# Confidence Intervals for Population Estimates: Just In Case? [duplicate]

I was thinking about the following question today:

Suppose I visit a university and want to determine the average height of a student at this university. Let's say I spend the whole week at the university and measure the height of every student on the entire list of students at the university (i.e. population). By the end of the week, I have measured every student on this list - I then take the average height from all these measurements.

In this case, I have was lucky enough to access the population of students instead of a random sample of students. Therefore, in a theoretical sense, there should not be any "risk" or "uncertainty" associated with my average measurement.

However, in reality, there is likely always going to be some source of error. For instance, it's possible that I might have made some measurements incorrectly, I didn't notice that some students were wearing shoes thus adding to their height, perhaps in reality the list might only contain 99% of the students and some students were not on this list, etc.

Thus, in such instances, even though I believe I am dealing with the population, there still might be errors associated with my data - some of these errors might be related to sampling errors because I might be only dealing with 99% of the population whereas some of the errors might be caused by other reasons (e.g. experiment related, measurement error, etc.).

This leads me to my question: In such cases when you think you are dealing with the entire population, does it still make sense to calculate the Confidence Interval for what you believe to be the population estimate ... as doing this might serve to somehow add a useful level of uncertainty to your knowledge? Or would calculating a Confidence Interval in such an example still be meaningless and this sense of uncertainty would be both misleading and meaningless as Confidence Intervals do not "magically safeguard" your estimates from all possible sources of error?

Thanks!

References:

• You need to make assumptions about your sources of error and their scales, and perhaps try to estimate them (e.g. remeasure some of the students to see how different the values are) Apr 25 at 10:58
• May I recommend a title change? Right now, it looks to be asking about confidence intervals for the usual situation of using a sample from the population. Perhaps consider a modification to reflect the fact that you measured the entire population (but with the possibility of errors).
– Dave
Apr 26 at 11:40
• This is a popular question. Here are some previous variants: Statistical inference when the sample "is" the population, Do we need hypothesis testing when we have all the population?, Can the sample equal the population?. Apr 26 at 17:12

While it makes sense to acknowledge that even population data might have errors or uncertainty for various reasons, you can't quantify that error using the specific statistical procedure we refer to as "calculating confidence intervals." Confidence intervals are designed to do one thing and one thing only: to quantify the amount uncertainty associated with using a random sample to generalize about the population from which it is drawn (that is, the amount of uncertainty due purely to sampling error). The reason this is actually possible to do (the reason we can quantify this uncertainty precisely) is due (in part) to a particular mathematical result called the central limit theorem (CLT), which is specifically about the using random samples to generalize to a population. The math used to generate confidence intervals stems directly from the CLT and related assumptions.

The different kinds of errors you are talking about are not related to the issues of using a sample to generalize about a population, and thus the CLT, and associated procedures like confidence intervals (or things like p values), can't help you quantify how big of a problem they are. In particular, these errors are likely to be non random - they are likely to affect certain kinds of observations more than others, leading not only to error, but bias, and confidence intervals don't help you to address issues of bias.

Now, if you try to calculate confidence intervals on a population, using the simple version of the formula

$$CI=x \pm } z \frac{s}{ \sqrt{n}}$$

And you put in the population N for "n" and the population standard deviation for "s" then you will get confidence intervals. But these will be meaningless, because one of the assumption of this formula is that the population you are drawing the sample from is infinitely large (or rather, that your sample is an infinitesimally small proportion of the population). In practice, this is usually a reasonable assumption (the proportion of all Americans included in any one election poll is basically zero). But if your sample is a non-trivial proportion of the population, then to get the correct confidence intervals you need to adjust that formula with a "finite population correction" which will make the confidence interval smaller, since there is less opportunity for un-sampled population members to contribute to error. In your case, where the sample size equals the population size, this version of the CI formula will correctly tell you that the size of the confidence interval is zero - since there is no possibly of the result being wrong due to sampling error, which is the only thing confidence intervals are concerned with.

In short: there are lots of reasons why data (whether it is from a sample or population) might be biased or wrong. But the procedure of confidence intervals is only designed to quantify one of those sources of errors - sampling error.

• @ Graham Wright: thank you so much for your answer! Apr 26 at 15:22
• It is not in principle a problem that no confidence interval can be calculated. It is a practical problem, because no good estimate can be made out of only a single measurement. You can repeat the same measurement several times and use the variations among those measurements to estimate the sampling error. Apr 26 at 19:01
• FYI, what the other answer (that closed your question) is referring it is this: Aside from what you were proposing (using CIs to quantify measurement error etc) you could basically "pretend" that your population actually is a sample of some even bigger population (so pretend that the pop of students at your school in 2022 is a sample of all possible student populations) and then calculate CIs - but this would still just quantify sampling error associated with this hypothetical sample, not other sources of error. Apr 28 at 0:28

does it still make sense to calculate the Confidence Interval for what you believe to be the population estimate

You already answered this yourself. Yes it makes sense because of sources of error/variation in the measurements.

• @ Sextus Empiricus: thank you so much for your answer! I think this is unrelated, but I have been trying to understand more about this question here : stats.stackexchange.com/questions/614128/… if you have time, can you please take a look at it? Thank you so much! Apr 26 at 15:21
• @stats_noob I will add a comment to that question. Apr 26 at 19:14
• @ Sextus Empiricus: thank you for your comment there! I think taking the logarithm of the response could be helpful in such a problem ... Apr 26 at 20:10

Confidence intervals grow out of a school of thought called Frequentism, in which we assume we could hypothetically repeat such a job an unlimited number of times.

So all the students of that university are just a sample of all those unlimited numbers of universities that might have collected students from the same background as your specific university has. Does that make sense? Not really, bug kind of:

Whenever the university that you studied stands in as an example for universities in general, you may consider reporting things that may not be sensible for the one actual thing you are investigating.

It can be misleading though, if someone took that as a measure for different universities that take in different students.

• @ Bernhard: thank you so much for your answer! Apr 26 at 15:22