correct description of confidence interval

doc/tutorials/error_analysis/error_analysis_part1.ipynb

@@ -102,7 +102,9 @@
     "At first glance, it might seem to be very expensive to compute the SEM, because one would have to repeat the whole simulation many times. However, under the right circumstances, the SEM can be estimated from *a single series* of $N$ measurements. We will discuss how this can be done.\n",
     "\n",
     "### Confidence interval\n",
-    "A common confidence interval (CI) is the $95~\\%$ CI. If we were given a number and a range around it, called the $95~\\%$ CI, we would know that, with probability $95~\\%$, the true mean value lies within that range. If the SEM is known, the $95~\\%$ CI is roughly $\\pm1.96s$.\n",
+    "A conficende interval (CI) specifies a range of numbers within which the unknown true mean value lies with a certain probability. A common confidence interval (CI) is the $95~\\%$ CI, which would contain the true value with probability $95~\\%$. Care must be taken interpreting the CI, since the lower and upper bound of a CI are themselves random variables. Just as a simulation run drafts samples from the overall ensemble, determining a CI from a simulation run is drafting a CI from all possible CIs. When the upper and lower bound of a CI have been calculated, this range either contains the true value or not, so there no longer is a probability attached to it. However, for repeated simulations with subsequent computation of the corresponding CIs, on average $95~\\%$ of CIs will contain the true value, while $5~\\%$ won't.\n",
+    "\n",
+    "If the samples are normally distributed and the SEM is known, the upper and lower bound of the $95~\\%$ CI are $\\overline{X}\\pm1.96s$.\n",
     "\n",
     "### Interquartile range\n",
     "The interquartile range denotes the range, within which the central $50~\\%$ of all samples lie, if one were to order them by their size. This leaves one quarter of all samples lying below the interquartile range, and another quarter of all samples above it.\n",