# What is the definition of "population" [duplicate]

I have a basic question on the definition of "population" in statistics.

Use this as example: the height of everyone in U.S.

In my study of probability and statistics course long time ago majoring engineering, the population means all of the heights of all people in U.S.

Now I'm taking courses to get certification as high school math teacher. In my math education class, the professor says I'm wrong. She wrote:

You say "population refers to the data being studied, which is the height of each student. The “population” is a collection of the heights of the students." This is not quite correct - the heights are considered the variable of the population/observational units/individuals being studied. The population is not the data values.

I talked to my professor again, and am sure that she says that in this case the population is all people.

I checked on internet, and saw both ways (mine and my professor's. I hope to get some clarity and answer here.

– Carl
Commented Apr 26, 2017 at 0:27
• I'd say population is generally used to refer to an ensemble of individuals, or things. Using this term to refer to all hypothetical measurements of that ensemble seems like a stretch to me. Commented Apr 26, 2017 at 0:38
• Krieger, N. (2012). Who and what is a “population”? Historical debates, current controversies, and implications for understanding “population health” and rectifying health inequities. The Milbank Quarterly, 90(4):634–681. Commented Apr 26, 2017 at 1:41
• My sense, from reading many textbooks, papers, and Web sites, is that those who claim a population is the set of numbers (such as heights) are those whose knowledge of statistical theory has not yet advanced to the point of clearly distinguishing sample spaces from random variables. The confusion may be benign in elementary settings, but it does a disservice to those who eventually think more deeply about such things.
– whuber
Commented Apr 26, 2017 at 13:21

There shouldn't be a debate about all these things. There is only one answer because all the things you mention are clearly defined.

Your population is all the US citizens.

You take a random sample of your population and now the variable "height of each individual" of your population is a random variable i.e: X1, X2, ...Xn are random variables because they depend on the sample (if you and I take 2 different samples of size n we both have the same X1, X2,...Xn random variables that indicate the height of each individual.

x1, x2,...xn are just observations which means that they are the specific/recorded heights of a specific sample. For instance 1.78 cm, 1.80 cm, 2.05 cm etc

Again in my sample X1 can take the value 1.78 cm and X2 can take the value 1.80 cm but in your sample X1 can be 2.10 cm and X2 can be 1.70 cm

I think your professor is more correct.

Typically, multiple items (variables, whatever) are measured on each member of a population.

If we only measure height then you might make an argument that the population is one of heights. But these have no meaning without the individual people.

There are some rough edges when it comes to population. Some are as follows.

We often refer to "the population of X on April 1, 2017." Of course, some people died and some were born that very day, as on every day. We have to say what to do with them.

Some people can't be measured, eg, those in hospitals, nursing homes, or prisons and jails. Some people are incompetent to be interviewed, eg, with dementia. Others do not speak the languages that are interviewed in. The most common survey languages are English and Spanish. People who only speak Navajo are not included. People who have no fixed address are often excluded, even if they are rich people who reside permanently in a hotel.

What is the height of someone who has had both legs amputated? You have to decide whether or not to include them and how to measure them. Any measurement you make is probably an estimate of their biological height.

"The population of the US in 1920" is even more unclear because of the time interval.

Our full definition of population must take these into account.

The notion of a well-defined population is not easily definable. We just live with it. Population is a term we like to throw off. It deserves more respect and attention.

While I think your professor is more right than you, I think there are many more important issues.