4 An example of application

In this section, we illustrate the new test by an application to real data. The data employed are the $V^{\prime }/V_m^{\prime }$ values from Table 5 of Schmidt (1968), where the data were believed to be uniformly distributed in the interval [0,1]. Indeed, the mean value of the data is 0.50 (see Schmidt 1968), and one may verify that the sample passes the K-S test.

In this sample, there are 33 data in total. Because some data share the same value, we have only 28 different values. Let x denote the random variable $V^{\prime }/V_m^{\prime }$. For a uniform distribution from to 1, the cumulative distribution function is

f(x)=x.

The distribution function of the sample can be calculated by Eq. (3). We can then build Table 1. In Table 1, Col. (1) gives the values of the random variable x. Column (2) presents the corresponding values of the cumulative distribution function f(x), from Eq. (29). Column (3) gives the values of the distribution function $f_N\left( x\right)$, calculated by Eq. (3). Column (4) presents the allowed $1\sigma$ deviation of the assumed distribution, ${\sqrt{{\frac{f(x)[1-f(x)]}N}}}$, and Col. (5) presents the deviation of the sample from the distribution, $\left\vert f_N(x)-f(x)\right\vert$.

  

Table 1: Data and deviations

.D..-1

From Table 1 we find that there is one point, i.e. x=0.035, where condition (9) is not satisfied. Therefore, the sample does not pass the $1\sigma$ distribution function deviation test for the assumed distribution according to definition 1.