Predictive power of a dataset
Usually predictive power refers to the model, rather than the data. I've occasionally seen some people use it in the way that the author of your book uses it (see this for example).
In the context of your book, yes, predictive power refers to whether input can be mapped to target output $X\rightarrow Y$. We can infer a dataset's &quot;predictive power&quot; by trying to model it (e.g. linear regression). If the model performs poorly, then there are two possibilities as the book says: either the dataset is not predictive (i.e. it does not offer a clean mapping from input to target output) or the methods we are using are unsuitable to model the mapping.

Some examples of both situations:

If you generated random data for $X$ and $Y$, the resulting dataset would (probably) have no predictive power as no model could reasonably generalize the mapping $X\rightarrow Y$.

If you have a nonlinear mapping, then linear regression would not fit it well. For example, if our dataset was such that $y_1$ is mapped to by $||\vec{x}||&lt;\alpha $ and all other inputs map to $y_2$, then our dataset is extremely predictive, but our linear regression model cannot fit it (since the mapping is nonlinear). In this toy example, it's easy to see the predictive power of the dataset, particularly if the input is in 2D/3D since we could just plot it. However, manually observing such trends in highly dimensional space using actual data can be very difficult, hence we use the tools that you are learning to help interpret the data. Also, when there's nonlinearity, it's difficult to statistically evaluate the dataset itself. Variables with linear relationships are simple to correlate (e.g. Pearson's correlation coefficient) but nonlinearities can make correlation difficult. I assume that this is why your book defers to vague terminology as it's probably for pedagogical, rather than pedantic, purposes. After all, it gets the point across without needing to discuss the ongoing research into quantifying nonlinear correlations.