Data Analysis, Lecture Notes - Engineering - 1, Study notes of Engineering Physics

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LECTURE NOTES

Part II

Keele University Physics/Astrophysics Laboratory 2

The Laboratory Notebook

(2.1.a) Introduction Whenever a scientific experiment is performed the results of that experiment must be recorded in some way. However the “record” of an experiment is more than just a list of numerical measurements, it contains information on how, when, why and even where! you did the experiment, what happened during the experiment, in fact all pertinent information. For the experimental strand of this module you are expected to record all such information in a single book, your Laboratory Notebook. Your lab. notebook should be A4 or foolscap size with a sufficient number of pages to last for both semesters of the academic year. The book can be hardback or softback as you like, however you should remember that hardback will stand up to any knocks it receives in the laboratory better than a softback. At the end of each of your experiments your notebook will be examined by one of the staff supervisors of the laboratory, who, based upon the content of your notebook, will discuss your experiment with you and award you a mark for that experiment. Your notebook will also form your source of information when writing up your laboratory reports.

(2.1.b) What Should You Record in Your Notebook? The simple and glib answer to such a question is everything. This is unrealistic of course but you should make every effort to record as much as is practically and sensibly possible about your experiment. It is not possible for anyone to predict the future and it is often very surprising what information you may need to know at a later date. An obvious thing you should record at the front of your notebook is your NAME, ADDRESS and that this is your SCHOOL OF PHYSICAL AND GEOGRAPHICAL SCIENCES notebook. You should take great care not to mislay your notebook but in the event that it is lost this can only increase the probability of its recovery. Each new experiment you start should be started on a new page in your notebook leaving one or two blank pages from your previous experiment. You should record the title of the experiment and at the start of each laboratory session you should record the day and date at the relevant point in the experimental record. The laboratory notebook is by definition a note book, it is not expected to be a beautifully written piece of prose, quite the reverse is often the case. You should record events and results as they happen. However each of your notes should be in the form of a short sentence which is sufficiently explanatory that you or someone else can understand it. The sort of information you record in your notebook will vary from experiment to experiment however some points that should be borne in mind in all cases are;

• Record all of your data/measurements as you take them.
• Always record the units of your measurements along with the measurements themselves.
• If your data is taken for a certain period of time or a certain number of oscillations etc. then always record this fact along with the measurements themselves.
• If you plot or fit your data using one of the computer programs, make a printout of the program output (usually a graph) and attach (glue, sellotape, staple) it into your notebook.

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to combine (add, multiply,.. etc.) them together to get another value still. In section(e) there are some comments on the scale and importance of errors in experiments.

(2.2.b) The Relevance of Errors to the Scientific Process If we have an experimentally measured value X for some quantity with an error bar ΔX (usually written in the format X ± ΔX) then what does this mean? It means that we have a ~ 66%

confidence that the “true” value of the quantity lies somewhere within the range X − ΔX to X + ΔX and a ~95% confidence that it lies within X − 2 ΔX to X + 2ΔX and a ~99% confidence that it lies within X − 3 ΔX to X + 3ΔX. Thus experimentally we don’t determine a specific value for a quantity but rather a range of values in which we believe that the “true” value lies.

What happens if we want to compare our experimentally measured value X ± ΔX with another value for the same quantity? This “other value” could be a value determined by a different method or by the same method but by a different person or it could be the result of a theoretical calculation. Let’s say (for example) that the other value is Y ± ΔY then the question that we ask is

does the value of X lie within the range Y − ΔY to Y + ΔY or does the value of Y lie within the range X − ΔX to X + ΔX

If the answer to this question is YES then we say that the two values “agree within the limits of experimental error”, otherwise they disagree. Two examples would be

i) A measured value of a resistance by Tom of 5 ± 1 Ω agrees within the limit of experimental error with the measured value of the same resistance of 4.7 ± 0.3 Ω made by Harry. ii) The measured value of the density of the Sun of 1410 ± 10 kg m-3^ agrees within experimental error with the theoretical value of 1407 kg m-3^ predicted by the theory of Dr. Who but disagrees with the theoretical value predicted by Mr Spock of 1457 kg m-3.

The conclusions we draw from the size of the error are not however just limited to agreement between two values their implication goes further. In example (i) above Harry’s value has a much smaller error bar than Tom’s and we’d conclude that Harry’s value was more precise than Tom’s. In example (ii) we’d conclude that Dr. Who’s theory was “correct” while Mr Spock’s was incorrect. Although if future experiments measured the mass of the Sun more precisely (smaller error bar) and Dr. Who’s value then lay outside the error bar range we’d conclude that his theory too was now incorrect. What this would probably mean was that Dr. Who’s theory was not sophisticated enough to predict the mass of the Sun to the same accuracy with which it could be measured! The interplay between experimental and theoretical physics revolves around this role of experiment to test the predictions of theory and of theory to accurately predict the results of experiments. In doing this, the error bar, the quantified accuracy of an experiment clearly plays a vital role. (2.2.c) Types of Error The sources of errors in an experiment tend to be classified into one of two types, either random or systematic errors. In the next two subsections each of these two classes are discussed in more detail.

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(2.2.c.i) Random Errors This class of error applies to the situations where a measured value “fluctuates” randomly about its “true” value. Each time a measurement of the quantity is made a slightly different result is obtained. An example used earlier was where electrical noise in a circuit causes a measured voltage to vary. Another example is the nuclear beta particle decay process considered in experiment A where the number of decays in a second fluctuates randomly about a mean value. It is also possible for some measurements which do not possess an intrinsically random nature to be effectively random. Imagine a situation where a “large” group of people are all asked to take a reading off of a scale. It is quite possible for different people, due to parallax errors, to read slightly different values off of the scale so that a random set of values fluctuating about a mean is obtained. A random error is one where in a series of measurements the values obtained are distributed in an unbiased and independent way about the “true” value. This type of error can be dealt with by repeating the measurement many times and then applying the ideas of statistical analysis. If we measure a quantity N times and obtain a set of values x 1 , x 2 , x 3 ,.....,xN then we can estimate the “true” value for x from the mean value, i.e.

x =

(x + x + x +...x ) N

N

n x i=

N i

The random values that we have measured were “generated” by a probability distribution function, which causes the underlying fluctuation in the values. We usually consider that for random independent errors this underlying probability function is a Gaussian or normal distribution given by the equation

P x

x x (^) T ( ) = exp −

⎣⎢^

2

πσ σ^

where x (^) T is the “true” value for x and σ is the standard deviation for the probability distribution. The standard deviation is the quantity that sets a scale for the range of fluctuations that we observe. We have estimated a value for xT from the mean of our measured values, we can also estimate a value for σ from our measurements by using the equation

σ = N

x (^) i x i

1 N

2 − 1

=

However equation(3) is not our estimate of the error bar. This is because we know that there are random fluctuations in the values of x and that our best estimate of the true value of xT was the mean. Our error bar is therefore a value for how accurately we have determined the mean value, not what is our estimate of the size of the fluctuations. The formula for our estimate of the error bar Δ x is

Δ x N

which is the “error in the mean”. If one considers equations (3) and (4) then one can see why they are different. If we took an infinite number of measurements we would be able to determine the mean value of the distribution and hence the true value of x exactly and hence have Δ x =0. However we would still have a distribution of x values, i.e. a non-zero value for σ, because of the underlying fluctuations in the measured values of x. It is obviously not possible to take an infinite number of measurements of a quantity in an experiment, however one should take enough measurements to ensure that Δ x is small enough for the purposes of an experiment. The formulae above give us a way of evaluating the mean, standard

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Some specific examples are

( ) ( ) ( ) ( ) sin( ) sin( ) cos( ) cos( ) cos( ) sin( )

log ( ) log ( )

log ( ) log ( )

log ( )

exp( ) exp( ) exp( ) ( ) log ( )

X X X X X

X X X X X

X X X X X

X X X X X

X X X

X

X

X X X X

X

X X X X X

X X X X X

−

±

2 2 1

10 10

n n n

e e

e

e

n

(2.2.d.ii) Combining Errors A slightly different problem of error propagation in a calculation is how do we find the derived error when we combine (add, subtract, multiply or divide) two quantities where each has an error bar. Clearly the resulting combination will also have an error bar and without going into detail the equations for calculating this resultant error bar are ;

Addition Z = X + Y Δ Z = ( Δ X ) +( Δ Y )

2 2 (6.1)

Subtraction Z = X − Y Δ Z = ( Δ X ) 2 +( Δ Y )^2 (6.2)

Multiplication Z = X × Y

Δ Z Δ Δ

Z

X

X

Y

Y

⎟ +^

2 2 (6.3)

Division Z = X ÷ Y

Δ Z Δ Δ

Z

X

X

Y

Y

⎝⎜^

⎠⎟^

⎝⎜^

2 2 (6.4)

(2.2.d.iii) Combining More than Two Quantities What if we want to combine more than two quantities ?, for example to multiply 3 quantities together. The solution to this problem is to break the operation down into parts which only involve pairs of quantities and then just keep applying the rules above to each pair until one obtains the final answer. For example let us consider the equation

Z

XY

UV

then we break this down into the equations

Z

A

B

A XY B UV

Z

Z

A

A

B

B

A

A

X

X

Y

Y

B

B

U

U

V

V

⎟ +^

⎟ =^

⎟ +^

⎟ =^

⎟ +^

Δ Δ 2 Δ 2 Δ Δ 2 Δ 2 Δ Δ 2 Δ^2

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which leads to the final result

Scales and verniers

(2.3.a) Introduction The minimum separation between graduations on a scale sets a limit to the precision with which a reading can be made. A vernier can be used with a scale to improve the precision of reading by a factor of 10 say.

SCALE

VERNIER

(^4 )

(^0 5 )

Fig. 1

Fig. 1 shows a scale marked in centimetres and subdivided into units of 0.1 cm (i.e. millimetres). The vernier has a similar subdivision with markings at 0, 5 and 10. The 0 mark indicates the desired reading on the scale but as indicated this can only be read easily to a precision of ± 0.1 cm. In Fig. 1 the mark 0 is lined up exactly with a subdivision of the scale, 4.3 cm. However the 10 mark on the vernier lines up exactly with the scale subdivision at 5.2 cm, i.e. ten division of the vernier correspond with nine subdivision of the scale. Clearly the markings on the vernier between 0 and 10 do not align exactly with any scale graduations. The interval between markings on the vernier is 0.09 cm; hence the first vernier mark is 0.01 cm short of a scale line (4.4. cm), the second mark is 0.02 cm short of the next scale line (4.5 cm) etc.

Δ Z Δ Δ Δ Δ

Z

X

X

Y

Y

U

U

V

V

2 2 2 2 ⎛ ⎝

(2.2.e) Using Judgement About the Scale and Importance of Errors In the previous sections we discussed how to calculate and how to propagate errors. Inherent in any good experimental practice is the desire to minimise the size of the error bar in the final result and hence to obtain as precise a value as is practically possible. Since any experiment must be done in a limited amount of time we cannot reduce every error bar in the experiment to “zero”. Instead we must exercise judgement in deciding which are the important errors to reduce. In this context important means those errors which have the largest effect on the final result of our experiment. This means thinking about how that final result is obtained and how each source of error contributes to the error in the final result and then minimising the biggest contributor(s).

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derived values and their errors. If, as in the gyroscope experiment, there are a large number of such values to be calculated a spreadsheet can provide a very efficient way to do this.

(2.4.b) Basic Points about a Spreadsheet

1) Getting data into the spreadsheet In figure 3 an example spreadsheet is shown. After either starting MS-Excel, or MS-WORKS and selecting the spreadsheet program, the user is faced with a “grid” of the form shown in figure 3. If the user has selected an existing spreadsheet then the cells of the grid will probably already be filled in whereas if the user selects a new spreadsheet the cells will be empty. Each cell can be referred to by its column letter and its row number, e.g. A3, C7, etc. The user can select a particular cell by using the mouse to click into that cell. Once in a cell one can perhaps more easily move to other cells by using the up/down/left/right arrow keys. The current active cell is (a) highlighted by a bold border and (b) has its co-ordinates displayed in the leftmost box beneath the toolbox line. A B C D E F 1 t in s T in s dt in s dT in s wW d(wW) 2 0.355 11.40 0.005 0.05 9.755 0. 3 0.330 27.50 0.005 0.05 4.350 0. 4 0.380^ 33.30^ 0.005^ 0.05^ 3.120^ 0. 5 0.390 8.66 0.005 0.05 11.689 0. 6 0.275^ 15.85^ 0.005^ 0.05^ 9.057^ 0. 7 0.310 22.25 0.005 0.05 5.724 0. 8 0.375^ 6.50^ 0.005^ 0.05^ 16.196^ 0. 9 0.320 9.80 0.005 0.05 12.589 0. 10 0.230 23.00 0.005 0.05 7.463 0. 11 0.345 4.40 0.005 0.05 26.007 0. 12 0.290 9.60 0.005 0.05 14.180 0. Figure 3 In order to enter data into a cell simply type in the numbers (or text) that you want to put in that cell. Once finished type a return (enter) and the data will be placed into the cell. You can edit (correct) the value in a cell by selecting that cell with mouse or arrows then clicking into the “cell contents” box on the same line as the cell co-ordinates box. The left and right arrows and delete keys can then be used along with the normal alpha-numeric keys to correct the value.

2) Manipulating cell values via simple formulae Once the data is in the grid cells it can be manipulated. The usual procedure is to write the answer from this manipulation into another grid cell. So for example if we want to multiply the element in A2 by 2 say then (assuming column B is unused) we can click into cell B2 and then in the cell contents box type = 2A2* , the = is important since it tells the spreadsheet that this is a formula. When we press return the value in B2 will be the numerical value of 2A2. What if we wanted to do this for all (or a range of) the values in column A and write the results in column B? Well we must first do one element (e.g. B2 as above) and then use the fill down command (which is under Edit in the bar at the top). First one must select the first element and also the other grid cells for which one wishes to calculate. This is down by clicking onto the first cell/element with the mouse, holding the left bottom down and then dragging down the elements of the column for which one wishes to calculate. Once this is done click on Edit and select fill, and then down or up as appropriate. The values for each element are immediately calculated, so that B3=2A3, B4=2*A4, etc. The example above was very simple and we can carry out more complicated operations. We can of course, add, subtract, multiply, divide ( symbols +, - , * , / ) by constants and also by other cell

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elements. For example if we select cell E2 we could enter in the contents box = A2+B2*(C2/D2)+ which uses the values in columns A, B, C and D to calculate the values in E.

(2.4.c) Function Names We can also take functions of the cell values, for example if we select cell B2 then in MS-Excel in the contents box we can type = SQRT(A2) to take the square root or =LN(A2) to take the natural logarithm. While most of the MS-Excel and MS-Works function names are the same, some are unfortunately different. The following is a list of some of the MS-Excel and MS-Works spreadsheet functions which may be useful to you;

Function MS-Excel MS-Works Function MS-Excel MS- Works Square root sqrt(a1) sqr(a1) Raise to power N

a1^n

Natural log. ln(a1) Log. to base 10 log(a1) Trig. functions

sin(a1), cos(a1), tan(a1) Inv. trig. func. asin(a1), cos(a1), atan(a1)

Sum a1 to a7 sum(a1:a7) Mean of a1 to a

average(a1:a7) avg(a1:a7) Standard dev. of a1 to a

stdev(a1:a7) std(a1:a7)

(2.4.d) An Example --- The Gyroscope Data In the gyroscopic motion experiment you measure the fast and slow periodic times for the

rotational motion ( call these t and T) and from them must calculate the angular frequencies ω and Ω. Furthermore there are errors Δt and ΔT in each of the measured times and what one wants in the end is the product ωΩ and the error in this product ΔωΩ. The formulae one wishes to use are therefore

t T

t

t

T

T

t

t

T

T

ω

π π ω

ω

ω

ω

ω

ω

⎟ +^

⎟ =^

⎟ +^

2 2 2 2

Let us assume that the t values are in column A, the T values in column B, the Δt values in column C and the ΔT values in column D. Then in column E we can get the ωΩ values by selecting E2 and entering the formula = 39.478 / ( A2 * B2 ) which we can then “fill down” into the remaining cells in column E. The next step is to get the error values. We can select F2 and use the following formula = E2 * SQRT( (C2/A2)^2 + (D2/B2)^2 ) which we can then again “fill down” into the remaining elements of the column. Hence we have the values we wanted, worked out for us by the computer.

(2.4.e) Saving, Updating and Printing Save your spreadsheet regularly into a file, you don’t want to have to type it all in again if you just need to change one value. If once you have completed your spreadsheet you find that a value is wrong (perhaps incorrectly entered) then when you change (edit) that value the spreadsheet is

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window and select the particular label to change, entering the new label into the input window in the same way as for the limits. In order to update your plot with these new limits/labels you will need to click on the Plot button again.

(2.5.d) Fitting the data In order to fit the data you have loaded you must first select a function for fitting, click on Function at the top of the window. For straight line fits there are two choices, y=mx and y=mx+c in the drop down menu, click on whichever function is appropriate for your data to select it. Now click on the Fit button at the bottom of the window and the fit results will be displayed in a box at the bottom right of the window. These values are m (the slope), c (the intercept), along with their error bar values, and χ^2 (the goodness of fit parameter). The definition etc. of the goodness of fit parameter and the least squares method used by the Linefit program is described in section 2.8. In order to see the best fit straight line plotted on top of the data you now need to click the Plot button again.

(2.5.e) Printing the plot In order to obtain a hardcopy of your plot click on File at the top of the window and then on Print Plot in the drop down menu. This will activate the windows printing system appropriate for the particular PC you’re using and you should answer the questions as per normal.

(2.5.f) Exitting from the program In order to exit from Linefit click on File at the top of the window and then on Exit from the drop down menu. A confirmation window will appear, click on the OK button to exit the program.

(2.5.g) Taking your own copy of Linefit If you have access to a PC running MS-Windows 95, 98 or NT4 you can take a copy of the Linefit program for your own use if you so wish. On the PC’s in the Physics PC Laboratory and in those in the first year experimental laboratory there should be a directory C:\P1LAB in which there is a file LINEFIT_INSTALL.EXE. This file is 1.24Mbyte in size and will therefore fit on a single blank diskette. It is a “self-extracting exe file”, if you copy the file to a directory on your PC and double click on the file then it should extract a series of files needed to run Linefit on your PC. Note one of the files extracted is called README.TXT and you should read that file for instructions on the installation process. Alternatively the LINEFIT_INSTALL.EXE frile can be downloaded from the Physics Department website. See Dr. A.Mahendrasingam, PHYSICS LABORATORY Module supervisor if you have any questions.

Preparation of laboratory reports

(2.6.a) Introduction In order to be of worth a piece of scientific research must be communicated to the rest of the world. The method for doing this is by “reporting” the work in the scientific literature. In the fields covered by the fundamental sciences this is usually done by publishing a paper in one of the relevant scientific or technical journals which in Physics would for example be the Journal of Physics or Physical Review. For scientists working in a commercial environment their work will have to be reported within the scientific management structure of their company. The format of such a report takes on the universally accepted style of a scientific paper.

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∗ Abstract ∗ Introduction

∗ [ Theoretical Background] ∗ Experimental (or Theoretical) Method ∗ Results ∗ Discussion and/or Conclusions

This scientific paper style is used when reporting the outcome of a single (or a small number of) experiment(s). If a whole program of experiments is being reported then a different format will probably be used, which may be somewhat more akin to the structure of a book. Since in this module you will only be tackling “single” experiments your laboratory reports should be written in this scientific paper format. The following subsections discuss the various aspects of how you should write your laboratory reports. In subsection(b) various general points pertinent to the whole of the report are given. Subsections (c.i) to (c.v) describe what are the expected contents of each of the sections of the report. Since you are expected to produce word processed laboratory reports subsection (d) discusses various points associated with this.

(2.6.b) General Points About Writing a Report There are some general points about writing a laboratory report, or any technical document, which apply to the whole of the report and which are best discussed before more detailed points are considered in subsection(2.4.c). i) All sections of the report must be written in proper English, it is not acceptable anywhere in the report to simply use “notes” or just to have a collection of figures and tables. ii) There should be a progressive logical flow in the content of the material presented in each section and in the report as a whole. In particular a reader should not have to search ahead for an explanation of a sentence or paragraph that they have just read or for the definition of a symbol which has appeared for the first time in the text. In order to do this you would be well advised to make a plan of each section of your report listing, in the order that you wish to make them, the pertinent points for that section. Each new point you make should then correspond to a new paragraph in that section. Depending on the exact content of your report it may be sensible on occasion to go further than this and to subdivide the section into subsections for different sub-topics dealt with within the section. iii) Write in the third person, past tense;-e.g. "The temperature was measured ...." NOT "I measured the temperature ..." iv) Labelling Equations, Figures and Tables :- A particular feature of a technical document is the inclusion of equations, figures and tables in the text. A figure could be either a diagram of apparatus or a graph of experimental results. All 3 of these types of included item should be labelled and in the text you should refer to them by those labels. The label for an equation is usually just a number at the end of the line on which the equation occurs. For a figure or table the label is usually on a line above or below the figure/table and the numerical label is prefixed by either the word Figure or the word Table as is appropriate. The numbering of equations, figures and tables should be sequential in the order that they appear in the text of the report. When referring to them in the text they should be referred to for example as equation(1), figure(1) or table(1). Examples of the labelling of equations, figures and table can be found throughout this manual. v) Always quote the error bar when you quote a result in your text.

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horizontal axis (abscissa). Each of the axes should have a label explaining what the variable plotted represents and what its units are. It is conventional to label an axis by;- name of quantity (symbol) × power of ten unit e.g. mass M × 10 -3^ kg This is taken to mean that, for example, a value on the mass axis of 2.7 denotes a value of 2.7 x 10- kg. If there is not enough space on the axes for a clear explanation or if further explanation is required then a figure caption (a short paragraph of text) should be added below the graph providing the necessary information. A derived result is one that has been obtained by processing or analysing the set of raw data points. An example would be the value of the slope from a straight line fit to the raw data. These results will normally just be quoted in the text of the results section. Only quote the derived result itself do not display any arithmetic performed in arriving at the derived result. If the derived result is taken from a graph then it should be clearly stated which figure it is taken from. All derived results should always have both their error bar and their units quoted with them.

(v) Discussion/Conclusion Comment on the significance of your results. Has the aim of the experiment been achieved? If your aim was to verify a theory, over what range of values has this theory been verified? Did the results deviate from theory outside this range? Are the numerical results of the expected order of magnitude? Do they agree with the generally accepted values (e.g. in tables of physical data) within your estimated experimental uncertainty? Comment briefly on any important uncertainties of measurement and how they might be reduced.

Word processing of laboratory reports

(2.7.a) Introduction The four laboratory reports that you submit in this module must all be word processed and there are facilities for doing this available in the Physics Department, the University computer centre and at other locations around the University campus. If you have facilities of your own for producing word processed documents then please use those if you so wish, there is no particular preference for any one or other word processing package.

(2.7.b) Equations, figures and tables In this manual the equations have been word processed and the figures and tables have been inserted at the appropriate points in the text. Your word processed reports do not have to be as sophisticated as this. It is quite acceptable for you to write in equations and also special symbols (e.g. Greek letters etc.) by hand. There is however an equation editor available on the Physics Department PC’s and if you wish to use this facility it can only improve the presentation of your report. Figures and tables do not have to be inserted in the text of your report but can be attached at the back of the pages of text. Since they will be labelled as figure(1) etc. and referred to as such in the text it should be clear to the reader which particular figure is being discussed. As an aside, when scientific papers are submitted to scientific journals for publication, the journal companies in fact insist that figures are attached at the back of the text rather than inserted.

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(2.7.c) Graphs and diagrams Further to this your graphs and diagrams do not have to be created by the computer, hand drawn diagrams and graphs are acceptable. However in a number of experiments you are asked to fit a straight line to your data and this can be done using the Linefit program on the Physics Department PC’s. In this case you should print off a copy of the resulting graph for inclusion in your report.

(2.7.d) Report style The text of your report should be in 12pt characters, in either Times, Times New Roman or Courier font. It should be double spaced and the pages should be numbered with your name on the front page. Again, this is a format upon which the scientific journals insist when submitting papers.

Fitting data to straight lines and least squares theory.

(2.8.a) Introduction It is very common to display experimental raw data in a graphical form, whereby we plot the measured values yi against the values of the “independent” variable x (^) i which we have set in the experiment. In order to signify our confidence (uncertainty) in the yi values we usually plot an error bar ± ei on the yi values. An example of such a plot is shown in figure(4).

Figure 4

In many of the experiments described in this manual and in the case of the data plotted in figure(4) we can intuitively see by looking at the data that we expect that underlying the raw data there is a straight line relationship between the values of y and x. The data points in figure(4) don’t lie on a perfect straight line however because of the effect of random errors. In figure(4) a “best fit straight line” has been drawn and again intuitively we’d say that we thought this was a good description of the

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(2.8.d) Minimising the Goodness of Fit Parameter

If we examine equation(9) then we can see that χ^2 is only a function of the two parameters m and c since the values yi , x (^) i and ei are the values measured in the experiment, not variables. A plot of χ^2 against m and c would look something like figure (5)

Figure 5

The best fit is when χ^2 is a minimum and therefore mathematically the best fit values for m and c must satisfy the equations

2 2 1 2 2 1 0 2

m

x y mx c e

c

y mx c e

i i i i i

N

i i i i

N

=

=

(10 a).

(10.b)

which can be re-written as two simultaneous equations in the unknowns m and c as

m

x e

c

x e

y x e

m

x e

c e

y e

i i i

i i i

i i i i

i i (^) i i i

i i i

2 2 2 2

2 2 2

⎟ +^

⎟ =^

⎟ +^

⎟ =^

(11.a)

(11. b)

which can be solved to get the values for m and c given by ;

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m e

y x e

y e

x e

c D

y e

x e

x e

y x e

D

e

x e

x e

i i

i i i i

i i i

i i i

i i i

i i i

i i i

i i i i

i i

i i i

i i i

D

2 2 2 2

2

2 2 2 2

2

2 2 2

2

⎟ −^

⎟ −^

⎟ −^

(12.a)

(12. b)

(12.c)

Equations(12a-c) give us a formula for determining the values of m and c which are the best

straight line fit to the data. How good this fit actually is depends upon the value of χ^2 with the best fit values of m and c substituted. Remember since we have minimised χ^2 this is the lowest value that can be achieved by fitting a straight line to this data. As noted earlier statistical theory would suggest that if the data “truly” lies on a straight line but that there are random errors in our measurements of the y- values then we should have a value of χ^2 ∼ 1. We can’t expect to get a value for χ^2 of exactly 1 in a real experiment but we’d hope to get a value which is comparable to 1. In any report we make we can

always quote the value of χ^2 as a measure of how well a straight line describes our experimental data. Since our data points have error bars (i.e. uncertainties) so too must our values for m and c. These error bar values are given by the equations ;

Δ

m e

c

x e

i i

i i i

D

D

2

2 2

(13.a)

(13. b)

In practical use we always evaluate equations(12a-c) and (13a-b) through a computer program. We enter the data into the program and the program then evaluates the equations and usually plots a graph as well as printing out the values of m , Δ m , c and Δ c.