Richard Dawkins, in his early book The Extended Phenotype, describes what he means when he says "statistically, X occurs". His original motivation was addressing a comment about gender, but it applies more generally:

If, then, it were true that the possession of a Y chromosome had a causal influence on, say, musical ability or fondness for knitting, what would this mean? It would mean that, in some specified population and in some specified environment, an observer in possession of information about an individual's sex would be able to make a statistically more accurate prediction as to the person's musical ability than an observer ignorant of the person's sex. The emphasis is on the word “statistically”, and let us throw in an “other things being equal” for good measure. The observer might be provided with some additional information, say on the person's education or upbringing, which would lead him to revise, or even reverse, his prediction based on sex. If females are statistically more likely than males to enjoy knitting, this does not mean that all females enjoy knitting, nor even that a majority do. [emphasis added]

I really enjoy his precise description of what statistics is. Ignore distributions, modelling, p-values and other statistical ideas for a moment: what statistics is really interested in is

For some event $$A$$, what characteristic $$X$$ allows $$P( A | X ) > P(A)$$.

For example, in Dawkins case, $$A$$ = person is a female, and $$X$$ = person enjoys knitting. Thus $$P(\text{person is female}\; |\; \text{person enjoys knitting}) > P(\text{person is female})$$.

Of course, in reality I don't have this much perfect information: I may have the ideal X, but not have enough data points to determine that $$X$$ is indeed significant. Conversely, I may have lots of data, but not the correct covariate $$X$$.