**4. Results**

First, let us examine the raw data, looking at the scatter plot between frequency and length in Figure 1, where each point is a word type represented by an associated value of *n* and an associated value of - (note that several or many types can overlap at the same point, if they share their values of - and *n*, as these are discrete variables). >From the tendency of decreasing maximum *n* with increasing -, clearly visible in the plot, one could arrive to an erroneous version of the brevity law. Naturally, brevity would be apparent if the scatter plot were homogenously populated (i.e., if *f*(-, *n*) would be uniform in the domain occupied by the points). However, of course, this is not the case, as we will quantify later. On the contrary, if *f*(-, *m*) were the product of two independent exponentials, with *m* = ln *n*, the scatter plot would be rather similar to the real one (Figure 1), but the brevity law would not hold (because of the independence of - and *m*, that is, of - and *n*). We will see that exponentials distributions play an important role here, but not in this way.

**Figure 1.** Illustration of the dataset by means of the scatter plot between word-type frequency and length. Frequencies below 30 are not shown.

A more acceptable approach to the brevity-frequency phenomenon is to calculate the correlation between - and *n*. For the Pearson correlation, our dataset yields corr(-, *n*) = −0.023, which, despite looking very small, is significantly different from zero, with a *p*-value below 0.01 for 100 reshufflings

of the frequency (all the values obtained after reshuffling the frequencies keeping the lengths fixed are between −0.004 and 0.006). If, instead, we calculate the Pearson correlation between - and the logarithm *m* of the frequency we ge<sup>t</sup> corr(-, *m*) = −0.083, again with a *p*-value below 0.01. Nevertheless, as neither the underlying joint distributions *f*(-, *n*) or *f*(-, *m*) resemble a Gaussian at all, nor the correlation seems to be linear (see Figure 1), the meaning of the Pearson correlation is difficult to interpret. We will see below that the analysis of the conditional distributions *f*(*n*|-) provides more useful information.
