The statistical and structural characteristics of 13 matrices of random numbers in which both the cells and the entries were randomly chosen are discussed. Each matrix was explored considering row means, standard deviations, and correlations as well as column means, standard deviations, and correlations. A study concerning the sequential arrangement of digits was performed by finding out in tables of random numbers how many times the values 0 to 9 are followed by any other digit. Analyses indicate clear factor structures when factor analyzing correlations of rows and of columns and when examining sequential arrangements, concluding that for a given set of digits it is possible to assert both randomness and nonrandomness depending on how the data are examined.