each day), so is likely to follow Benford's law quite well. Multiple probability distributions Anton Formann provided an alternative explanation by directing attention to the interrelation between the distribution of the significant digits and the distribution of the observed variable. He showed in a simulation study that long-right-tailed distributions of a random variable are compatible with the Newcomb–Benford law, and that for distributions of the ratio of two random variables the fit generally improves. For numbers drawn from certain distributions (IQ scores, human heights) the Benford's law fails to hold because these variates obey a normal distribution, which is known not to satisfy Benford's law, since normal distributions can't span several orders of magnitude and the mantissae of their logarithms will not be (even approximately) uniformly distributed. However, if one "mixes" numbers from those distributions, for example, by taking numbers from newspaper articles, Benford's law reappears. This can also be proven mathematically: if one repeatedly "randomly" chooses a probability distribution (from an uncorrelated set) and then randomly chooses a number according to that distribution, the resulting list of numbers will obey Benford's law. A similar probabilistic explanation for the appearance of Benford's law in everyday-life numbers has been advanced by showing that it arises naturally when one considers mixtures of uniform distributions. Invariance In a list of lengths, the distribution of first digits of numbers in the list may be generally similar regardless of whether all the lengths are expressed in metres, yards, feet, inches, etc. The same applies to monetary units. This is not always the case. For example, the height of adult humans almost always starts with a 1 or 2 when measured in metres and almost always starts with 4, 5, 6, or 7 when measured in feet. But in a list of lengths spread evenly over many orders of magnitude—for example, a list of 1000 lengths mentioned in scientific papers that includes the measurements of molecules, bacteria, plants, and galaxies—it is reasonable to expect the distribution of first digits to be the same no matter whether the lengths are written in metres or in feet. When the distribution of the first digits of a data set is scale-invariant (independent of the units that the data are expressed in), it is always given by Benford's law. For example, the first (non-zero) digit on the aforementioned list of lengths should have the same distribution whether the unit of