# What is the difference between sample space and population?

I wonder why people cannot differentiate between sample and population? It is rediculous, IMO, not to see the difference between Universe and one of its subsets. Meantime, nobody seems to question about the difference between sample space and population. How do I know if I take a sample from sample space or from a population? Both seem like concepts of Universe of discourse. Is one defined in probability theory whereas another comes from statistics and one does not care much about the other but they still mean the same thing or there is a bigger difference?

• As according to the wikipedia links you gave, a population is a complete set of items that share at least one property in common that is the subject of a statistical analysis, whereas the items in a sample space do not need to share any properties in common. So in my opinion that is a big difference. – user30490 Oct 1 '15 at 13:44
• @ZERO Should I conclude that the population is a subset of sample space and sample space is a unconstrained population (common property is vacuous conjunction of properties)? – Valentin Tihomirov Oct 1 '15 at 13:46
• Also, I think you could make a reasonable argument that the population could be viewed as a subset of the sample space. – user30490 Oct 1 '15 at 13:46
• @ZERO Are you saying that the fact that I take sample from population is not enough to say that population is a sample space? – Valentin Tihomirov Oct 1 '15 at 13:48
• Nope, I never said that. – user30490 Oct 1 '15 at 14:05

The population is the set of all units a random process can pick. The sample space S is the set of all possible outcome of a random variable.

For example, the population can be the complete population of the US. Then your random process picks a person, John Smith.

If your random variable asks the color of hair of a person, then S={black, brown, blonde,...}. If your variable asks the age, S = [0,130[. If your variable asks the number of letters in the last name, then S=N.

In some examples, they are the same, like if you ask for the number of points on the dice. Then the population is {1,2,3,4,5,6} and the event space is also {1,2,3,4,5,6}.

In the case of one random variable, this concept is a bit tedious. It becomes very clear and important when you have multiple variables. Then one realization, John Smith, can answer all these questions, X_1 ... X_n.

In both Probability and Statistics, we refer to the (probability-theoretic) experiment of drawing a SAMPLE, size n, from a (statistical) POPULATION. The SAMPLE SPACE for this experiment is therefore the set of all n-element samples. If n = 1, then the sample space is the same as the population.

Now, I understand that both of these last two concepts have been considered equivalent to the notion of "universe". It can be said that we begin with the population as our universe, and then modify the universe so that it consists of n-element samples (subsets/combinations) of the population. The difference between Probability and Statistics is that the population is fully known in the study of probability theory, while statistics is about using the chosen sample to make an inference about the population.

• A sample with n=1 is not the whole population. Your answer doesn't seem to make any sense. – Michael R. Chernick May 20 '17 at 1:56
• @michael chernick: its about sample space, not one single sample. – Michael M May 20 '17 at 20:14
• It is worded very poorly. What does the word n-element samples mean and why does it need to be introduced? How does extending a population to our universe clarify anything about a statistical population. – Michael R. Chernick May 20 '17 at 21:04