Formulae for Statistics and Error

Mean

where N is the number of data points, i ranges from 1 to N, and xi is each data point.


Standard Deviation


where N, i, and xi are defined the same as above, and μ is the mean.

Standard Deviation of the Mean

where N is the number of data points, and σx is the standard deviation.

Total error


where σsys and σr are the systematic and random error, respectively. The standard deviation can be treated as a random error.


Reduced Chi-Squared


where yi,thа is the theoretical value (expected value), yi,obs is the observed value, σi is the error associated with the yi,obs value, N is the number of data points, and ν is the degrees of freedom given as ν = N - p, where p is the number of free parameters.


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