TITLE: Is a data set really a set?
QUESTION [8 upvotes]: Originally I thought that in statistics, a data set is just a set of real numbers, and that was it. But in the case of a set, there can only be one instance of any given entry, e.g. in  set theory
$$\left \{ 1,2,2 \right \}=\left \{ 1,2 \right \}$$
A set of this form in set theory may also be called an unordered pair.
But from the point of view of statistics the objects on both sides of the equation are distinct, e.g. the left one has a mode, while the right one doesn't; the left one's arithmetic mean is $\frac{5}{3}$, while the arithmetic mean of the right one is $\frac{3}{2}$.
Question: are the concepts of data set in statistics and set in set theory really different things? Are data sets of real numbers in statistics really $n$-tuples of set theory in disguise, or taken to be ones most of the time implicitly?

REPLY [5 votes]: In most cases the data set will be a true set if you view it as a set of observations.  In the example of temperatures, there are only a few different temperatures, but each one corresponds to a different day.  Your data set consists of ordered pairs (day, temperature on that day) and no pair is repeated.  The only way you get repetition is if you observe the same data more than once.  If you have a data set of the number of legs on horses, your observations are (horse name, number of legs).  If you have a repeat, you have observed the same horse more than once, so you might want to delete one observation.  Alternately, you might worry that a horse had lost a leg, in which case your data is (horse name, date observed, number of legs) and again you won't have any duplicates.