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Find the mean and standard deviation from a c2 fit of a Gaussian to a histogram of the data, assuming the variance in each bin is just the number of counts. Use a normalized Gaussian with amplitude N*binwidth/(sigma*sqrt(2*pi)).

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Homework 

Here are 19 of the 20 “perfect” g measurements (m/s2) after removing one outlier:

9.792

9.768

9.809

9.800

9.829

9.821

9.837

9.817

9.802

9.815

9.813

9.758

9.795

9.836

9.822

9.818

9.806

9.800

9.811

 

  1. Compute the mean and standard deviation from Equations 4.9 and 4.13.
  2. Find the mean and standard deviation from a c2 fit of a Gaussian to a histogram of the data, assuming the variance in each bin is just the number of counts. Use a normalized Gaussian with amplitude N*binwidth/(sigma*sqrt(2*pi)).
  3. Now fit a Gaussian to the histogram by minimizing the negative log-likelihood (eg -1*M from 10.7). Assume the histogram contents are Poisson distributed (2.11) when calculating the likelihood.  You can use the minimize function from scipy to find the best fit values for mean and sigma.
  4. Find an upper and lower limit for each parameter from the likelihood fit by stepping the parameter away from the best fit value until the likelihood increases by 0.5.
  5. Now find mean and sigma from an unbinned fit of a Gaussian PDF to the data. Minimize the negative log of the likelihood computed from the values of the PDF evaluated at each data point.  Repeat step 4 to find the uncertainties for the best fit values obtained this way.
  6. Compare the mean and standard deviations found in these four ways. Do they agree within uncertainties?
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