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Error Analysis
Physics 122Shirley ChiangWinter 2014
Measuring Errors
• Precision vs. Accuracy– Precision : reproducibility, repeatability– Accuracy: how close the result is to the correct value
Systematic vs. Random Errors
• Systematic Errors – biases in the measurement, leading to result different from the correct one– Calibration error– Environmental effect on measurement tool– Imperfect method of observation
• Random Errors – Statistical Errors– Fluctuations (noise) in measurement apparatus – Random operator errors– Noise in Nature– Johnson noise, shot noise– Usually can be improved by repeated measurements
Quoting Uncertainties in Measurements
• (measured value of x) = xbest x• Significant digits
Ex. 1(measured g) = 9.82 0.02385 m/sec2
is absurd estimate of the undercertainty(measured g) = 9.82 0.02 m/sec2
Ex. 2. (measured speed) = 6051.78 30 m/sec => (measured speed) = 6050 30 m/sec
• Fractional uncertainty =
Sample vs. Parent (Limiting) Distribution
Sample distribution • Histogram of measurements• Becomes the Parent (limiting) distribution in the
limit of infinite number of measurements
Properties of Distributions
• Mean of N measurements, • Mean of parent distribution
→
• Median / half of measurements are above this value, and half are below
/ ) = / ) = 0.5• Mode – most probable value, the one which will occur most often with repeated measurements
Comparison of Mean, Median, Mode
Standard Deviation
lim→
1
lim→
12
lim→
12
lim→
1
For sample population, ∑ ̅ , use factor (N‐1) in the denominator since ̅ is determined from the data and not independently
Useful Probability Distributions in Physics
• Binomial• Poisson• Gaussian• Lorentzian
Binomial Distribution
Poisson DistributionBinomial ‐> Poisson when p‐>0,
Gaussian Distribution
Gaussian (Normal) Distribution
Normal Error Integral or Error Function erf(t)
Gaussian vs. LorentzianLorentzian:
Error PropagationUse partial derivatives
Summary of Error Formulas
Fitting Curves
• Method of least squares to fit straight line
Judging Quality of Fitted Curves• Compute 2 ,
where oi = observed value, ei = expected value
• Reduced 2reduced =2 , where = number of degrees of
freedom = N‐n‐1,where N=number of observations
n=number of fitting parameters reduce by 1 for assuming the mean value is another fitted parameter
• A good fit has 2reduced =1, i.e., match between observationsand estimates is in accord with error variance
Example from Melissinos, Chap. 10
2 = 0.846, and plot shows that 93% of the cases would have a larger 2 than obtained here, i.e., this fit is too good!