Statistics and meta-statistics in the concept of complexity

A crucial distinction between different classes of complexity measures is found to consist in their statistical structure. While first-order statistics always provides one or another kind of a measure of randomness, second-order (meta-) statistics provides more subtle measures of complexity. An example of a second-order measure is constructed and applied to specific cases. Some formal and conceptual issues concerning the general relevance of meta-statistical concepts of complexity are pointed out.