1. Introduction to Error Analysis
Atomic Lab
Introduction to Error Analysis

2. 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: x= > x=0.02
g= 9.82 ± 0.02
Right and Wrong
Wrong: speed of sound= ± 10 m/s
Right: speed of sound= 330 ± 10 m/s
Always keep significant figures throughout calculation, otherwise rounding errors introduced

3. Statistically the Same
Student A = 30 ± 2
Student B = 34 ± 5
Since the uncertainties for A & B overlap, these numbers are statistically the same

4. Precision Mathematical Definition
Precision of speed of sound= 10/330 ~ 0.33 or 33%
So often we write: speed of sound= 330 ± 33%

5. Propagation of Uncertainties– Sums & Differences
Suppose that x, …, w are independent measurements with uncertainties x, …, w and you need to calculate
q= x+…+z-(u+….+w)
If the uncertainties are independent i.e. w is not sum function of x etc then
Note: q < x+…+ z+ u+…+ w

6. 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

7. 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

8. Power Law
Suppose q= xn and x ± x

9. 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

10. 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

11. Mean (or average)

12. 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

13. 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

14. Histograms
Number of times that value has occurred
Value

15. Distribution of a Craps Game
Bell Curve Or
Normal
Distribution

16. Bell Curve Centroid or Mean x+s x-s 68 %
Between x-2s to x+2s, 95% of population
2s is usually defined as Error

17. Gaussian X0
In the Gaussian, x0 is the mean and sx is the standard deviation.
They are mathematically equivalent to formulae shown earlier

18. 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)

19. 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 FWHM=2.354*sx

20. Weighted Average
Suppose each measurement has a unique uncertainty such as
x1 ± s1
x2 ± s2 …
xN ± sN
What is the best value of x?

21. 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 w=weight= 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

22. Least Squares Fitting
What if you want to fit a straight line through your data?
In other words, yi = A*xi + B
First, you need to calculate residuals
Residual= Data – Fit or
Residual= yi – (A*xi+B)
When as the Fit approaches the Data, the residuals should be very small (or zero).

23. Big Problem Some residuals >0 Some residuals <0
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

24. 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

25. Chi-square, c2

26. Using Determinants

27. A Pseudocode Dim x(100), y(100) xsum=0 x2sum=0 Xysum=0 N=100 Ysum=0
For i=1 to 100
xsum=xsum+x(i)
ysum=ysum+y(i)
xysum=xysum+x(i)*y(i)
x2sum=x2sum+x(i)*x(i)
Next I
Delta= N*x2sum-(xsum*xsum)
A=(N*xysum-xsum*ysum)/Delta
B=(x2sum*ysum-xsum*xysum)/Delta

28. 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 doesn’t vary too much

29. Without Proof

30. 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

31. Extending the Method
Power Law
Multivariate multiplicative function

32. Uglier Functions q=f(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

33. 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 y=f(x)?
If r2 <0.5 then there is no correlation.