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Cramér-von Mises's test

Cramér-von Mises's test is an empirical distribution function omnibus test for the composite hypothesis of normality. It uses the summed squared differences between observed and expected cumulative proportions as the test statistic. For the one-sample test the procedure is as follows:

  1. Arrange the data in rank order.
  2. Estimate the mean () and standard deviation (sY) of the data (variable Y).
  3. Standardize each of the values of Y by subtracting the mean and dividing by the standard deviation.
    (N.B. it is only necessary to do this if you only have access to tables of the standard normal distribution.)
  4. Calculate the test statistic T as follows:

    Algebraically speaking -

    T    =    1  +  Σ [ 2i − 1  −  F(Yi) ] 2
    12n2n
    where
    • T is Cramér-von Mises's statistic,
    • n is the number of observations
    • F is the specified normal cumulative distribution function. If values have been standardized, it is the standard normal.

  5. If T is larger than the tabulated value, then the null hypothesis that the data conform to a normal distribution is rejected .