When I plot my Regression Line and the corresponding lines for 2 standard deviations, sometimes it happens, that no values lie outside the 2 standard deviations. Is that possible/correct or must there values outside 2 standard deviations per definition of the standard deviation? I am not sure about this.

  • $\begingroup$ Can you please clarify, did you actually compute 2 SD's and if so, of what set of values? Or are you referring to the 95% confidence interval of the regression slope? In general, it is possible for there not to be values outside of 2 SD's from the mean, if the values are all relatively close to the mean, but the number of values is large. $\endgroup$ – AlexK May 6 '19 at 20:37

OK, given the comment by @whuber, I've revised my answer to make it more comprehensive and hopefully informative.

By 95% confidence intervals (CI) I mean 95% point-wise CI-s. There are also simultaneous CI-s, which are in general wider than point-wise CI, but they will not be discussed here.

Consider a simple regression setting: $ E[ y| x, \beta] = x^T\hat{\beta}$.

Then for each value of $x$, the corresponding 95%CI can be read as a confidence that the interval will contain the population mean for the specific value of $x$.

In other words, a specific 95%CI corresponds to an imagined scenario of many times repeated data gathering/sampling and re-analysis, where in (about) 95% of cases the calculated mean will be contained within such CI.

We know that the conditional distribution, for each $x$, of the mean response is normal with 5% point mass outside the CI of approximately $\pm 2SD$-s (obviously, the CI is the 95%CI).

Imagine for a moment that at each of the $x$ values the response was measured a large number of times.

Then, it should be clear that in case where the repeated measurements are many, about 5% of the points will be outside the corresponding CI. Otherwise, that CI wouldn't be well calibrated.

Now, back to the original problem of (supposedly) not seeing points outside the CI often enough.

When there are a few response data points per $x$, the extreme ones (let's call them so, that is, outside 95% CI) will show rarely, but in a large number of repeated scenarios, this will still occur for about 5% of the points. This means that one should see about one extreme response point per about 20 response points (for a specific $x$).

  • $\begingroup$ Some confidence intervals are constructed using standard deviations of the sampling distribution of the sample mean. That isn't what the question seems to be referring to. Regardless, it does not imply "5% of points may lie beyond:" that is an incorrect interpretation of a confidence interval. Your interpretation reads more like a prediction interval than a confidence interval. $\endgroup$ – whuber May 6 '19 at 16:49
  • $\begingroup$ @whuber -- I meant conditional mean and point-wise CI--s. $\endgroup$ – dnqxt May 6 '19 at 17:32

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