A Calculation of

The reduced statistic, , is normalised by the number of degrees of freedom, . Given N measurements of the same quantitity ( where ), the assumption of a particular underlying probabiltiy distribution of the measurements may be verified with . The data is first binned (n bins) to create a frequency distribution with for each 'th bin (where ). For k separate sets of observations, there will be a distribution of such that there exists a standard deviation over k for each . The assumed distribution function, say for the Gaussian distribution, will have some difference from when evaluated for each . If that difference is normalised by , it is simply a standard deviation measurement to the ``fitting function'', , where s denotes that it is from the sample. Taking the sum of the ratio of the standard deviations therefore gives a quality figure which may be used to ascertain the likelihood of the assumed distribution, i.e.

From this elaboration, implies that the assumed distribution is very good. This corresponds to P (integral probabilities of with ) that approach , depending on . Extremely large or extremely small P values in general indicate that the assumed distribution is inadequate because it is either too perfect (there must be some noise in the random variation or else some other factors are affecting the method of measurement) or too imperfect (any other function may be assumed as well). The calculation in practice was carried out as

for one distribution with where , the assumed distribution, is given by

is the normalized expected distribution of for the total number of data points, is the separation between data points, is the mean, and for (standard deviation). The details concerning the normalisation of for use in the calculations of may be found in Bevington [Bev69] or any other standard statistical text [MSW86].