The idea of plotting confidence intervals on data, which is discussed in a number of posts elsewhere on this blog, should be straightforward. Everything we observe is uncertain, but some things are more certain than others! Instead of marking an observation as a point, its better to express it as a ‘cloud’, an interval representing a range of probabilities.
But the standard method for calculating intervals that most people are taught is wrong.
The reasons why are dealt with in detail in (Wallis 2013). In preparing this paper for publication, however, I came up with a new demonstration, using real data, as to why this is the case.