The definition of confidence intervals is somewhat unintuitive. Usually when people try to reformulate this definition into something more intuitive, they run into trouble. (That is, their new definition isn't quite true.)

I like the explanation given by Dr. Nic in this video:

https://www.youtube.com/watch?v=tFWsuO9f74o

I think the Wikipedia article is also good, although I didn't vet the whole thing.

https://en.wikipedia.org/wiki/Confidence_interval

Naeem Aslam If you use a sample of a specific size (N) to estimate a population parameter - your estimated population parameter will fall within this range or interval 95% of the time.

*Note that I wrote "estimated" population parameter, not "true" population parameter. This is a widespread misunderstanding.

Hello Naeem Aslam. This excerpt from the article linked below may help.

“An exact 95% confidence interval is calculated such that it includes the true value of the estimated parameter 95% of the time. We do not know, however, if the interval we have is one of those that are correct or not. It is like a person who tells the truth 95% of the time, but we do not know whether a particular statement is true or not.”

Roberto Behar, Pere Grima & Lluís Marco-Almagro (2013). Twenty-Five Analogies for Explaining Statistical Concepts, The American Statistician, 67:1, 44-48, DOI: 10.1080/00031305.2012.752408

Article Twenty-Five Analogies for Explaining Statistical Concepts

Notice that contrary to what Blaine Tomkins posted, the "true value of the estimated parameter" would be contained within 95% of 95% CIs if one had the CIs for all possible samples of size N.

For a nice visualization of CIs, and the fact that 95% of 95% CIs contain the true value of the estimated parameter, take a look at this page:

• https://rpsychologist.com/d3/ci/

Finally, many attempts to reformulate the correct definition into something more intuitive (as Sal Mangiafico put it) end up producing something that sounds more like a Bayesian credible interval.

• https://en.wikipedia.org/wiki/Credible_interval

HTH.

PS- Here are some more YouTube videos you might find helpful.

Parts 1 & 2

• https://www.youtube.com/watch?v=U59Rbpus824
• https://www.youtube.com/watch?v=lmCWaAEfPhE

Parts 3, 4, and 5

• https://www.youtube.com/watch?v=zMndXTnZjbs&t=3s
• https://www.youtube.com/watch?v=EWK1uOPniaw&t=22s
• https://www.youtube.com/watch?v=Dg8GbnyEOz8

@BruceWeaver This is true in theory, but it would be nonsensical to obtain the 95% CI for every possible sample of size, N. If you could measure every possible sample, you would have the true population parameter - defeating the purpose of calculating the interval in the first place.

see the attached article for a better sense of what the confidence interval does and does NOT describe

Thanks for the article, Blaine Tomkins. For those who find it TLTR, here is a one-page letter that covers the basics of what frequentist 95% CIs do and do not convey.

• https://pubs.rsna.org/doi/abs/10.1148/rg.2016150248?journalCode=radiographics

The definition given in that letter is shown in the attached image. The (4) they cite is the Morey et al. article you shared, albeit with 2016 (not 2014) as the date. (Perhaps the one you shared is the accepted author ms rather than the final published article.)

Understanding a sampling distribution also helps here.

When one talks about a population parameter (in hypotheses testing), he actually makes use of a sampling distribution.

Blaine Tomkins, nothing in the definition I quoted suggests that one has to be able to compute the CIs for all possible samples of size N.

The correct definition of a frequentist CI is based on the hypothetical notion of drawing all possible samples of size N from a population and computing the x% CI for each sample. If one could do that, x% of those intervals would include the true value of the parameter being estimated, and 100-x% of the intervals would not.

In the usual case, we have just one sample and one of those possible intervals. But as the authors of the analogy put it, we do not know if it is one of the x% of intervals that tell the truth (i.e., they include the parameter) or one of the 100-x% of intervals that lies (i.e., they do not contain the parameter).

Bruce Weaver Ok, now I think I see the source of my confusion. I think we came at this question from different angles. You are describing what a confidence interval is in theory. I was coming at the question moreso from the practical side (i.e., when I calculate a confidence interval from my one sample statistic, what does that interval mean?).

In other words, you're talking about the confidence interval that would be created from all possible samples, I was talking about the confidence interval observed from just one sample. These are entirely different things.

This might be one of those bizarre situations where we're both correct because we're talking about different things. My apologies for creating confusion.