introduction to experiment: part 2nas2173/lab0_introtoexperimental_part2s18.pdf4 review:...

INTRO TO EXPERIMENTAL PHYS-LAB 1494/2699

Introduction to Experiment: Part 2

Nate Saffold [email protected]

Office Hour: Mondays, 5:30-6:30PM @ Pupin 1216

PHYS 1493/1494/2699: Introduction to Experiment – Part 22

Important news1. The lab sessions will start February 6th NOTE: The

experiment will not follow the same order as in the lab manual. Refer to the preceptors’ website for the full calendar

2. First quiz on error analysis next week! (1/30) 3. Check the course website for supplementary reading

on error analysis: http://www.columbia.edu/~nas2173/1493.html

4. I posted a tutorial on how to use Mathematica to perform data analysis. You can access a free copy of Mathematica Student Version with your Columbia UNI

5. Any comment on the tutorial or question about Mathematica are more than welcome!

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Review: statistics● For multiple (N) measurements of the same quantity, find:

− Mean:

− Sample standard deviation :

− Standard error on the mean :

● Always report result as:


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Review: interpretation of the results● Suppose the expected value of a result is .

● − Experiment and expectation are in good agreement.

● − Experiment and expectation are consistent.

● − Experiment and expectation may disagree. Grounds for further

investigation. ●

− Difference is statistically significant.


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Combining data and uncertainties


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Real life example● Suppose we want to measure the mass of the famous Higgs boson ● We have two different detectors (ATLAS and CMS) that measure the mass

of this particle. They measure the following values with uncertainties:

● ATLAS:

● CMS:


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Real life example● Suppose we want to measure the mass of the famous Higgs boson ● We have two different detectors (ATLAS and CMS) that measure the mass

of this particle. They measure the following values with uncertainties:

● ATLAS:

● CMS:

● Should we take the average and report 125.65 GeV? ● What should the error be? More or less than 0.7 GeV? More or less than

1.1 GeV? ● Which of the two detectors is more important?


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Weighted Averages● Question: What happens when you have different

measurements all with different precisions and you want to combine both of them into a unique result?

● We want something that: 1. Gives more ‘importance’ to those measures that are more precise

2. Returns to the usual average when the errors of the different measures are all equal to each other


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Weighted Averages● Question: What happens when you have different

measurements all with different precisions and you want to combine both of them into a unique result?

● We want something that: 1. Gives more ‘importance’ to those measures that are more precise

2. Returns to the usual average when the errors of the different measures are all equal to each other

● Following this idea one defines:

- Weighted mean:

- Weighted error of the mean:


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Real life example− ATLAS:

− CMS:

● Mean:

● Standard deviation:

● Combined result: ● The final error is smaller! It’s always better to combine

consistent results

Watch out for significant and

decimal figures!


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Error Propagation


Absolute vs Relative Uncertainty• Up to this point I have been reporting values in terms of absolute

uncertainty, but it is time to introduce relative uncertainty, which is useful when propagating error.

• Absolute Uncertainty:

• Relative Uncertainty:


Combing Errors - Addition/Subtraction

• When adding or subtracting two measured values that each have error, we add their absolute error in quadrature

• If: • Then:


Combining Errors - Products/Quotients• When multiplying or dividing quantities with uncertainties we add

their relative uncertainty in quadrature

• If:

• Then:


Combining Errors - Multiplication by a Constant• Suppose , where B is known exactly and we know

the error in x

• Then, the uncertainty in q is:

• Or, equivalently:

• So, the relative error stays the same when multiplying by a known constant


Combing Error - Exponents• If n is an exact number (no uncertainty) and:

• Then:

• These rules come from partial differentiation, which we will see in the next few slides


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Combining Errors - Functions● One more question: what happens if the final result is not measured

directly but it’s obtained from the combination of many variables? ● Suppose that we want to measure the error on some function

and we know the average and uncertainty of each variable, , and .

● Propagation of Errors:

● A small caveat: this formula is only valid if the errors on x, y and z are uncorrelated


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Example 1 (single variable)● We are asked to determine the area of a circle:

● We measure the radius with its uncertainty:

● Propagation of errors tells us that the uncertainty on the area is given by:

● So the result for the area of the circle is:

PHYS 1493: Introduction to Experiment – Part 2

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Example 2 (many variables)● We are asked to determine the area of a rectangle:

● We measure the length and width with uncertainties:

● The uncertainty on the area is then given by:

● The final result is then:


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Graphical Analysis of Data


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Graphical Methods

● When plotting data, you want to show the uncertainties in data points.

● Convention: Plot each data point using “error bars”.

● Graphical analysis of your data is an essential and intuitive way to present the results of the experiment

Avg.

Avg. + σ

Avg. - σ

( s2 )


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Linear regression● It is very common to have

two pairs of measurements, (xi, yi), that are linearly related:

● How do we find the parameters a and b that best describe our points? How do we find their errors?

= Exp. point with error


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Linear regression● It is very common to have

two pairs of measurements, (xi, yi), that are linearly related:

● How do we find the parameters a and b that best describe our points? How do we find their errors?

● Best Fit Line: ● Find line that passes as close as possible to the highest number of

points ● Usually a tedious task! Luckily we have statistical packages that can

do all the hard work for us. (e.g. SciPy, Mathematica) ● Python: curve_fit (in th scypy.optimize package) ● Mathematica: NonlinearFit function

= Exp. point with error


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Linear regression● Suppose now that your experimental points are while the

corresponding points on the line are . ● If the errors on your points are all the same or fairly similar the

use of unweighted fit is justified. ● In this case you want to find the a and b such that the following

quantity is minimized:

● If the errors are different (which is often the case) then you have to minimize the so-called “chi squared”:

● Since this is long and boring, let the computer do that for you! ● More details on the lab manual…


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Checking Fit● How do we know if our hypothesis is correct? In other

words, that is the correct function to represent the trend in the data.

● Plot residuals:

● Residuals are defined

as:

● If the residuals are randomly distributed this gives us confidence in the fit.


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Checking Fit● What if a pattern does arise? ● Could be one or a

combination of factors: − Wrong fit function − Presence of a

systematic error

− May be an indication that a weighted fit is needed.


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A general “rule”● There is a general “rule of thumb” that must be applied in Physics

to: ● Check the correctness of a fit ● Check for malfunctioning of an instrument ● Check for algebra mistakes in data analysis ● many other things…

Everything must be reasonable! (If not, you probably did something wrong…)

(or maybe your TA did…) (or maybe your instructor…)


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Tips on how to write a clear lab report


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Rules for lab reports● You are free to write your report at home. However, you must submit your

raw data to your TA by the end of the lab session. The raw data should either be sent by email or written using the “Carbon-copy notebook, National Brand #43-649 (available in Columbia bookstore)”.

● You can type your report on your computer as well as make plots and compute the parameters of a fit with the help of your favorite software/programming language

● You must submit the complete report by the beginning of the next weekly session

● Your TA will grade it and bring it back to you the following week


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Organization of the report● A good lab report should usually be organized in the following sections:

1. Introduction: Usually no more than half a page. A brief description of the goals of the experiment and the main physics behind it (including formulae you are going to use)

2. Method: More or less one page. Description of the apparatus and instruments (including their accuracy). A brief description of how the experiment is performed

3. Data: This section is essential! It contains your raw data and their explanation

4. Data analysis: Essential too! Some times it can be merged with the data section. Contains everything you need to go from the raw data to the final result (calculations, error propagation, …)

5. Conclusions: Usually no more than a page. It should report if your experiment succeeded or not, together with the main sources of error and, if present, a description of the systematic errors and how they affected the measure.


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What your TA wants● This is what a lab report must always have to be clear

and easy to understand:

1. A detailed description of what each raw data set represents from a physical point of view

2. A description of the uncertainties assigned to each measured quantity. Why did you assign a certain value to the error? If no error is assigned, why did you decide that it is negligible?

3. Raw data well organized in tables

4. When needed, data represented in clear and large plots

5. A clear presentation of what your final result is. It should always be in the form:

6. A description of the main sources of error affecting your measure


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What your TA does not want● This is what a lab report should never have:

1. Experimental data (either raw or analyzed) without their associated uncertainty or a statement about its absence

2. Experimental errors presented without any explanation

3. Tables or plots without labels

4. Any quantity without the right units

5. Messy presentation and/or numbers randomly distributed on the page without proper organization


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Summary● Weighted Averages:

− Combine independent experimental results based on the precision of the result.

● Propagation of Errors: − Propagating uncertainty through algebraic equations. − Uncertainties on derived values.

● Graphical Analysis of Data: − Minimization of residuals. − Evaluation of Fit; Plotting residuals

● How to write a lab report: − Organization of a report − What a report should have − What a report should never have


Comic Relief


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