• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Probability amplitude wikipedia, lookup

Coherent states wikipedia, lookup

Density matrix wikipedia, lookup

Quantum state wikipedia, lookup

Interpretations of quantum mechanics wikipedia, lookup

Hidden variable theory wikipedia, lookup

Bell's theorem wikipedia, lookup

Quantum entanglement wikipedia, lookup

Measurement in quantum mechanics wikipedia, lookup

Transcript
```Experimental Uncertainties:
A Practical Guide
• What you should already know well
• What you need to know, and use, in this lab
More details available in handout ‘Introduction to
• In what follows I will use convention:
– Error = deviation of measurement from true value
– Uncertainty = measure of likely error
Why are Uncertainties
Important?
• Uncertainties absolutely central to the
scientific method.
• Uncertainty on a measurement at least as
important as measurement itself!
• Example 1:
“The observed frequency of the emission line
was 8956 GHz. The expectation from
quantum mechanics was 8900 GHz”
• Nobel Prize?
Why are Uncertainties
Important?
• Example 2:
“The observed frequency of the emission line
was 8956 ± 10 GHz. The expectation from
quantum mechanics was 8900 GHz”
• Example 3:
“The observed frequency of the emission line
was 8956 ± 10 GHz. The expectation from
quantum mechanics was 8900 GHz ± 50 GHz”
Types of Uncertainty
• Statistical Uncertainties:
– Quantify random errors in measurements between
repeated experiments
– Mean of measurements from large number of
experiments gives correct value for measured
quantity
– Measurements often approximately gaussiandistributed
• Systematic Uncertainties:
– Quantify systematic shift in measurements away
from ‘true’ value
– Mean of measurements is also shifted  ‘bias’
Examples
• Statistical Errors:
True Value
0.45
– Measurements gaussian0.4
distributed
0.35
– No systematic error (bias)
0.3
– Quantify uncertainty in
0.25
measurement with standard
0.2
deviation (see later)
0.15
– In case of gaussian-distributed 0.1
measurements std. dev. = s in 0.05
formula
0
– Probability interpretation
-3 -2 -1 -0
1
2
(gaussian case only): 68% of
  x  x 2 
1

exp  
2
measurements will lie within ± 1
2
2s 
2s

s of mean.
3
Examples
• Statistical + Systematic Errors:
True Value
0.45
– Measurements still gaussian0.4
distributed
0.35
– Measurements biased
0.3
– Still quantify statistical
0.25
uncertainty in measurement with 0.2
standard deviation
0.15
0.1
– Probability interpretation
0.05
(gaussian case only): 68% of
0
measurements will lie within ± 1 s -3 -2 -1 -0 1 2
of mean.
  x  x 2 
1

exp  
2
2
– Need to quantify systematic error
2s 
2s

(uncertainty) separately  tricky!
3
Systematic Errors
• How to quantify uncertainty?
• What is the ‘true’ systematic
error in any given
measurement?
– If we knew that we could correct
for it (by addition / subtraction)
• What is the probability
distribution of the systematic
error?
True Value
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
– Often assume gaussian
distributed and quantify with ssyst.
– Best practice: propagate and
quote separately
0
-3
-2
-1
-0
1
2
3
Calculating Statistical
Uncertainty
• Mean and standard deviation of set of independent
measurements (unknown errors, assumed
uniform):
1
x0 
x

N
i
 x;
i
1
2


s 
x

x

i
N 1 i
2
• Standard deviation estimates the likely error of
any one measurement
• Uncertainty in the mean is what is quoted:
sx 
s

1
2
 xi  x  


N  N ( N  1) i

1/ 2
.
Propagating Uncertainties
• Functions of one variable (general formula):
df
F 
X
dx
• Specific cases:
 
 x 2  2 xx
   nx
x
n
n 1
x or
sin x   cos x x
1
ln x   x
x
or
 
 x2
x

2
x
x2
 xn
x
n
n
x
x
 
Propagating Uncertainties
• Functions of >1 variable (general formula):
f 
2
• Specific cases:
2

 f
  f
  x    y  .
 x   y 
2
f=
Apply equation
Simplify
x y
xy
f 2  x 2  y 2
f 2  x 2  y 2
x y
f 2
 y 2 x   x 2 y 
2
2

x 
f   2
2
y
x2
2
 4 y 
y
2
 f

 f
 f

 f
2
2

 y 
 x 
  

  
x
y





2
2
2

 x   y 
      
 x   y 

2
Combining Uncertainties
• What about if have two or more
measurements of the same quantity, with
different uncertainties?
• Obtain combined mean and uncertainty with:
x
2
x
s
 i i
i
1 s
i
2
i
1
s
2

i
1
s
2
i
• Remember we are using the uncertainty in the
mean here:
s
si 
N
Fitting
•
Often we make measurements of several
quantities, from which we wish to
1. determine whether the measured values follow a
pattern
2. derive a measurement of one or more parameters
describing that pattern (or model)
•
•
•
This can be done using curve-fitting
E.g. EXCEL function linest.
Performs linear least-squares fit
Method of Least Squares
ln eta [cPs]
• This involves taking
measurements yi and
comparing with the
equivalent fitted value yif
• Linest then varies the
model parameters and
hence yif until the
following quantity is
minimised:
 y
N
i 1
i
 yi
f

2
In this example the model is
a straight line
yif = mx+c. The model
parameters are m and c
1
0.5
0
-0.5
-1
-1.5
0.0026
• Linest will return the fitted
parameter values (=mean)
and their uncertainties (in
the mean)
0.0028
0.003
0.0032
0.0034
0.0036
0.0038
1/T [1/K]
In the second year lab never use
Trendline’ or linest to estimate your
parameters!!!
Weighted Fitting
• Those still awake will have noticed the least
square method does not depend on the
uncertainties (error bars) on each point.
• Q: Where do the uncertainties in the parameters
come from?
– A: From the scatter in the measured means about the
fitted curve
• Equivalent to:
1
2


s 
x

x
 i
N 1 i
2
• Assumes errors on points all the same
• What about if they’re not?
Weighted Fitting
• To take non-uniform uncertainties (error bars) on
points into account must use e.g. chi-squared fit.
• Similar to least-squares but minimises:
 y i  yi
   
si
i 1 
N
2
f



2
• Enables you to propagate uncertainties all the way
to the fitted parameters and hence your final
from Second Year web-page)  this is what we
expect you to use in this lab!
General Guidelines
Always:
• Calculate uncertainties on measurements and plot
them as error bars on your graphs
• Use chisquare.xls when curve fitting to calculate
• Propagate uncertainties correctly through derived
quantities
• Quote uncertainties on all measured numerical
values
• Quote means and uncertainties to a level of precision
consistent with the uncertainty, e.g: 3.77±0.08 kg, not
3.77547574568±0.08564846795768 kg.
• Quote units on all numerical values
General Guidelines
Always: