under an element of the distribution and the area

under . where in this case =0 and =10 .

Figure 5.2 shows the mathematical connection between the Gaussian distribution and someother statistical distributions that are commonly encountered in physics. The mean and spread(standard deviation or half-width) of the distributions are also described. They will bediscussed in the laboratory sessions and are referred to specifically in some of theIntroductory Physics Experiments. Further details are discussed by Bevington [2].

Binomial

Gaussian

Poisson

n np >>1

n , np =

Figure 5.2 Schematic diagram showing the relationship between some of the probabilitydistributions discussed in the following sections. E.g. the Poisson distribution looks the sameas the Gaussian distribution in the limit of large .


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5.4 Discrete probability functions

Variables can be discrete as well as continuous, and in these cases P ( x) represents a probability not a probability density. This changes the way we calculate normalisation,expectation values, and expectation variances. The normalisation looks similar to equation 5-2, but the integral is replaced by a sum:

10

n

x

x P (5-8)

where there aren different possible results for x. Similarly the expectation value is given by:

n

x

x xP x0

(5-9)

and the expectation variance is:

n

x

x P x x0

222 (5-10)

5.5 Binomial Distribution

If in a trial involvingn elements the probability of a successful event is p and of a failureq (so that p + q = 1) then the probability of x successes is

xn xq p x xn

n pn x P

!!

!,;(5-11)

Note that this probability distribution is asymmetric unless p = q = 1/2.

As x varies from 0 ton the expressions for P ( x;n,p ) given by (5-11) are in fact the successiveterms in the binomial expansion of ( p+q )n, and since p+q = 1 the distribution is normalised:

11,;0

nnn

x

q p pn x P (5-12)

The expectation value is given by:

np pn x xP xn

x 0,; (5-13)

A measure of the spread of this distribution about the mean value is given by the standarddeviation as described previously. In this case:

npq (5-14)

Example: What is the probability of throwing a 6 five times in ten throws of the dice?

Ten throws of the dice means thatn = 10. p = the probability of throwing a six = 1/6,q = the


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probability of failing to throw a six = 5/6. Putting these numbers into equation (5-11) givesthe probability distribution shown in Figure 5.3, below. By calculation and from the figurethe probability of throwing a 6 five times in ten throws of the dice is 0.013.

x

0 2 4 6 8 10

P ( x ; n , p

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

n =10, p =1/6, q = 5/6.

Figure 5.3 Binomial Probability Distribution for throwing a number x of 6's from ten

throws calculated using equation 5-11.

5.6 Poisson Distribution

This is appropriate for the statistics of random events, such as lightning strikes etc. where theexpectation number of events, , is known. The probability of having x events is given by:

!

; xe

x P x

(5-15)

The expectation standard deviation is given by:

(5-16)

This distribution represents an approximation to the binomial distribution for the special casewhere the average number of successes is comparatively rare, very much less than the possible number, i.e.


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follows from equation (5-11) that the Poisson distribution is also discrete and asymmetric, but the asymmetry becomes less apparent with increasing expectation value . This is shown by the two distributions in Figure 5.4 with different values of .

x

0 2 4 6 8

P ( x ;

)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

x

0 5 10 15 20 25

P ( x ;

)

0.00

0.04

0.08

0.12

0.16

= 1.7 = 10

Figure 5.4 Two Poisson distributions. The asymmetry decreases as the mean increases. A

continuous curve is shown although the function is only defined at the integer values shown

by the dots.For large the Poisson distribution closely approximates a Gaussian distribution described inthe next section. The mean value is given by:

np (5-17)

One very important aspect to note here is that the ration of the standard deviation to the meanis:

1

(5-18)

This means that the percentage error in the mean decreases as the mean increases - a usefulfact to keep in mind for experiments where the Poisson distribution applies.

Example: Radioactive counting.

Suppose we have 0.5mg of 238U which contains about 1.3x1018 nuclei. This number is n.These undergo -decay with a half-life of about 4.5x109 years which means that the probability p of decay of one nucleus in one second is about 5x10-18. Thus in 0.5mg the meanvalue in a one second interval of the number of counts isnp=6.5. P(x; 6.5) is the probability of observing x counts in a one second interval.


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6. Propagation of Errors

6.1 Mathematical Background

In this section we set out the general rules for determining the error or precision of a derived

quantity in terms of the errors of each of the directly measured properties. This is known asthe propagation of errors.

In the last two chapters we have seen how best estimates of the mean and the standard error in the mean can be calculated and the types of distributions that are frequently encountered.However, in experimental science the best values of each of several direct measurements of different quantities are often used to derive the value of another property. For example, thevolume of a box is found by multiplying the lengths of its sides while a velocity can bederived from a direct measurement of a distance and of a time. These calculations involvesimple multiplication and division. If our best estimates for the dimensions of the box werethat it had a widthw0, a height h0, and a depthd 0 then the best estimate of the volume of the box is just:

000 d hwV (6-1)

If the errors in the sides were w, h and d , respectively then we could estimate the error involume by expanding the expression for V about the point (w0 , h 0 , d 0) in a Taylor series. Thisgives the change in value V as:

000000 ,,, hwd wd h d

V d

h

V h

w

V wV

(6-2)

where we have neglected higher order terms. The partial differential term:

00 ,d hwV

(note the use of / x instead of d /dx) means differentiateV with respect tow keep h and d constant with values of h0 and d 0, respectively:

0000, 00

d hd whwd

d wV

d h

Substituting the appropriate derivatives in to equation 6-2 gives:

d hwhd wwd hV 000000 (6-3)

Equation 6-3 gives the change in volume of the box for small changes in the lengths of eachdimension. You can see that it is correct, because it is just the area of each face times the

extra thickness perpendicular to the face. In writing equation 6-2 we have assumed that thechanges are small. If w, h and d all change there are some cross-terms missing, but negligible,as can be seen by calculating the change in volume exactly:


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d hwd hwhd wwd hd hwhd wwd hd hwd hwd hwhd wwd hd hwhd wwd hd hw

V d d hhwwV

000000000

000000000000000

000 ))()((

where the round bracket contains the first-order terms which we keep in equation 6-3 and thesquare bracket contains the second and third order terms we neglected.

The above equations discuss small changes in general, which can be positive or negative. Inthe case of an error, the exact magnitude and sign is not known, and if we are interested in thelikely error in the volume of the box due to small uncertainties in the lengths we should notsimply add all three terms in equation 6-3. This is because it is actually quite unlikely that allthree lengths will have a positive error of one standard deviation. In all probability onequantity might have a positive error, one a very small error, and the other a negative error,and this would tend to cancel out a bit giving a smaller error inV than 6-3 would suggest. In

fact, it can be shown (quite simply, ref [2]) that the error terms should be squared and addedthen square rooted, rather than simply being added. For the example of the box:

200200200 d hwV shw sd w sd h s (6-4)

where we use the notation V s to mean the standard error in the mean value of V etc.

In general, if the derived quantityu is related to some independent directly measuredquantities ( x, y, z, . . . ) by a functional relationship:

,,, z y x f u (6-5)

where the function f may be additive, multiplicative, exponential or some other combination.The function f is assumed to be continuous and differentiable.

We also assume that the probability distributions for x, y,z etc are well behaved so that theexpectation or mean value of the observed quantityu is defined by:

,,, z y x f u (6-6)

where the bar is used to signify the expectation or mean value. The expectation error in themean value of u is given by

2

,,,

2

,,,

2

,,,

z y x z

z y x

y z y x

xu z u

s yu

s xu

s s

(6-7)

In other words differentiateu with respect to each of the variables that contain significanterror and evaluate the differentials at the mean values, multiply each differential by itscorresponding error, square each error term, and add them up and take the square root.

Note that it is only necessary to differentiate w.r.t. variables with significant error. Supposethe error in z is insignificant, the z term in 6-7 will also be insignificant, and therefore may be


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neglected.

Sometimes you may come across a function that is very long and tedious to differentiate. Inthis case it may be quicker to calculate the terms in equation 6-7 from the following:

),,,(),,,(,,,

z y x f z y s x f x

u s x

z y x x

(6-8)

which is the change inu due to a change in x of its error. Do the same for each of the other terms which have an error, and then substitute these terms into equation 6-7, i.e. square andadd the changes inu due to each variable, then square-root. Do NOT simply find the changein u when all the variables increase by their error all at once.

You should note the assumptions made in deriving equation 6-7 (eg see Taylor [4]). In particular it assumes that the fluctuations in x and y are small and uncorrelated.

6.2 Specific ExamplesIn the sections that follow a number of examples are given. In each caseu is a function of x and y, whilea and b are constants.

6.2.1 Addition and Subtraction

Consider that the quantities x and y are related by.

byaxu (6-9)

The best value of the derived quantity is given by:

yb xau (6-10)

If the errors in x and y are uncorrelated then, using equation 6-7, the standard error in themean value of u is given by:

2222 y xu sb sa s (6-11)

Notice that the error terms in equation 6-11 are added irrespective of whether there is a plusor minus sign in 6-10. It is also important to note that it is the absolute errors in x and y which are relevant here and not the fractional or percentage errors.

Example: Evaluate z = 3x y for x = 0.8 0.1 and y = 3.0 0.3.

For the sum the best estimate of z is 2.4 + 3.0 = 5.4. For the difference the best estimate of z is 2.4 - 3.0 = -0.6.

If the errors in x, and in y, are uncorrelated then the standard error in the mean in both casesis given by:

424.03.011.09 22 z s


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So the answer is 5.4 0.4 for the sum and -0.6 0.4 for the difference.

Notice how important the errors can become when subtracting two quantities and also thedifference when the measured quantities are correlated.

6.2.2 Multiplication and Division

For multiplication let the functional relationship have the form shown below:

axyu (6-12)

Then the best value of the derived quantity is:

yau (6-13)

For division let the functional relationship have the form shown below:

y x

au(6-14)


y x

au(6-15)

For the product putting the differentials into equation 6-7 gives:

22 y xu s xa s ya s (6-16)

Now dividing through by the best estimate of u gives:

2

2

2

2

y xa

s xa

y xa s ya

u s y xu

22

y

s

x s

u s y xu (6-17)

For the quotient putting the differentials into equation 6-7 gives:

22

y xu s xa

s ya

s (6-18)



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22

y xu s xa y

xa

s xa y

ya

s

22

y

s

x s

u s y xu (6-19)

From equation 6-18 and 6-19 it is clear that the fractional standard error in the mean is thesame in both cases, irrespective of whether u involves products or quotients.

Thus in the case of a product or quotient the fractional standard error on the mean depends onthe fractional standard errors of the directly measured quantities.

Example: Evaluate z = 3x/y for x = 0.8 0.1 and y = 3.0 0.3 .

The best estimate of z is 2.4/3.0 = 0.8. The fractional errors in x and y are 0.125 and 0.10,

respectively. For the standard error in the mean:

16.00.33.0

8.01.0 22

z s z

So the answer is 0.8 16%, i.e. the error is 16% of the best estimate of z . Writing it as anabsolute error gives 0.80 0.13.

Notice the difference between quoting the result with a fractional error and an absolute error.

6.2.3 Power laws

Suppose u varies with a power law, which may be any power - positive, negative (for reciprocals) or fractional (for roots):

baxu (6-20)


b xau (6-21)

Putting the differential into equation 6-7 gives:

21 xbu s xab s (6-22)


b x

bu

xa s xab

u s 1

sb

u s xu (6-23)


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So the fractional error is just multiplied by the power.

6.2.4 Logarithms and Exponentials

If u is found by taking the logarithm of x, i.e.:

axu ln (6-24)

then x xu 1

, so that

x s

s xu (6-25)

Alternatively, if u is related to x by:

bxu exp (6-26)

then bubxb

xu exp

so that:

xu bs

u s (6-27)

Example: Evaluate z = ln(2x) and z = exp(2x) when x = 1.0 0.1 .The best estimate of z in the two cases is 0.693 and 7.389, respectively. For the first examplethe error in the mean is 0.1 giving the answer 0.7 0.1 using equation 6-24. In the secondcase using equation 6.26 the fractional error in the mean is 0.2 giving the answer 7.41.5.


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7. Representing Data

7.1 Drawings, Photographs and Tables

Graphs, line drawings and tables are all concise and easily assimilated ways of presentinginformation if they are properly prepared; they should be used in preference to burying the

information in the text.

7.1.1 Photographs

Photographs should not be used unless there is no other way of conveying the information;line drawings are often to the preferred. For example a photograph of a piece of apparatus can be confusing or useless because of the amount of irrelevant detail it contains, whereas a linedrawing can and should be made helpful by showing, schematically, only the essentialfeatures.

Graphs, line drawings and photographs are all classified as figures. The figures should benumbered serially in Arabic numerals in the order in which they are mentioned in the text andshould be referred to by their figure numbers; for example, A graph of stress against strain isshown in Figure 1. Tables should be serially numbered in the same way as figures, and arereferred to as Table 1, Table 2, and so on. Tables and figures should be numberedindependently of one another.

Each figure and each table should have its own caption. The purpose of the caption is to addenough information to what is in the figure or table itself to enable the reader to get the wholemessage you want the figure or table to convey without having to search through the text of the report. The caption starts with the figure or table number. For example, the caption to a

graph might read: Figure 4: Closed-loop voltage gain in dB of the amplifier shown in Figure3 plotted against frequency in Hz on a logarithmic scale, for different amounts of feedback.The value of the feedback resistor R is shown against each curve.

7.1.2 Tables

Tables should not be used to present data which are included in graphs, unless the data are toa higher precision than can be shown in a graph. In that case you should consider whether thegraph should be omitted.

However, a table is a useful way of grouping data together where they can easily be foundand, if relevant, comparisons between them can easily be made.

The same rules regarding the displaying of units and the avoidance of superfluous zerosapply to tables just as they do to graphs. The precision of a quantity, which on a graph would be shown by an error bar, is shown after the number either with asign or by enclosing theerror in brackets. An example of part of a table is shown in Table 7.1.

Sample Number Length ( m) Length ( m)1 1.1 0.1 1.1(1)2 3.8 0.2 3.8(2)

Table 7.1 Fictional data: the caption should explain in a more-or-less self-contained way themeaning of the columns and rows and where the errors and uncertainties come from.


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Like a graph, a table must have a caption. Apart from any other necessary information itgives the caption should state what the error limits represent, for example standard deviation,or estimated maximum error or whatever.

7.2 Graphs

It is usual to plot the independent variable (that is the one you set to a chosen sequence of valuesin the experiment) along the horizontal axis (the abscissa) and the dependent variable along thevertical axis (the ordinate).

7.2.1 Error Bars

It often happens that the value to be plotted represents the mean of several repeatedmeasurements. The extent of the spread of these measurements about the mean can beexpressed numerically in various ways, for example by calculating the standard deviation,and can be represented on the graph by an error bar drawn through the plotted point and of length equivalent to the spread. An example of a graph with error bars was given on page 11.It is important to inform the reader about the precision of your measurements, and the use of error bars on graphs is a convenient way to do so. What the error bar represents must beexplained in the caption. For example it may represent the standard deviation (calculated bytaking multiple measurements for one of the data points) or it may be your best estimate of the precision likely to have been obtained.

You will find that many papers published in the scientific press contain graphs without error bars. There may or may not be good reasons for their omission but you are not to follow their example: as an undergraduate you must regard yourself as an apprentice physicist and must put error bars in your graphs. If you think you have a good reason for not putting error barson your graphs this should be stated in the graphs caption.

7.2.2 Drawing the Line

Most, although not all graphs have in them not only plotted points representing themeasurements but also a line or lines (either straight or curved) relating to the plotted pointsin some way. Such a line may be based on theory, it may be used to indicate an apparentmathematical relationship between the quantities for which no theoretical basis is known tothe author, or it may simply be a smooth curve drawn as a guide to the eye. In any case the basis for drawing the line should be stated, preferably in the caption.

7.2.3 More than One Line on the Same Graph

A convenient way to make comparisons is to plot two or more lines or curves in the samegraph. In such a case the lines must be distinguished in some way, for example by usingdifferent plotting symbols, by using full ______ , dashed - - - - , or dotted lines ....... and/or bylabelling the lines.

You should not use different colours for this purpose: scientific journals require black andwhite, and colours are not distinguished by the usual photocopiers. Make sure you can

distinguish between different lines and different data points if they come close or cross. Donot obscure the lines in the graph by writing on the graph itself the explanation of what theyrepresent. Put this information in the caption, and keep the labelling on the graph to a


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minimum. An example of a caption where this is done has already been given: part of thecaption reads The value of feedback resistor R is shown against each curve. Then the labelin the graph itself is just the resistance value. If for some reason this technique cannot be usedthen the curves can simply be labelled A, B, C etc. and the significance of the labelsexplained in the caption.

Label the axes to show what quantities are plotted and what units are being used. Units withany prefixes attached to them should be enclosed in curved brackets. This is a conventionemployed by many but not all scientific journals.

7.2.4 Linear Scales

Length (10-5m)

0.0 0.5 1.0 1.5

Length (m)

0.000000 0.000005 0.000010 0.000015

Length ( m)

0 5 10 15

(i)

(ii)

(iii)

Figure 7.1 Three linear scales showing different prefixes for the same range.

A linear scale is one where equal increments of a quantity are represented by equalincrements of distance along the scale. Logarithmic scales are discussed in a later section. For a linear scale, mark graduations at equal spacing along the axis. Insert numerical values atregular intervals which are spaced widely enough to avoid the axis becoming cluttered withnumbers. It is helpful to the reader (and it may help you when plotting) if a scale is chosenand marked so as to make numerical interpolation easy.

Superfluous zeros in the axis markings should be eliminated by changing the unit in whichthe value is expressed to a suitable multiple or submultiple of the unit. This should be done by using the appropriate SI letter prefix. If, for some reason, none of the SI letter prefixes isconsidered suitable, a numerical prefix can be employed; this is usually a power of ten. Asimple example of three equivalent representations is shown in Figure 4.1.

These three scales all convey exactly the same information. For example, 0.000015 = 15m(= 15x10-6m) = 1.5x10-5m. However, (ii) and (iii) convey it more quickly than (i) andtherefore are to be preferred. Method (ii) uses a standard SI prefix () and is therefore to be preferred to (iii) unless there is some positive reason for using (iii).


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Note that the unit, including prefix where used, is placed in curved brackets. This is particularly useful where numerical prefixes are used in emphasising that the prefix belongsto the unit. Avoid ambiguities about whether a scaling factor is attached to the unit or to themeasure; thus Length (10-5m) could also correctly be written Length x 105 (m).

Note also that if the lengths in the above example had run from 1.000000, to 1.000015minstead of from 0.000000m to 0.000015m and assuming that the precision of themeasurements justifies all these digits being given, then the zeros are not superfluous andcannot be removed simply by changing the size of the unit.

Where the quantity to be plotted is dimensionless the option of using multiples or submultiples of the unit is not available as there is no unit. In such a case zeros whichindicate the decimal point position can be removed by plotting a stated multiple or submultiple of the quantity; for example:

Birefringence x 104

0 1 2 3 4 5

Figure 7.2 Scale for a dimensionless quantity.

This means that the numbers marked on the axis represent birefringence values of 0x10-4,1x10-4, 2x10-4, 3x10-4, and 4x10-4.

7.2.5 Logarithmic Scales

A logarithmic scale is one on which the logarithm of the quantity is plotted; thus equalincrements of distance along the scale correspond to equal increments in the logarithm of thequantity. It is convenient here to consider separately scales where natural (Napierian)logarithms are used; logarithms to base e denoted by ln, and scales where the logarithms areto base 10 denoted by log.

7.2.5.1 Natural logarithmic (ln) Scales

Suppose ln p is to be plotted onone axis of a graph and let prepresent, for example, pressuremeasured in pascals (Pa). It isusual to mark on the scale thevalues of ln p ; these values aredimensionless and have no units.However, the unit in which p ismeasured must be shown.Unfortunately there is no generally agreed way of doing this; for the purpose of your

laboratory reports you should do as follows:The inner brackets, round Pa, show that Pa is the unit, and the outer brackets, round p (Pa)

ln [ p (Pa)]

3.0 3.2 3.4 3.6 3.8 4.0

Figure 7.3 : Natural logarithmic scale.


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show that Pa belongs to p and not to the logarithm. The mathematical propriety of taking thelogarithm of a dimensional quantity is not discussed here!

7.2.5.2 Log Scales

Two methods are in common use for logarithms to base 10. Suppose that frequency f

measured in hertz (Hz) is to be plotted on a log scale from 0.1 Hz to 1000 Hz. The firstmethod is the same as for logarithms to basee. That is, the values of the logarithm aremarked on the axis as shown in Figure 7.4(i).

In the second method, the axis is labelled f (Hz) and marked with values of f in Hz rather thanwith values of the logarithms of f . Thus equal increments of distance along the axis do notrepresent equal increments of f : they represent equal increments in log f. It should be evidentfrom the frequency values shown that it is log f which is plotted rather than f . But you shouldalways state in the caption that it is log f which is plotted.

In your laboratory reports you may use specially printed log graph paper. Such paper isavailable with log scales on both axes and with a log scale on one axis and a linear scale onthe other, and with one or more than one decade on the log axis. The log axis is marked off torepresent equal increments of the quantity whose log is being plotted - hence the markingsare not equally spaced, as indicated in Figure 7.4 (iii) which gives an expanded version of two decades of the axis above. You would not insert all these numbers on your graph; theyare given here to emphasise the non-linearity of the scale and the manner in which the scalerepeats from one decade to the next with corresponding values multiplied by 10.

log [ f (Hz)]

-1 0 1 2 3

f (Hz)

10-1 100 101 102 103

f (Hz)

1 10 100

(i)

(ii)

(iii)

Figure 7.4 Log scale where in (i) the log of f has been plotted on a linear scale while in (ii) and (iii) f has been plotted on a log scale.


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8. Fitting Data using the Least Squares Technique

8.1 Introduction

It is often helpful to test the validity of a theoretical model by manipulating it so that a linear data

plot can be fitted using a straight line with an expression of the form:cmx y (8-1)

This is shown schematically in Figure 8.1.

x

y

A

B

C

D

E

F

c

Figure 8.1 Linear fit to data with error bars showing best straight line (solid curve). The

extremes of fit consistent with the error bars are shown by the dashed curves.

How do we estimate the uncertainty in the gradient and intercept?

For most laboratory experiments a graphical method will suffice. The best straight line is firstdrawn AB. Try looking down the line to check this. The line has a gradientm and interceptc.Then draw two additional lines CD and EF so as to go through the limits of the data and their experimental errors. These lines are used to give the limits of the gradient and the intercept.

This is obviously a very subjective procedure, but to help make it rather more objective a few points should be borne in mind;

1. All the experimental data should be given the same weight unless there are specialreasons for rejecting one or more points.

2. If the error bars have been estimated in the recommended way so that the true value istwice as likely as not to lie within the range of the error bars the best line should go throughtwo thirds of the error bars and miss the remaining one third. On this same basis the lines CD

and EF should correspond to the extremes of the line AB that still go through about twothirds of the error bars.


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8.2 Best Straight Line Fit: Linear Regression

There is a commonly used procedure for computing the best fit of a straight line to a set of data points using the principle of least squares. It is a good idea to be aware of this principleeven when drawing the best straight line by eye as describe above. If the data points are

assumed to only have significant error in their y values (NB if they have significant error in x and not y then the axes must be swapped) then the principle is to choosem and c in equation8-1 so that the mean square deviation is a minimum. The deviation of each point is measuredvertically to the fitted line, and is weighted by the size of its error bar. The sum of the squaresof the deviations is also known as 2:

n

i i

ii

scmx y

1

22 (8-24)

This criterion leads to the formulae for the best estimates of the gradientm and interceptc onthe y-axis and additionally the best estimates of the standard deviations of the gradient andintercept. The expressions below have been arranged so that they involve various sums whichare straightforward to tabulate. These are given for the special case when the uncertainties inthe y values are all the same and equal to s y.

22 x x

y x xym (8-3)

xm yc (8-4)

The denominator in 8-3 is just the variance in the x data points, and the numerator is thecovariance of the x and y values (see next section). The error in the gradient and intercept aregiven by

ym s x xn s 221 (8-5)

mc s x s2 (8-6)

You can see from equation 8-5 that in order to obtain a small error inm then the number of data points should be large, the variance of the x values should be large (i.e. there should be alarge spread in x), and the error bars should be small. From Eqn 8-6, for a small error inc, theerror in the gradient should be small but in addition the root mean square value of x shouldalso be small, i.e. there should be plenty of values near x=0.

Many calculators and almost all good graphical software will do this type of data fitting.Remember that in equatios 8-3 to 8-6 that the error in all the y data is assumed to be the same.When this is not the case then the terms in the above equations have to be weighted. Further details are given in a number of texts [2,3].


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8.3 Correlation

In the above section we assumed that there was a linear relationship between the x and y data pairs. However, an important experimental question often asked is whether there is arelationship at all between two measured quantities. A parameter that provides a criteria to

assess this is called the coefficient of correlation given by [3]:

2/122222122 y y x x

y x xy

y y x x

y y x xr

ii

ii (8-7)

where r 2 can take any value from 0 to 1. The numerator is the covariance of x and y and thedenominator is the product of the s.d.s. Whenr is 0 there is no correlation while for perfectcorrelation r 2 is 1. At what value of r should doubts about any correlation change toconfidence? One possible rule of thumb is that the error in the gradient is less than 33%, i.e.

sm < m/3. It can be shown that:

2

22

21

r nr

m s m

(8-8)

Hence our rule of thumb give that:

2173

nr

(8-9)

and the minimum value of r that satisfies this condition decreases as the number of observationsn, increases.

8.4 The 2 Distribution: Testing the Goodness of Fit

The minimum value of 2 obtained during fitting has a probability distribution that can becalculated assuming Gaussian statistics for each data point. The distribution has expectationvalue and variance 2 . This can be used to test if our fit is likely to be the true function if we are confident of the data and the error values we gave. If 2 is within (2 ) then wecan be reasonably confident that our fit is good. Conversely if we dont know the size of theerror-bars, we can assume that the 2 value is equal to , we can work backwards to find theerror in our y-measurements.

The value of 2 which corresponds to the least-squares straight-line fit is related to thecorrelation coefficient and the spread in y values.

r s

y

y

1)var(22 (8-10)

It can be seen that to have a very good (tight) fit the correlation must be very good (r near to1), the error bars should be small, and there should be a large range of y values.


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8.5 Final Remarks

In this section we have only been able to discuss briefly linear least squares fitting of experimental data. The details of fitting data where the error bars are not all the same size hasonly been mentioned in passing and we have not discussed how the data should be analysed

when both the x and y data have significant errors associated with their measurement. Evenmore important is the fact that not all experimental data can be fitted by a straight line. Non-linear least squares fitting is also possible. All these subjects are discussed in more detail byBevington and his book provides a route to a more detailed analysis of this subject [2].


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9. The Laboratory Report

9.1 General Comments

In the first semester you will be required to write up some experiments as formal laboratoryreports. The following notes are intended to help you in structuring your report. Whenever

possible, it is preferable to prepare your report using a word processor. A report shouldconsist of no more than 10 pages in total, including title page (with date experiment started), brief abstract, text, tables, diagrams and references. The text should follow standard format,i.e. aims of experiment, brief theory, description of apparatus and method (to the extent thatthis differs from the laboratory script, which must accompany the report), results, discussionand conclusions. The report should be written so that someone who is familiar with thePhysics course, but not with the particular experiment, can understand it. Try to carry thereader with you so that at all stages of the report they will understand the relevance of whatthey are reading. Be grammatical, concise and lucid. Poor spelling should be avoided; have agood dictionary available during the writing of the report, and ensure you spell-check documents thoroughly. Write legibly and strive for a tidy presentation. Avoid abbreviations.Though some people choose to write reports in the first person it is more usual to write in thethird person.

Writing a report is intended to give you practice and prepare you for a likely future situationin which you have to report on your work to a section leader or to colleagues, or write ascientific paper. So keep it concise, and dont waste the readers time!

The following sections will be found suitable for most of the reports that you will need towrite. If you wish to depart from the suggested arrangement, you may well devise your ownsection headings to suit the material; it is however important to impose a good structure on

your report. The Abstract, Introduction, Discussion and Conclusion sections areessential. Any lengthy material, such as an algebraic derivation or details of a technique thatinterrupts the flow of a report may be relegated to an appendix.

Your report should be presented for marking by handing it in at the FEPS student supportoffice (08AA02). Remember that the academic responsible for marking your report may alsowant to see the associated entries in your laboratory notebook.

9.2.1 Plagiarism and Copying

Plagiarism, or copying other peoples work, is treated very seriously by the University andany student found guilty of committing plagiarism will be subject to the penalties set out inthe Universitys regulations. The guidelines that this Department applies to plagiarism areexplained in the Undergraduate Student handbook, the main features of which are reproduced below.

As part of a degree programme students are required to submit various types of coursework for assessment (examples include essays, laboratory reports, computer programs anddissertations). Whilst researching work students will normally read other people's work in books, journals, conference papers and lecture notes and therefore students should be awarethat plagiarism occurs in the following cases:-

Reproduction of all or part of the work of any other student or external author;


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Inclusion of portions of another text in your own work;

Copying of phrases or sentences, or direct paraphrasing of these;

Copying previously assessed work of your own without the agreement of your lecturer.

In many cases it is necessary to include quotations, sentences and paragraphs of other people's work, be it published or unpublished, in order to highlight a particular point. In suchcases, any included text from another source (apart from that containing common knowledge)must be indicated by quotation marks or indented paragraphs that clearly identify the exactextent of this borrowed text, together with appropriate references.

Within the context of writing your laboratory report, these guidelines apply in just the same wayas for writing an essay etc. In particular you should realise that your laboratory report must bewritten by yourself and in your own words, and should be distinctly different from that of your laboratory partner. We expect partners to discuss together the analysis and presentation of their

experimental data, but when it comes to writing the laboratory report this should be doneseparately. We do not expect to see large elements of the laboratory script reproduced in thereport.

9.2.2 Title, Authors and Affiliation

You should think carefully about the title of your report. Use the title to give the reader someinformation about what they are about to read. Remember to include the names of all the authorsand where the work was done. It may be obvious now, but it is not always the case.

9.2.3 AbstractThis should not be much longer than about one hundred words and should mention both the kindof measurements made and the methods used. It should also summarize the numerical resultsobtained, or, if relevant, state the qualitative results. It is most important that the prospectivereader should be able to find out whether the report is of interest to him or her based on thissection alone. This section is independent of the rest of the report and should be written when thereport itself is completed. It should make sense if it is read on its own in complete isolation fromthe report itself. It should be concise and informative.

9.2.4 Introduction

This section should set the work in context, both in terms of previous research in the subjectand reasons why the experiment is interesting, or useful or relevant. The purpose of theexperiment and the choice of method should be mentioned. In a research report proper theintroduction would include a mention of previous work that is relevant. It is best to write theIntroduction last when you are quite clear what it is you are introducing. Do not copymaterial directly from the lab scripts but do try to be selective.

9.2.5 Theory

Give the necessary theoretical background for the understanding of the experiment. Do notderive results available in standard texts or derive the theory of any standard instrument. Insteadyou should cite the appropriate references. Modifications of standard theory should be discussed


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and, in all cases, the expression which is used to derive your final result should be stated andsymbols defined. Again this section is most easily written after you have written the results anddiscussion sections as then you will know what parts of the theory are relevant.

9.2.6 Experimental Arrangements and Techniques

Include brief explanations of any diagrams of apparatus - remember that well thought outdiagrams are far superior to lengthy descriptions. Give detailed specifications of apparatuswherever it is of critical importance. Call all diagrams "Figure 1, 2 . . ", and refer to them as suchin the text, drawing the reader's attention to their existence at the earliest point at which it would be useful to do so. Provide all figures with a caption.

9.2.7 Procedure

Describe the experimental procedure adopted to obtain the data - but don't include theobvious or the trivial. Include any precautions which were necessary.

9.2.8 Results

Results should be presented graphically wherever possible, otherwise numerical resultsshould be arranged in as convenient and concise a form as possible. A brief verbal account,referring to the g

MSc Laboratory Handbook 2012-13

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