Transcript Errors2013

Experimental Errors
&
Writing Reports
Prof. D. Evans
October 2013
Layout of Lecture
• Part 1 Errors
– What is an experimental error/uncertainty?
– Presentation of errors
– Calculation of errors
• Part 2 Writing a physics Report/Essay
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–
–
–
–
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Layout of a talk
Abstract
Referencing in text
Captions for figures and Tables
Copying text word-for-word
List of references
What is an Experimental Error?
• Every measurement or calculation derived from
a measurement will have an uncertainty or error,
.
• Therefore, when we write the value of a
measurement we should also include the
uncertainty
– E.g. mass of ball, m = 29.7  0.3 g
• What does this error actually mean?
– Does it mean that the mass of the ball is definitely
between 29.4g and 30g ?
• No, it doesn’t.
What is an Experimental Error?
• If I were to carry-out the experiment, to measure the ball,
a million times and plot my measured results, it might
look something like this:
• But what is the error?
• It’s the standard deviation of this plot
 i.e. half the width of the
distribution at half its height
• As you can see, there are still quite
a large proportion of the data outside
the range 30  .
• Ok, but what does this error (or
uncertainty) actually mean?
No. of measurements

30
Mass, g
What is an Experimental Error?
• The plot below is a “normal” (or Gaussian) distribution –
similar to my mass plot.
• What it shows is that about 68% of the data are within
 of the mean value.
• Or, put another way, nearly a third of the data are more
than 1  from the mean.
• 95% of the data
are within 2 and
99.7% within 3.
• only about 0.3% of
the data are > 3
from the mean.
Fitting Data
• This means that if you are fitting a load of data
points to the equation y = mx + c (and the data
should fit a straight line)
– About a third of your data points will lie more than 1
from the line and
– 1 in 20 points will be more than 2 from the line.
y
So, this looks like a good
fit, even if not all the data
points lie on the fitted line
x
Chi Squared (2) per Degree of
Freedom
• Most fitting programs usually return a 2
per degree of freedom
– This is an indication of how well the data fits
the curve
– A high value means it’s not a very good fit
– A very low value means your errors are too
big (i.e. you have over estimated them)
– An ideal 2 should be between 1 and 1.5
Chi Squared (2) per D of F
y
2 ~ 10,000
y
2 ~ 0.001
Not a good fit
x
Errors too big
y
x
Looking good!
1 < 2 < 5
x
Interpretation of Errors
• So, if you measure the charge of an
electron in the lab and get the results
– e = (1.42  0.08) x 10-19 C
– Is this compatible with the actual value of the
electron charge: e = 1.61 x 10-19 C ?
– The difference is 0.19 x 10-19 C which is about
2.4 from the actual value.
– A bit of a judgement call but I would usually
say two values are compatible with each other
if they are within 3 - so, yes it is compatible.
Presentation of Errors
• So, values should also have their errors
(and units) shown but what’s the best way
to present these?
– Here’s an example from a student’s lab book:
– f = 109.3584923 Hz  3.48320294 Hz
– What’s wrong with this?
This adds no information as the error is 3 Hz
This adds no
useful information
Errors should only have 1 significant figure or sometimes 2.
The result should only be given to the precision of the error.
E.g. f = 109  3 Hz or f = 109.4  3.5 Hz
Presentation of Errors
• Spot the mistakes:
f = 109.358  5 Hz (wrong)
f = 109  5 Hz (correct)
L = 56  0.4 m (wrong)
L = 56.2  0.4 m (correct)
M = 14.3  0.13 g (wrong)
M = 14.30  0.13 g (correct – worth keeping 2 s.f. here as 1st is a
1 but then give result to same precision)
– Q = 3.42 x 10-18 C  3 x 10-20 C (not wrong but bad style)
– Q = (3.42  0.03) x 10-18 C (can now clearly see where the error
lies)
–
–
–
–
–
–
• I think you get the idea
• Any question on presentation of errors?
Common Error Calculations
• You get much more detail from lab ‘bible’ and
lectures by Dr Tungate
a 2  b 2
Error on a+b is
Error on a-b is also a 2  b 2
Error on ab is
 a   b 
ab     
 a   b 
2
Error on f = anbmcl
is  f  2 na 2 mb
2

 
  lc 
   
 
 

 f   a   b   c 
2
2
Error on Function of x
• f(x) is some function of x and x has an error of
x, error on function is:
f
df
f  x
dx
f
x
x
gradient
f
f
x
x
Error on function of x - examples
1
f  2
x
df
2
and
f  x   3 x
dx
x
f 1  2 
2x
2 2
   3 x   x 3 x  
f
f  x 
x
x
The minus sign just means f gets larger if x gets smaller
End of Errors Part
Reports - layout
• Your report should have a title, date, and your name on
it. The structure of the report should then be as follows:
• Abstract (1 paragraph) which explains what the report is
about is gives the important result(s) (with errors).
• Introduction - sets out the context of the physics behind
the report
• Theory – obviously sets out the theory behind the
physics.
• Description of experimental apparatus – with a figure.
• Note: sometimes you may put the description of the
apparatus before the theory if the theory requires
knowledge of the experimental set up first.
Reports – Layout Continued
• Experimental procedure – this is where you explain in
more detail exactly how you carried out the experiment.
• Results - The results section is not a list of numbers.
Instead it is a passage written in your very best English.
– It should include a description of how the results are analysed,
as well as containing critical values.
– Results of error analysis should also be presented.
– Figures and Tables showing key results should also be included.
• Conclusions - Give your final numerical results here
along with their uncertainties.
– Compare if possible with values obtained by other authors.
Make substantiated comments on any discrepancies
– Did the experiment achieve its original aim? How could it be
improved?
– Remember, in a publication, many readers just read the
conclusions.
Reports – Layout Continued
• References – give references of books and
papers which you use for your work.
– Try to avoid web references as they tend to disappear
over time – most of your references should be books
or published papers.
• Appendix (appendices) – you may not wish to
include a long list of tables and results or a
complex error analysis in your main report. In
which case, put them in the appendix and refer
to them in the main report.
School of Physics & Astronomy
PHYSICS YEAR 1 LABORATORY
Title Page
First Semester Report
December 2013
The Use of the Fresnel Biprism to Determine the
Mean Wavelength of the Sodium D-Lines
by T.O.P. Student
(Group X)
Tutor: Dr. Ivor Pain
Referencing
• The references should be referred to in the main
text and not just a list at the back e.g.
The energy densities reached in lead-lead collisions at the CERN
LHC [1] are believed to be well in excess of those required for a
phase transition, from hadronic matter to a Quark-Gluon Plasma, to
take place [2].
Reference to a paper or book describing the LHC
Reference to a theory paper describing the
conditions for quark-gluon plasma formation.
These references are then listed in the Reference Section e.g.
References
[1] P. Brown et al. Physics Letters B (vol 45) pg 257 (1995).
[2] B. Cool and A. Theorist, Physical Review Letters 63 pg 123 (2005).
[3] …….
Referencing
• The text should always be in your own words –
NEVER COPY even if giving a reference.
• Best to read references and make notes. Then
put the references away and write in your own
style.
• If you want to quote a sentence from a text,
make it clear that it’s a quote and put it in italics
or between quotes “To be or not to be”. Or both:
“To be or not to be”. Don’t quote whole sections
of text.
Figures & Tables
• All figures and tables should be referred to in the
text e.g.
The experimental layout is shown in figure 1and
consists of …..
• Figures should have a figure caption under them
and tables should have a caption on top e.g.
Table 1: Measured voltage and
current for different frequencies.
Figure 3: A normal distribution showing the
percentage of data in each standard
deviation.
Frequency
(Hz)
Voltage
(V)  0.1V
Current
(amps) 
0.02 A
100
4.5
0.34
1000
6.7
0.11
Style
• The report should be a continuous piece of prose.
– Do not write it like a shopping list or recipe
• It used to be the firm convention that all experiments
were written up in the passive voice and past tense (e.g.
"the diameter was measured"). Although stilted, it is still
by far the most common, because "I measured" sounds
too pushy! Non native-English speakers also find the
stilted style easier to comprehend.
• Mathematical equations should be set out as in good
books, i.e. they should be included into the grammatical
structure of a normal sentence thus (note the comma!):
"The area, A is given by
b
A   ydx ,
(6)
a
where a and b are the x-co-ordinates of the two wires."
Finally …..
• If you’re not sure about anything – just
ask!
• It is always a good idea to get someone
else to have a quick read of your
report/essay – it is sometimes difficult to
spot your own mistakes.
• Any Questions?