Some of the most common **misconceptions about confidence intervals** are:
* “There is a 95% chance that the true population mean falls within the confidence interval.” *(FALSE)*
* “The mean will fall within the confidence interval 95% of the time.” *(FALSE)*
While I do know the true definition of confidence intervals, I wonder why the above are not true?
In: Mathematics
If you think about a bell curve, the average is in the middle of bell. Going out +/- 2 standard deviations gives you the range within 95% of the bell curve. The mean always lies within the distribution, not 95% of the time.
Confidence intervals are calculated from samples of a population, centered around sample mean of interest and ranging up or down from it by the standard deviation scaled by the confidence level. Every sample produces a different mean/SD, and thus a different interval. If you calculated all possible sample CIs for some population, the proportion of them that would contain the true population mean somewhere in the CI would be the same as the confidence level (e.g. calculating a 95% CI for every possible sample -> 95% of the resulting CIs would contain the true population mean). So then the statements given would not be true for any given sample CI, as you would not have any capacity to judge whether a given CI is among the 95% (or whatever confidence level) or not based on that single sample alone.
Aleatory vs epistemic probability goes kinda beyond ELI5 stuff, so let’s put that in a box and lock the box up and put a hungry tiger on top to guard it.
The first statement is false because the probability that this confidence interval includes the true mean isn’t 95%, it’s either exactly zero or exactly one — you just don’t know which.
The true population mean doesn’t fall within the CI you just calculated 95% of the time. Rather, if you repeated your procedure many times, you would find that in 95% of cases, the population mean fell within the CI you calculated in that instance.
For instance, if I simulate many samples of 10 data points that I draw from a Normal distribution with mean 0 and variance 1, and for each sample I calculate a CI around my sample mean, I will find that in approximately 95% of samples, the calculated CI contains 0 (the more samples I draw, the closer this proportion will be to exactly 95%).
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