  Data collection is a fundamental problem in the scenario of big data, where
the size of sampling sets plays a very important role, especially in the
characterization of data structure. This paper considers the information
collection process by taking message importance into account, and gives a
distribution-free criterion to determine how many samples are required in big
data structure characterization. Similar to differential entropy, we define
differential message importance measure (DMIM) as a measure of message
importance for continuous random variable. The DMIM for many common densities
is discussed, and high-precision approximate values for normal distribution are
given. Moreover, it is proved that the change of DMIM can describe the gap
between the distribution of a set of sample values and a theoretical
distribution. In fact, the deviation of DMIM is equivalent to
Kolmogorov-Smirnov statistic, but it offers a new way to characterize the
distribution goodness-of-fit. Numerical results show some basic properties of
DMIM and the accuracy of the proposed approximate values. Furthermore, it is
also obtained that the empirical distribution approaches the real distribution
with decreasing of the DMIM deviation, which contributes to the selection of
suitable sampling points in actual system.
