> Statistics is "the practice or science of collecting and analyzing numerical data in large quantities"

That's not true.

Statistic can analyze data in small quantities you can read up on it with nonparametric statistic. At least half of nonparametric statistic deal with small data (mostly through use of ranking). With Bayesian stat you can just assign a prior distribution.

I think the best way to describe statistic is that it uses data to infer the population. A subset data via sampling and infer a statistic (mean, median, whatever) about the population at large.

More often than not I'm just surprise as to why Data Science is so big when seemingly it seems like statistic does every freaking thing with data. From how to correctly sampling, designing experiment to collect the correct data to answer a hypothesis, making sure the data aren't bias, etc.. It deals with designing the data from inception to end, including either collecting the data or data given to you already. On top of dealing with problematic data such as missing data, imputation, etc..