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Soluciones de análisis de errores

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ERROR ANALYSIS ACTIVITY 1: STATISTICAL MEASUREMENT UNCERTAINTY AND ERROR BARS
LEARNING GOALS
At the end of this activity you will be able…
1.
…to explain the merits of different ways to estimate the
statistical
uncertainty in common measurements like a DC voltage reading. 2.
…to explain which of the methods actually give an estimate of the standard deviation, which
over/underestimate the uncertainty, and which are just a rough estimate of the uncertainty. 3.
…compute the uncertainty in a derived quantity using the u
ncertainties in the measured quantities (error propagation). 4.
…add error bars to a plot as a visual representation of uncertainty.
INTRODUCTION TO MEASUREMENT UNCERTAINTY
When attemping to establish the validity of our experimental results it is always important to quantify the uncertaint
y. Measurement uncertainty wasn’t invented to make lab classes tedious, rather it is a core part of any
experimental work that gives us a way to quantify how much we trust our results. A simple and rigorous way to make a
measurement and estimate it’s uncertainty is to take
measurements
and estimate the value by the mean:
̅∑
The estimated uncertainty (standard deviation,
, or variance
) of any one measurement is given by
∑
̅
While the uncertainty in the mean value
̅
is smaller and is given by
̅
The remainder of this activity will discuss a variety of practical considerations about using uncertainty in the lab.
ESTIMATING THE MEAN AND UNCERTAINTY
The next four questions cover a basic measurement that was essential in the Gaussian Laser Beams lab.
In a group of 2 or 3, find a lab bench with a laser and turn it on immediately. It might take 5 minutes for the laser to warm up.
Use the photodetector to measure the DC optical
power. It doesn’t matter whether you detect the whole
beam or not, just so you have some signal to record.
Question 1 Measurement and uncertainty using the Multimeter
Note: It may be possible to answer questions 1 and 2 at the same time if you use a BNC “T” to
send the voltage to the multimeter and oscilloscope at the same time.
Make a table of estimated DC voltages from the laser and the corresponding uncertainties using the following methods a.
“
Eye
ball” the mean.
“
Eyeball
”
the amplitude of the random fluctuations. b.
If you multimeter has the capability
, set the multimeter on max/min mode to record the
and
fluctuations over a certain time period. You can estimate the mean by
and the uncertainty by
. c.
Record the instantaneous voltage reading on the multimeter
times d.
What is the resolution intrinsic to the multimeter according to the spec sheet on the course website? (No measurement required) How does this compare to the observed uncertainty in parts a-c?
Question 2 Measurement and uncertainty using the oscilloscope
Continue the previous table of estimated DC voltages from the laser and the corresponding uncertainties using the following methods. For each method comment on if and how it depends on the setting for the time scale or voltage scale on the oscilloscope. a.
“Eyeball” the mean.
“
Eyeball
”
the amplitude of the random fluctuations (no cursors or measurement tools). b.
Use the measurement function on the scope to record the mean and RMS fluctuations. c.
Use the cursors to measure the mean and size of fluctuations. e.
Record the voltage reading on the oscilloscope
times. Take the mean and standard deviation. d.
A comparison with the data sheet is difficult because so many factors affect the observed noise in the oscilloscope. You can find some information on the data sheet on the course website. There is information about the resolution and the DC measurement accuracy.
Question 3 Summary of Questions 1 and 2:
a.
Which method(s) should give a true estimate for the standard deviation? b.
Did any methods overestimate the uncertainty? c.
Did any methods underestimate the uncertainty? d.
How reliable was “eyeballing”?
e.
Did the time scale or voltage scale affect any of the oscilloscope measurements?
Question 4 Measurement using the NI USB-6009 DAQ
a.
Is the USB data acquisition device more like the multimeter, or more like the oscilloscope? (Just thinking back to your experience using the DAQ last time. No data taking for this part.) b.
How would you go about estimating the uncertainty when using your DAQ?
WRITING NUMBERS AND THEIR UNCERTAINTY
The convention used in this course is that we 1)
only display one significant digit of the uncertainty and 2)
display the measurement to the same digit as the uncertainty. The numbers 154
2, and 576.33
0.04 both follow the convention. However, numbers copied from the computer
are usually always displayed as “machine precision” with no regard for significant digits.
Question 5
Mathematica generated the following fit parameters and corresponding uncertainties:
How should the two Mathematica fit parameters in the above table be rewritten?
ERROR PROPAGATION: FROM MEASURED TO DERIVED QUANTITIES
Often times, the quantity of interest in an experiment is not actually measured, rather it is derived from a set of measured quantities. An example is estimating the resistance of a circuit element from measurements of the current through it
and voltage across it. In this case we could use Ohm’s law
(
to convert our measured quantities (voltage and current) into a derived quantity (resistance). Error propagation comes in when we want to estimate the uncertainty in the derived quantity based on the uncertainties in the measured quantities. Keeping things general, suppose we want to derive a quantity
from a set of measured quantities
,
,
, etc. The mathematical function which gives us
is
. In general, any fluctuation in the measured quantities
,
,
, …
will cause a fluctuation in
according to
()()()
(1)
This equation comes straight from basic calculus.
It’s
like the first term in a
Taylor series. It’s the linear
approximation of
near
.
However, we don’t know the exact magnitude or sign of the
fluctuations, rather we just can estimate the spread in
,
,
, which we often use the standard deviations
,
,
. In this case, the propagated uncertainty in
is:
()
()
()
(2)
There are standard equations provided in courses like the introductory physics lab for the error in the sum, difference, product, quotient. These are all easily derived from the general formula.
Question 6
Review things from long ago…
Apply the general error propagation formula in Eq. 2 to calculate the derived uncertainty
in terms of the measurement uncertainties
and
when a.
b.
c.
Question 7
In the case where a voltage
and current
are measured to derive the resistance
, use Eq. 2 to calculate the uncertainty in
in terms of the uncertainties
and
.
Question 8
A diffraction grating can be used to measure the wavelength of light according to
where
is distance between the grating spacing, and
is the angle of incidence and reflection, and
is the “order” of the diffraction peak. Note that
is an integer. If we derive
from measurements of
and
, use Eq. 2 to calculate the error in
in terms of the uncertainty in
and
,
and
.
ERROR PROPAGATION IN MATHEMATICA
So far in this class we have explored the use of Mathematica for some basic data analysis and plotting, but Mathematica also has powerful symbolic math capabilities. One example where this can be helpful is for complicated error propagation calculations. The following bit of Mathematica code can calculate the propagated variance in the derived quantity symbolically. The code is available and explained in a Mathematica notebook on the course website.

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