Department of Physics, University of Windsor

Laboratory Handbook

Prepared by Dr Tim Reddish (for 64 - 311)

Containing General Information on Laboratory Procedures, Report Writing and an Introduction to Error Analysis.

1. Aims and Objectives: What is the Laboratory for?

There are several reasons why Laboratory is part of Physics programs. An obvious one

perhaps is that the laboratory provides a way of learning physics that sitting in lecture theatres or

poring over books in a library does not. Often an intellectual idea presented in the lecture or

textbook is given a more tangible reality by the experience of the laboratory. To say that "light

consists of photons" is all very well, but to sit with some apparatus for a couple of hours, in

which this very phenomenon is happening, can lead to a much deeper appreciation of what this

statement really means.

However, the laboratory is not intended to be simply an extension of lecture courses.

Spending time in the laboratory helps the scientist to become comfortable in making

measurements and observations, and provides an important reminder of where "physics" comes

from. This reminder can be especially valuable to aspiring theoreticians. For all physicists the

laboratory fosters an understanding of the relationship between the models of physics and the real

physical world.

In the last two paragraphs, the justification for undergraduate laboratory work has been in

terms of "education". The laboratory can also provide "training". There are several aspects to

this training: it leads to generally being able to feel at home in a technical environment and one

becomes acquainted with some ‘simple’ technology (for example: amplifiers, lasers, circuits,

spectrometers, power supplies, computers, oscilloscopes, multi-meters, etc...) during the early

part of an undergraduate program. In addition to these technical aspects some of the training has

more general application. The preparation of a scientific report is an exacting task, although it is

not too difficult to master the form, which is highly prescribed. The content is another matter,

and explaining the rationale and the outcome of a physics experiment is an excellent

communications exercise.

A further and valuable element of general training - often not fully appreciated by experts in

"personal transferable skills" and which is particularly developed by the practice of science - is

the ability to access the significance of numerical data. This skill is equally applicable to

assessing the data of social trends, sales figures or sports performance, for example. A scientist's

concern about the "noise" in data, and the effect this has on the conclusions that may be drawn,

plays a large part in this skill. (For example: Are the two numbers 90 and 100 significantly

different? Answer: It depends on their uncertainties, i.e. what we mean by ‘significant’) In

2

addition there is the scepticism and the probing which are second nature to a scientist before data

are accepted at face value. Although an ability to quickly see when the results of a calculation

are arithmetically sensible is also part of the skill, the skill is not just "numeracy". By analogy

with "computer literacy" this competence with data might be called "data literacy".

In conclusion, approach the laboratory in a positive way. Try to see it not as a series of

hurdles that must be cleared to gain assessment marks, but rather as an interesting way of

learning about the essence of physics and acquiring useful skills. Believe that you are in the labs

not because it is a requirement but because of the curiosity, which you share with your professors

and all those involved in science, about the physical world.

2. Some Mechanics of the Laboratory.

Students will generally work in pairs. This introduction to small ‘team work’ is important, as

working effectively with others to produce a successful outcome is increasingly an expected and

necessary skill in the workplace. Nevertheless, any irreconcilable problems should be taken up

with your professor.

The laboratory programme is designed so that the student can acquire and develop

competence in laboratory work, in coping with "errors" and preparing reports. This competence

is assumed - and will be built upon - in later years. It is vital that full advantage is taken of the

laboratory time and that the student learns as much as possible from the TA’s during that time.

Unlike the lecture courses, no specific textbook is recommended for the laboratory aspect,

(although a list of useful books appears on the last page). There is usually adequate coverage of

theory in the laboratory handouts and some of the theory related to the assignments may be found

in the textbooks recommended for the lecture courses. Sections 4, 5 and 6 of this handbook

should be regarded as the course text specific to this course. Make good use of them throughout

the year.

3. In the Laboratory

3.1 Teaching Assistants (TA)

The TA will give any specific instruction needed in the use of equipment, explain the

experiment, and provide help when needed. While demonstrating, the TA’s are not concerned

with assessing the students' work, so students should not feel inhibited about consulting the TA

3

during the laboratory session. Towards the end of an experiment, students should check with the

TA that data has been processed in the approved manner and discuss, in general, the outcome of

the experiment. Experiments will end with a brief interview, where the TA and/or professor will

check your understanding, measurement method and analysis, and the use of your laboratory

notebook.

3.2 Laboratory notebooks

Good use should be made of a ‘ bound laboratory notebook’ (see Section 4). It does not

have to be a paragon of neatness but should be a genuine "day book" which contains sufficient

notes and sketches together with the data, calculations and graphs. The notebook should contain

the "raw" measurements and your estimate of their uncertainties. Ensure it is legible and contains

all the necessary information to enable a proper report of the experiment to be written up – even

at a later date if required. It is not good practice to record readings on sheets of paper to be

copied up later, or to do mental arithmetic, or some other processing, and record only the

processed data in the book.

3.3 Report writing

Students will be asked to write up formal reports for certain the experiments they have

completed. In preparing the report the student should try to put into effect the recommendations

of Section 5 of this handbook. Students may discuss writing up an experiment with the TA or

professor, but ultimately the student is responsible for the content and style of the report.

3.4 Uncertainties in measurement

An essential part of any quantitative observation or experiment is the estimation and

processing of the uncertainties or "errors". It is often felt that the error is a poor relation to the

result itself and that somehow in an ideal world we would not have to bother with the tedious

operation of calculating the errors. This is a misconception: the result and the error are of equal

importance. The course of science is, at least in theory, directed by the interplay of random and

systematic error. With experience, the process of dealing with random errors becomes a quick

and relatively painless operation. Students should try to apply the contents of Section 6 of this

handbook to the experiments during the year. It is recognised that to the novice, "errors" seems

4

an abstract, dry and difficult matter, and that it takes time to become comfortable and expert with

errors.

3.5 Assessment

The marks obtained from the laboratory interview (weighted 25%), together with marks from

the reports (weighted 75%) both contribute to the laboratory portion of the course mark. The

mark for an ‘interview’ is essentially a measure of achievement as science practitioners: do the

data look reliable; have they been correctly processed including, of course, the errors; does the

notebook contain adequate commentary and give, in general, the impression of at least the

beginnings of professional competence, and do you as individuals (within the pair) appreciate the

‘physics’ within the experiment?

The report, which is to be word processed, is marked in terms of science and as an exercise

in communication. The latter involves both the technicalities, such as having properly numbered

and labelled diagrams complete with captions, and the general quality of the report in terms of

the material being presented logically, clearly and concisely. After finishing a report-designated

experiment, there is a 1 week period by which the report must be handed in - after which NO

credit will be awarded, unless by prior arrangement with the Professor or with documented

medical evidence. All reports must be handed in to TA. For their part, the TA will undertake to

return marked work in good time so that any deficiencies in that work can be remedied in the

next assignment.

3.6 Safety in the laboratory

Everyone in the laboratory has responsibilities for their own safety and the safety of others.

The laboratory safety document (see Appendix) outlines good practice. All accidents are

avoidable, which makes the fact that the victims may be permanently disadvantaged by the

accident that much more tragic.

4. Keeping a record of work in your laboratory notebook.

(i) Date and title. Start by entering the date and the title of the assignment.

(ii) Getting started. After studying the lab script, and possibly discussing with the TA

what you are going to do, make a brief note about what readings you intend to take. A sketch

might be useful here. Also briefly note what will be done with the data, e.g. one variable might

5

be squared and plotted against another and the slope or intercept of the straight line fitted to the

data used to find some interesting result.

Set up the tables in which you will record the readings, put the units at the head of each

column and provide for the recording, with units, of any parameters which will remain constant

while the varying readings are taken. Include or leave space for any extra columns for the

processed raw data, e.g. if logs or averages or squares are to be taken.

Also make provision for recording the associated errors, with units. If the ultimate error

associated with each data point will be different, for example if the data are to be squared,

provide a column for the errors. You might make a brief note about what you are doing here, so

that if, at some later time, you were to come back to your notebook you would not have to waste

too much time trying, and possibly failing, to decipher what went on. As a professional scientist

your note book is the all important source you will need in order to prepare a report, or give a

presentation, or explain the project to a third party (perhaps your boss).

(iii) The experiment. Record the data and estimate any appropriate errors and consider

anything else which may be relevant, for example, it may become obvious during the experiment

that there could be differences between the observation and the associated ideas or theory.

(iv) Processing the data. This may involve 3 stages. The initial processing of data might

involve taking logs, or averaging or squaring the data and entering the derived quantities in the

appropriate column of a table. Make sure to include units. In parallel with this the errors will be

propagated through the same process and also entered in the table.

The second stage may involve the least squares fitting of a straight line to the pairs of data

points, weighted appropriately using the derived errors, and the determination of the slope and

intercept together with their uncertainties. Record all of this clearly in the notebook.

The final stage may involve inserting the parameters of the best fitting straight line in a

physical formula together with other measured parameters to produce a final result. Carry out the

calculation and the parallel error propagation. For this final statement first round the error to one

(or possibly, two) significant figure(s) and then round the result appropriately. Check that the

numbers seem sensible, for example, if the typical errors associated with the measured variables

are about 1% and the final error is 0.001%, is this OK/reasonable?

6

(v) Discussion and conclusions. Say something intelligent and interesting about the

experiment; compare to theory, etc.

Final reminder: is everything you need to write up your work at some later stage (when you have

forgotten all the subtle details) in your notebook? (Sketches, units, constants, x10? etc...)

5. Writing Laboratory Reports

5.1. Who are you writing for?

When writing something, it is a good idea to ask yourself why you are writing it. The

answer usually is that you are expecting someone to read it, and once you have decided who the

reader is going to be, the style and content of what you write should largely follow from that

decision. In the undergraduate laboratories you are in a dilemma, as the person who will read

your lab report is, of course, a TA (or professor) who knows the experiment well. What can you

write for the TA except a basic presentation of results, calculations and error estimate? If you are

to do more than this, you have to accept that there is an artificial element in writing the report. It

is an exercise in communication, not the real thing; but it is a very important exercise. No piece

of research is complete until it has been reported in some form, even when the results are

negative. Otherwise the work has been pointless. Writing your undergraduate lab reports is an

important preparation for professional life, and for communicating in general with your fellow

human beings.

If the TA is not the best person to have in mind when writing the report, who should it be?

Professional scientists almost invariably write for their peers, so you could do the same in writing

your reports. The reader would be someone with a degree of knowledge and experience similar

to your own, a fellow student who is not necessarily going to do the experiment you are

reporting, and whose knowledge of the work will rest largely on your report. You should

imagine that your peer is interested in the results and may want to use then, so it is vital that he or

she knows what your results are worth. The reader will generally not wish to repeat your

experiments, but you should give enough information about your methods that the reader has

confidence in them, and could, using his or her experience and knowledge and with a certain

amount of effort, get into a position to do the same experiment.

7

The most important element of successful communication is to be able to put yourself into

the shoes of the person with whom you are communicating. How does your presentation of your

ideas and information look through their eyes? What do they need, or want, to know? When do

they want to know it? What should be put in or left out? In attempting to communicate you must

be continually making decisions. Make sure every element of your report deserves its place

there, i.e.: be concise and informative, not long-winded waffle! If you are in doubt about whether

something should be in or out, do not take the easy route of including it "just in case". It takes a

lot of experience and practice to become an effective communicator. There is no reason why, to

accelerate the process, you should not discuss a report you are preparing with the demonstrators.

Ultimately, of course, the responsibility for the content and style of the report is yours.

5.2. The Mechanics

Endeavour to obey the rules of grammar and spelling. This is important because too

eccentric a use of language will distract the reader from the substance of your message. Write in

a formal style; you have to be very much in command of your subject to write successfully in a

casual or humorous way about it. Present things in a logical way, sentence by sentence,

paragraph by paragraph. Woolly writing comes from woolly thinking.

Although there will be some variation from experiment to experiment, most of your reports

should include the following headings: Introduction; Theory; Method; Results; Analysis and

Discussion; Conclusions; References. In addition there may be Appendices and

Acknowledgements. It is quite common when preparing a scientific paper to first write the

Theory, Method, Results, Analysis and Discussion sections, and then write the Introduction and

Conclusion. You may find this approach helpful too. With the exception of References and

Acknowledgements you may find it useful to number the sections as this can make cross-

referencing easier.

(i) Introduction. This should be concise and informative, not just short and vague. It

should place the work in some sort of context and explain why the work was done. In a

particular experiment the reason may be one or more of the following, not necessary exclusive,

list:

8

Objective

to test a theory

to observe a phenomenon

to determine fundamental constants

to determine the values of useful or interesting properties

to investigate the characteristics of devices

to look at techniques

(ii) Theory. This section should logically guide the Reader to the principles, assumptions

(e.g. small angle approximation), and the necessary equations that are relevant to the experiment.

It should make clear what aspect of the theory the experiment will test. This section should not

contain complex derivations, which could be referenced from a book or possibly put in an

Appendix.

(iii) Method. This should not be a copy of the instruction sheet details. Rather, describe

the general method concisely, including only details that you consider important. Diagrams are

very valuable, but do not spend time laboriously drawing them if they can be conveniently

photocopied from some source. You can add your own labels to the diagram to help link it to the

text. You must "call up" or refer to the diagram at the appropriate place in the text (i.e. “as seen

in figure 1”...). All diagrams should be numbered and have captions.

(iv) Results In the valuable space of a learned journal it is usually considered

undesirable to present data in duplicate as both graphs and tables. Graphs give the best

impression of what the data look like and are to be preferred. In the context of the undergraduate

Laboratory, however, the raw data may be helpful to the demonstrator, so it is a good idea to

include them somewhere in tabular form in an Appendix. Sometimes the data are not best

presented in graphical form in which case a table must be used in the main report. Such results

may be presented in the form of a histogram. Make sure all graphs and tables are referred to in

the text and properly furnished with numbers and captions, labelled axes and columns, units and

any legends and footnotes needed for clarity. Before going on to the calculations, analysis and

discussion of the next section of the report, write a short description of what the data look like,

9

pointing out any notable features. Figures in this section of the report may contain features, such

as error bars, fitted lines or curves, which anticipate the analysis of the following section. A

word or two about this in the caption to the figure, for example "The line is the least squares best

fit to the data ", will keep the reader in the picture.

(v) Analysis and Discussion This may be the longest section of your report so you

should cast the text into paragraphs and into titled (and possibly numbered) subsections, so the

reader has signposts, landmarks, and points at which to pause and think, to help with a smooth

journey through the report. The first subsection might be 4.1 Analysis of the data. This may

involve calculations from the data and putting the results of the calculations into the formulae of

physical models. It is at this stage that you should make use of what you have learned from

Section 6 of this handbook. The propagation of the errors through the calculation is an activity

on an equal footing with the calculation of the end result of the experiment. Until you have

calculated the error, you cannot write down your result on final form. In scientific papers, the

details of errors calculations would be taken for granted, but they should be included in your

laboratory report.

REMEMBER: The desired quantity = the result ± the (random) error (or uncertainty)

In the next subsection you should discuss the significance of your result in the context of

theory. If the theory and experiment agree, you might like to consider how the random errors

could be reduced, so that the theory could be tested more severely. If theory and experiment do

not agree, you should review the likely importance of various sources of systematic error, and

discuss how the method, or the theory, might be modified.

(vi) Conclusions. Do not vaguely write here, for example: "The theory and experiment

were/were not in agreement". A conclusion may often be in two parts: a summary of what your

data became after some sort of analysis and processing, and a comment on this. You might write

something along the lines of "Y was found to depend on X, and could be fitted to the function

Y(X), which is consistent with Hooke's Law", or "The magnetic permeability of monel metal fell,

as temperature rose, reaching a critical point at 2 ± 5 °C. This behaviour is consistent with ideas

about the effect of thermal agitation on magnetic order".

10

The conclusion may contain some comment on the method, perhaps how it could be

developed to reduce random or systematic error. The conclusion should not be about you: "I

found the experiment difficult", "The equipment was useless", "The experiment was easy to do,

and I learned a lot about oscilloscopes". Rather, talk about these things to your TA!

6. Measurement and Errors

6.1 Introduction

Any measurement or the result of a scientific experiment must be accompanied by an

estimate of its associated uncertainty or "error". Although quantitative statements found outside

the scientific context seem never to be accompanied by estimated uncertainties, one assumes that

this is taken into account in, for example, quoting the measured quantities of athletic

performances or in making statements about social trends (e.g. unemployment).

Involvement with the measurements and their uncertainties leads to ability in appraising data

generally so that one knows what quoted figures are worth. This is a "transferable skill" (like

literacy, numeracy, computer literacy) which could be called ‘data literacy’, and which must be

an essential skill for anyone having to decide a future policy on the basis of numerical evidence.

6.2 Random error

If we exclude careless error (like misreading a scale, forgetting to check a zero reading –

which is an error in ‘calibration’), errors are not really about the experimenter but about the

experiment. The errors (better named uncertainties) are as much a result of the experiment as the

result itself. An experiment does not take place in isolation from the rest of the world, but is

subject to the effects of a multitude of external influences. If equipment is sensitive it will

respond to temperature fluctuations, draughts, vibrations, varying electrical and magnetic fields,

and so on. The effect we wish to measure is the "SIGNAL" and the unwanted signals (telling us,

say, that a cloud has momentarily covered the sun and that the laboratory temperature is falling)

are "NOISE". When there are many independent contributions to the noise, the noise will be

random and produce a random error. The experimenter's response to random error is to improve

the screening of the experiment from external perturbations to reduce noise, and to extend the

measurement time to more effectively average out the noise: i.e. to apply some kind of

"filtering".

11

(i) High signal-to-noise ratio. In the undergraduate laboratories signal-to-noise ratios

are often high so that readings do not usually have to be taken from visibly fluctuating meter

needles or digital displays. Even so, there is still an experimental uncertainty. This comes from

the scale from which the reading was taken and is sometimes called the resolution error.

Suppose we read an analogue meter, marked in divisions of 1/10, and decide that we can

judge the reading to the nearest scale division. The reading is (say) 3.3. Evidently it is not 3.4 or

3.2, which suggests that the reading lies between 3.25 and 3.35, i.e. it lies within ±0.05 of the

quoted value. If a 3.3 value is not accompanied by the uncertainty, the ±0.05 error can therefore

arguably be assumed to be the upper limit in the value’s uncertainty (i.e. ± half the smallest scale

division). This should not encourage you to be lazy and omit error values, but it can help you

interpret numbers that are quoted without apparent errors. Moreover, it emphasises that readings

(or computed values) must be rounded to the appropriate level of accuracy.

But the question remains as to what does the 3.3 ±0.05 reading/uncertainty actually means

when, in this situation of high signal-to-noise ratio, the reading is not fluctuating? As the ±0.05 is

limited solely by the scale resolution of the instrument (whether analogue or digital), it tells us

that the measuring instrument is too crude for accurately determining the desired quantity. If this

were a length measurement, for example, we should not use a metre rule where a micrometer

would be more appropriate. If a better measuring tool is available, then use it. This would, in

turn, give a spread of values upon repeated reading, which could be processed to give a mean

value and a genuine statistical (random) error (as will be discussed in the next section).

Consequently, a resolution error should not be interpreted in a statistical manner, but seen as a

fundamental limit for a particular measuring device.

(ii) Low signal-to-noise ratio – least squares fitting.

Suppose we are measuring x in the presence of noise. The value of x fluctuates with time

(Figure 1) and we collect a set of n varying readings (Figure 2). How do we define what is the

signal? The criterion to be used is that the target value of x is the "least-squares" best fit. This

will be the value that is closest to each of the population of readings. We will call this target

value X, and the criterion to be satisfied is that the sum of the square of the deviations (di) of X,

which is defined as (X-xi), from each xi is a minimum, i.e. ( X xii

n− )

=∑ 2

1is a minimum.

12

0 50 100

1x(t)

Time (t) Figure 1. Time variation of x.

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40

1

x i

Figure 2. Distribution of 50 readings of x, showing their scatter.

We find what X satisfies this condition from evaluating ddX

X xii

n( )− =

=∑ 2

10 (i.e., by

determining the gradient of this function with respect to X and taking this turning point to be a

minimum.) This can be evaluated as: 21

( )X xii

− = 0n

=∑ , or by expanding the brackets as:

. By rearrangement so that X is the subject of the equation: nX xii

− ==∑

10

nXx

nX

ii == ∑

=1

1 n, i.e.

the target value (X) is the arithmetic mean of the data set. Alternatively, we have just shown that

the mean is simply that value which minimises the sum of it’s squared deviations - or the ‘least

square deviations’. This may seem a long-winded way of defining the ‘mean’, but, as we shall

see, the ‘principle of least squares’ can be extended to more complicated situations.

Having determined the signal, the next step is to determine the noise. We can construct a

histogram of the data (Figure 3). If our data set is very large so that we can make the ‘bin’ width

very small, then the histogram will approximate to a smooth curve (Figure 4).

13

0.5 1 1.50

10

20

50

0.5 1 1.50

50

100

150

500

0.5 1 1.50

5000

1 104

1.5 104

50000

Figure 3. Histograms of the scattered readings of x, for increasing n.

0.5 1 1.50

5000

1 104

1.5 104

0.85 1.15

Figure 4. With an infinite number of values for x, the histogram becomes a smooth curve. (X = 1.0, σ = 0.15)

The higher the noise, the broader the curve will be.

A measure of the breadth is obtained by taking the

width (±σ) of the central section which contains

about 2/3 of the total area under the curve. σ is,

therefore, a measure of the noise, and will be the

average (root mean square –‘rms’) distance of an

individual measurement xi from the centre of the

distribution. Notice the symmetry about the mean

value; this is because there is an equal chance of a

specific deviation above or below the mean value

in an infinite set of measurements. In a finite set

of data (represented by a histogram), we get only

an imperfect picture of the distribution curve and

we can only get an approximate value of σ.

As mentioned earlier, σ is the ‘rms’ distance of an individual measurement xi from the centre of

the distribution, which can be expressed formally as: σ = −=∑1 2

1nX xi

n

n( ) . However, this

equation is not strictly valid for a real experiment with a finite number of data points and it has to

14

be modified to give the best estimate of the standard deviation: ∑=

−−

=n

nixX

n 1

2)(1

1σ . We

now have (n–1) independent values of di = ( ixX − ), rather than n, on the bottom of the

expression because we have already used the data set to get X . Notice that we do not reduce σ

by taking more readings, we simply determine it more accurately. It is also worth noting (for

future reference) that the symmetric continuous curve in figure 4 is called a ‘Normal’ or

‘Gaussian’ distribution. It is usually assumed that random errors follow such a normal

distribution, which can be interpreted in terms of probability. Therefore, a specific measurement

has a 68% chance of being within ±1σ of the mean value: 95% within ±2σ and 99.7% within

±3σ.

Note: (a) σ2 is also known as the ‘variance’, (b) a calculator generally gives σn not σn-1, unless

both buttons are indicated - use the proper function (if in doubt refer to its instruction manual

where the formulae should be given). Be aware that this ‘σ ‘ is not the standard deviation of the

mean (σm ) - discussed below.

We can now use the noise figure to find the error in the

mean ( X ), i.e. the average distance between the X

which has been determined from a finite set of n

readings, and the true value which could only be

determined by taking an infinite set of readings.

Suppose the set of n readings is taken, independently,

over and over again, then each histogram will be

slightly different and would have a different X

because each is only an imperfect version of the

complete distribution curve. We therefore get a set of

X which can themselves be displayed in the form of a

histogram or curve if we had a very large number of

sets of n

0.8 1 1.20

1

2

3

X σ X σX = 1

Figure 5. The distribution of X

from an infinite set of histograms,

compared with the broader

distribution of figure 4.

readings (Figure 5). The distribution of the values of X will be narrower than the distributions

of x because the calculation of each X has already averaged out the noise to some extent

15

(depending on how big n is). The width of the distribution of means (σ X = σm) can be shown† to

be:σ σm n

= and σm is called the standard deviation of the mean and is the typical distance

between the true value and any one determination of X . Thus the result of the experiment is

that the desired quantity is given by: x X m= ± σ and note that this quantity does improve with

larger number of data points (n). † This shown later below and also proved in “An Introduction to Error Analysis” by J. R. Taylor (p127-130)

(iii) More least-squares fitting. In many experiments we are not simply taking repeated

readings of x, but taking pairs of readings (x,y) of two variables which are connected together.

Often we want to draw the best fitting or average straight line y = ax + b to our set of (xi, yi). The

slope, a, and intercept, b, of the best fitting line are the signal (i.e. the quantities we desire to

know) and we also need to determine their rms errors, σa and σb. Similar to the previous section,

Figure 6. For the "best-fitting" straight line

the sum of the squares of the residuals, has a

minimum value.

the process involves minimising the

deviations, where the deviations are now

defined as: di = yi - axi - b. The analysis

takes the experimental errors to only depend

on yi, not xi. In practice, this means that y is

variable that has the dominant error.

Another feature of importance is that every

pair of data points (xi, yi) are used to find the

gradient and intercept, and not simply two

co-ordinates as often done in (unacceptably)

naive analysis to find the ‘best’ line through

a data set (see figure 6).

PC-friendly least squares fitting programmes exist and can be downloaded from the Web (eg:

“Gnuplot”) to carry out the error analysis whenever your experiment requires a straight line fit to

data points. Maple, Origin and other data analysis packages exist that can also perform this task.

16

For your information, the best values for a, b, σa and σb from N (xi,yi) are given by:

aN x y x y

N x xi i i i

i i=

−

−∑∑∑

∑∑( ) ( )(

( ) ( )2 2)

∑ ∑−=

22

22

)()( ii

ya

xxN

Nσσ

bx y x x y

N x xi i i i

i i=

−

−∑∑∑∑

∑∑( )( ) ( )(

( ) ( )

2

2 2i )

∑ ∑

∑−

=22

222

)()(

)(

ii

iyb

xxN

xσσ

where σ y i i iNy ax b

Nd2 2 21

21

2=

−− − =

−∑ ∑( ) and ∑ implies i

N

=∑

1.

The above formulae assume that the random error for each yi (i.e: σ yi) are approximately

the same. If they are very different for some reason (like if - in a certain experiment - it is

inherently more difficult to measure small quantities than large ones and so the corresponding

random errors are significantly different) then the equations can be ‘weighted’ so that more

emphasis is placed on the more reliable points in the least squares fit. ‘Gnuplot’ has this facility

and the appropriate formulae can be found in text books on error analysis.

When plotting graphs, the least squares best fit line is also shown along with the data points

and their error bars which (usually) correspond to ±1σ yi. Note that it can be anticipated that, if

the random errors follow the normal distribution, then 68% (or 2/3) of the error bars would touch

the best fit line. You should also be aware that the above least squares fitting method is the

simplest form of data analysis for a straight-line graph. More advanced methods exist that clarify

and extend this least squares fitting approach.

6.3 Propagation of Errors

(i) Functions of one variable.

Take the example of determining the area of a circle: A = πr2, given that we determine

(from a series of measurements) a mean radius ( r ) and the uncertainty in the mean: σ r . The

value of A

17

0 2 4 60

50

100

A,area

r, radius Figure 7. The area of a circle as a function of radius (curve). The dashed line shows the gradient at r = 4.

is simply πr 2 , but the uncertainty of A (σA ) needs to

take into account the sensitivity that small changes in

r would have on the value of A. This is equivalent to

knowing the gradient of A (at r ), which can be

found by differentiation (see Figure 7), so that:

σ σA rdAdr

= = π σ rr2 .

This expression can easily be generalised for any

function containing only one variable.

(ii) Function of several variables.

In the determination of physical quantities it is frequently necessary to measure several

parameters and then combine these in some formula. Each parameter - assumed to be

independent from one another - will have its corresponding random error (σ), as discussed above.

The question now arises as to how one finds the overall error in the combination; this is known

generally as the ‘propagation of errors’.

Consider the example: V = πr2 h (the volume of a cylinder) given that we know r r± σ and

h h± σ . We need to evaluate a method for determining what effect small changes in both r and h

would have on the value of V. The surface of V as a function of r and h is shown from two

perspectives in Figure 8. As the measurements of r and h are independent from each other, then

(A)

(B)

Figure 8.

The surface plot of a cylinders volume (V) as a function of both r and h as viewed from two positions.

by plotting them on orthogonal axes one can see that the gradients are different (for a specific

point on the surface) when viewed from different directions. The gradient in the h direction can

be evaluated at the point ( , , )r h V by keeping r constant, likewise the gradient in the r direction

can be found by keeping h constant. This procedure is called “partial differentiation” (denoted by

∂V/∂r rather than dV/dr), resulting in an overall uncertainty in V of:

( ) ( )σ σ σ π σ πV r h r hVr

Vh

rh r22

22

2 2 2 2 2 22=∂∂

⎛⎝⎜

⎞⎠⎟

+∂∂

⎛⎝⎜

⎞⎠⎟

= + σ

The terms are added as ‘squares’ as this represents the best estimate of the overall uncertainty

(the rigorous proof can be found in text books on error analysis.) To justify this in a

‘handwaving’ way the sketch of the triangle shows that these independent errors (shown at right

angles to each

other) - which also need to be weighted by the appropriate (gradient)2

- are combined using Pythagoras’ theorem. Note that because they are

added as squares, the errors can never cancel each other out.

If the quantity of interest, X is a function of n independent variables

(a,b,c....), the propagation of errors equation can be generalised to:

σ σ σ σX a b cXa

Xb

Xc

22

22

22

2=∂∂

⎛⎝⎜

⎞⎠⎟

+∂∂

⎛⎝⎜

⎞⎠⎟

+∂∂

⎛⎝⎜

⎞⎠⎟

+.....

This is a general equation, you should memorise it and know how to use it.

However, there are some special examples which can all be proved by applying the above

equation: Examples (a) and (b) are particularly useful as many equations can, with appropriate

simplifying substitutions, can have those forms.

a) If X = a ± b ± c ±... (sums and differences), then σ σ σ σX a b b2 2 2 2= + + +.....

(Regardless of the ± signs, the errors add in quadrature).

19

b) If X abcd

=......

(products and quotients) then

σ σ σ σ σX a b c dX a b c d

⎛⎝⎜

⎞⎠⎟

= ⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

+2 2 2 2 2

... i.e. the fractional or percentage errors add in

quadrature.

c) The error in the mean. As a final example we use the propagation of errors equation to

work out the error in the mean (σ X ) of a set of readings (xi) in terms of the error in each reading

(σ xi) and the number of readings (N).

As: X x x xN N

xNi

i

N=

+ + +=

=∑1 2

1

1......, then

σ σ σ σX x xN

xi

xi

NXx

Xx

Xx

XxN i

2

1

22

2

22

22

22

11 2

=∂∂

⎛⎝⎜

⎞⎠⎟ +

∂∂

⎛⎝⎜

⎞⎠⎟ + +

∂∂

⎛⎝⎜

⎞⎠⎟ =

∂∂

⎛⎝⎜

⎞⎠⎟

=∑...... σ

as each xi is a measurement of the same true value, their σ xiall have the same value - which we

denote with σ x , and the partial derivatives are simply 1/N for all xi. Consequently σ X becomes:

σ σσ

X xx

i

N

NN

N2

22

2

21

1= ⎛

⎝⎜⎞⎠⎟

==∑ , or σ

σX

xN

= as given earlier.

6.4 Systematic Error.

The random errors that have been discussed above are not the only source of error. Other

sources may contribute to systematic errors in the experiment which results in a discrepancy

between the ‘measured’ value and that given by the theoretical ideal or model. Is the theory or

experiment wrong? The answer is "neither" – they are different. To reduce the difference either

the experimental technique or the theory must be made more complicated (or both). So the

response of the scientist to systematic error is to do science, to carry out research to investigate

the nature of the systematic error and develop experiments and theories. Thus systematic error

can be regarded as the driving engine of science.

From an experimental point of view one must recognise that no apparatus is perfectly linear,

i.e. if the observable doubles in size the output does not. The accuracy of the calibration (and the

20

zero position) must match the desired accuracy of the experiment. The imperfection of man-made

instruments also contribute to systematic error, e.g. the eccentricity of a spectrometer turntable;

an imperfectly marked scale. Furthermore, the imperfections in the Experimenter in, say,

consistently starting the timer too late or too early, or in failing to reduce parallax error in reading

a scale can contribute to systematic errors. More generally, the differences between the way an

experiment (or system) is actually behaving and the way it is presumed to behave lead to

systematic errors. Carefully designed apparatus, independent monitoring or control of the

conditions which may produce systematic error, and the ability to identify unsuspected influences

lead to the reduction in systematic errors. A general and positive scepticism is a healthy trait in

any scientist. The physical ‘intuition’ in identifying and eliminating sources (or possible sources)

of systematic error is an important skill to acquire and develop.

Recommended Books for Further Reading

1) An Introduction to Error Analysis J R Taylor (University Science Books, OUP)

2) Experimental Methods L Kirkup (John Wiley)

3) Statistical Treatment of Experimental Data H D Young (McGraw-Hill)

4) Data Reduction and Error Analysis for the Physical Sciences P R Bevington (McGraw-Hill)

21

Summary of Main Error Analysis Formulae.

Mean and Standard Deviation:

The arithmetic mean is ∑=

=N

iix

NX

1

1 and the value which minimises the sum of it’s squared

deviations - or the ‘least square deviations’. The standard deviation, σ, which is the ‘root mean

square’ distance of an individual measurement xi from the centre, is ∑=

−−

=N

iixX

N 1

2)(1

1σ

Note: σ2 is also known as the ‘variance’. The standard deviation of the mean is :Nm

σσ = and

is the typical distance between the true value and any one determination of X .

Note: this quantity improves with larger number of data points (N).

Check the mathematical definitions of σ provided by your calculator (see its instruction book).

It is usually assumed that random errors follow such a normal distribution, which can be interpreted in terms of probability. Therefore, a specific measurement has a 68% chance of being within ±1σ of the mean value: 95% within ±2σ and 99.7% within ±3σ.

Propagation of Errors.

If the quantity of interest, X is a function of n independent variables (a,b,c....), the propagation of errors equation can be generalised to:

σ σ σ σX a b cXa

Xb

Xc

22

22

22

2=∂∂

⎛⎝⎜

⎞⎠⎟

+∂∂

⎛⎝⎜

⎞⎠⎟

+∂∂

⎛⎝⎜

⎞⎠⎟

+.....

The following two special examples can both be proved by applying the above equation:

a) If X = a ± b ± c ±... (sums and differences), then σ σ σ σX a b b2 2 2 2= + + +.....

(Regardless of the ± signs, the errors add in quadrature).

b) If X abcd

=......

(products and quotients) then

σ σ σ σ σX a b c dX a b c d

⎛⎝⎜

⎞⎠⎟

= ⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

+2 2 2 2 2

... i.e. the fractional or percentage errors add in

quadrature. These two simple equations can be applied to most experiments, with appropriate substations.

22

SAFETY IN THE LABORATORY All undergraduate students working in the laboratories must observe the following: FIRE PRECAUTIONS:

• Smoking is NOT allowed in any of the rooms. • No connection should be made to the gas mains without the permission of a TA. • Be very careful when using soldering irons - they are a frequent cause of fire.

ELECTRICAL SAFETY

• Do not attach any improvised apparatus to the 120volt a.c. mains without permission of a TA or Technician.

• Always give special attention to safety when using high voltages and/or high currents. COMPRESSED GASES

• Consult a TA, Professor or Technician before using compressed gases. • Always turn off the main valve when you are finished for the day and report this to the

TA. Report immediately any leaks or other defects. SOLVENTS

• Many solvents are toxic and/or highly flammable. Avoid breathing the fumes. • Before using solvents, check that there are no naked flames in the room.

LASERS

• Move the laser as little as possible, keeping the beam at bench height. Avoid specular reflections.

NOTE: You are required by the Health and Safty at Work Act and the University to:

• take reasonable care for your own health and safety and that of others who might be affected by what you do or do not do.

• cooperate with the University on health and safety. • not interfere with or misuse anything provided for your health, safety andwelfare.

IF YOU THINK THAT THERE IS A HEALTH AND SAFETY PROBLEM, PLEASE DISCUSS IT WITH THE TEACHING ASSISTANT OR PROFESSOR.

23