# Population vs Sample Space

In probability theory, it is my understanding that the sample space $$\Omega$$ is the set of all possible outcomes of an experiment. In my mind, I am thinking of it as a set of data points such as height, faces of a die, etc. depending on what your experiment is measuring. I am also of the understanding that these outcomes could really take any data form like tuples (0,1), single values (1), strings like "true" or "false", etc.

I am trying to understand how this relates to the concept of a population. For example, if my experiment is measuring the heights of NBA players, then would the population be the set of all those measurements while a sample space would be a (non-strict) subset of it? I have read other questions online about this, but am still confused.

In statistics, it depends on whether we are interested in sampling from finite populations. Usually we are not, except when we are in the survey literature. The concept of the population for finite population sampling is straightforward. It is the population from which you sample. For example, for NBA players, it would be the actual players themselves in a particular season.

More often, we assume we are sampling from an infinite population. This can be applied to the case even when the population you sample from is apparently finite. For example, with the sampling of NBA players, we can imagine a larger population of "potential" NBA players, including future players. Since this hypothetical population does not exist, we can imagine it to have infinite possibilities. In this sense, the population is more than just the sample space. It also has a population distribution associated with it. In classical statistics, this distribution is assumed "fixed but unknown", and the purpose of our sample is to make inferences (intelligent guesses) about it. We also tend to make assumptions to ensure we have a "nice" problem to work with. For example, we would typically assume height is on a continuous scale, even though measurements are of finite precision (discrete). We may even assume it follows a Normal distribution if that helps.

This is to say, a "population" is generally a hypothetical construct to allow us to do maths. It may be regarded as an approximation to the "real" population we are interested in making inference on.

Sample space is a concept in probability theory, its a component of the probability space $$(\Omega,F,P)$$, in your case you can say the sample space is the heights of all the NBA players.

While "population" is a term used in statistics. It's used in combination with the term "samples" to emphasize "samples" is a subset of the "population". The population is also the heights of all the NBA players.

It happens that these two concepts referring to the same thing in your case, but it's not generally true for all other cases.

• I think it is confusing to conflate elements of a population or sample space with the values of a random variable. For instance, when studying heights of NBA players, both of these sets would be sets of people, not heights. Which people are in them depends on one's model and objectives. For instance, they might consist of all current NBA players; all current and former players; or even a hypothetical "population" of all possible future NBA players. – whuber Apr 23 at 13:25