We study two randomness measures for the celebrated Kroneckersequence S(!) formed by the fractional parts of the multiples ofa real !. The first measure is the well-known discrepancy, whereas theother one, the Arnold measure, is less popular. Both describe the behaviourof the truncated sequence ST (!) formed with the first T terms,for T !". We perform a probabilistic study of the pseudorandomnessof the sequence S(!) (discrepancy and Arnold measure), and we giveestimates of their mean values in two probabilistic settings : the input !may be either a random real or a random rational. The results exhibitstrong similarities between the real and rational cases; they also showthe influence of the number T of truncated terms, via its relation to thecontinued fraction expansion of !.