Is there any assumption between confidence intervals and data normalty? I read this discussion on whether confidence intervals are useful and I am trying to figure out if there is any underlying assumption about the data that I used to construct a confidence interval.

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*Does my data need to come from a normal distribution?

*What does happen if the data used to construct confidence intervals is not normally distributed?

*Bonus question: why should I use confidence intervals and not just report mean/median and standard deviation?

 A: *

*No, although a lot of confidence intervals you may encounter are motivated by either transforming some statistic onto a scale in which the sampling distribution is something close to normal or by using asymptotic arguments (e.g. with enough data, the distribution is close enough to normal).
The sample mean for an exponential random variable is a good example of this.  If $X_1, \dots, X_n$ are iid samples from an exponential distribution with rate parameter $\lambda$, their sample mean distributed according to a gamma distribution with shape and rate $n$ and $n\lambda$ respectively.   The data are not normal, and yet we can obtain a confidence interval as explained here


*The consequence of non-normality is that it may be harder to compute a confidence interval pen and paper.  You might have to resort to using computational methods (like the bootstrap or profile likelihood methods).


*Confidence intervals communicate uncertainty in the estimate, whereas the standard deviation as a measure of uncertainty on the level of the observation.  Let's say I report a mean of 10 and a standard deviation of 2.  I am more certain that the estimate of 10 is closer to the truth if I used 10,000 data points as opposed to 100 data points (assuming I used a random sample and there is no bias in my data collection).  Confidence intervals account for this.  As you get more data, confidence intervals will get narrower, meaning there less uncertainty in our estimate.
