Atomic Lab

1
Atomic Lab

• Introduction to Error Analysis

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Significant Figures

• For any quantity, x, the best measurement of x is
  defined as xbest ?x
• In an introductory lab, ?x is rounded to 1
  significant figure
• Example ?x0.0235 -gt ?x0.02
• g 9.82 0.02
• Right and Wrong
• Wrong speed of sound 332.8 10 m/s
• Right speed of sound 330 10 m/s
• Always keep significant figures throughout
  calculation, otherwise rounding errors introduced

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Statistically the Same

• Student A 30 2
• Student B 34 5
• Since the uncertainties for A B overlap, these
  numbers are statistically the same

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Precision

• Mathematical Definition
• Precision of speed of sound 10/330 0.33 or 33
• So often we write speed of sound 330 33

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Propagation of Uncertainties Sums Differences

• Suppose that x, , w are independent
  measurements with uncertainties ?x, , ?w and
  you need to calculate
• q xz-(u.w)
• If the uncertainties are independent i.e. ?w is
  not sum function of ?x etc then
• Note ?q lt ?x ?z ?u ?w

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Propagation of Uncertainties Products and
Quotients

• Suppose that x, , w are independent
  measurements with uncertainties ?x, , ?w and
  you need to calculate
• If the uncertainties are independent i.e. ?w is
  not sum function of ?x etc then

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Functions of 1 Variable

• Suppose ? 20 3 deg and want to find cos ?
• 3 deg is 0.05 rad
• (d(cos?)/d? -sin? sin?
• ?(cos?) sin? ?? sin(20o)(0.05)
• ?(cos 20o) 0.02 rad and cos 20o 0.94
• So cos? 0.94 0.02

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Power Law

• Suppose q xn and x ?x

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Types of Errors

• Measure the period of a revolution of a wheel
• As we repeat measurements some will be more or
  some less
• These are called random errors
• In this case, caused by reaction time

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What if the clock is slow?

• We would never know if our clock is slow we
  would have to compare to another clock
• This is a systematic error
• In some cases, there is not a clear difference
  between random and systematic errors
• Consider parallax
• Move head around random error
• Keep head in 1 place systematic

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Mean (or average)
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Deviation
Need to calculate an average or standard
deviation To eliminate the possibility of a zero
deviation, we square di
When you divide by N-1, it is called the
population standard deviation If dividing by N,
the sample standard deviation
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Standard Deviation of the Mean
The uncertainty in the best measurement is given
by the standard deviation of the mean (SDOM)
If the xbest the mean, then sbest smean
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Histograms
Number of times that value has occurred
Value
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Distribution of a Craps Game
Bell Curve Or Normal Distribution
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Bell Curve
Centroid or Mean
xs
x-s
68
Between x-2s to x2s, 95 of population 2s is
usually defined as Error
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Gaussian
In the Gaussian, x0 is the mean and sx is the
standard deviation. They are mathematically
equivalent to formulae shown earlier
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Error and Uncertainty

• While definitions vary between scientists, most
  would agree to the following definitions
• Uncertainty of measurement is the value of the
  standard deviation (1 s)
• Error of the measurement is the value of two
  times the standard deviation (2 s)

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Full Width at Half Maximum

• A special quantity is the full width at half
  maximum (FWHM)
• The FWHM is measured by taking ½ of the maximum
  value (usually at the centroid)
• The width of distribution is measured from the
  left side of the centroid at the point where the
  frequency is this half value
• It is measured to the corresponding value on the
  right side of the centroid.
• Mathematically, the FWHM is related to the
  standard deviation by FWHM2.354sx

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Weighted Average

• Suppose each measurement has a unique uncertainty
  such as
• x1 s1
• x2 s2
• xN sN
• What is the best value of x?

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We need to construct statistical weights

• We desire that measurements with small errors
  have the largest influence and the ones with the
  largest errors have very little influence
• Let wweight 1/si2

This formula can be used to determine the
centroid of a Gaussian where the weights are the
values of the frequency for each measurement
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Least Squares Fitting

• What if you want to fit a straight line through
  your data?
• In other words, yi Axi B
• First, you need to calculate residuals
• Residual Data Fit or
• Residual yi (AxiB)
• When as the Fit approaches the Data, the
  residuals should be very small (or zero).

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Big Problem

• Some residuals gt0
• Some residuals lt0
• If there is no bias, then rj -rk and then
  rj rk 0
• The way to correct this is to square rj and rk
  and then the sum of the squares is positive and
  greater than 0

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Chi-square, c2
We need to minimize this function with respect to
A and B so We take the partial derivative of
w.r.t. these variables and set the resulting
derivatives equal to 0
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Chi-square, c2
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Using Determinants
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A Pseudocode

• Dim x(100), y(100)
• xsum0
• x2sum0
• Xysum0
• N100
• Ysum0
• For i1 to 100
• xsumxsumx(i)
• ysumysumy(i)
• xysumxysumx(i)y(i)
• x2sumx2sumx(i)x(i)
• Next I
• Delta Nx2sum-(xsumxsum)
• A(Nxysum-xsumysum)/Delta
• B(x2sumysum-xsumxysum)/Delta

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c2 Values

• If calculated properly, c2 start at large values
  and approach 1
• This is because the residual at a given point
  should approach the value of the uncertainty
• Your best fit is the values of A and B which give
  the lowest c2
• What if c2 is less than 1?!
• Your solution is over determined i.e. a larger
  number of degrees of freedom than the number of
  data points
• Now you must change A and B until the c2 doesnt
  vary too much

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Without Proof
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Extending the Method

• Obviously, can be expanded to larger polynomials
  i.e.
• Becomes a matrix inversion problem
• Exponential Functions
• Linearize by taking logarithm
• Solve as straight line

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Extending the Method

• Power Law
• Multivariate multiplicative function

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Uglier Functions

• qf(x,y,z)
• Use a gradient search method
• Gradient is a vector which points in the
  direction of steepest ascent
• ?f a direction
• So follow ?f until it hits a minimum

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Correlation Coefficient, r2

• r2 starts at 0 and approaches 1 as fit gets
  better
• r2 shows the correlation of x and y i.e. is
  yf(x)?
• If r2 lt0.5 then there is no correlation.