What possible relevance can statements about the sample be in helping us draw a conclusion about the population?
In response to your question about commenting on distributional shape, we refer you to the following three things:
1. Part of a talk we gave at Teachers Day in 2009:
T: Tell me Matt (Regan), why are we getting students to describe and infer about shape, spread and so on? It is just so long-winded and laborious getting students to see and write down all this stuff.
A: Good question! It’s important. The general answer is that it is all about training, we are trying to train students to become good data detectives, to use their eyes and to learn what to look at and what to look for in plots. Why would we want to do that? Well, one good reason is that at higher levels, students will need these skills when they use formal tools of analysis. All formal methods of analysis make assumptions and often these assumptions involve properties like shape and spread of the underlying distribution (or population distribution). The primary way of checking these assumptions is through the use of plots and so they will need to have had lots of experience in looking at, and commenting on, features they see in plots.
More than this though, it is simply part and parcel of what a good data detective is about. Sure, we want them to make a call on a specific issue at hand, (e.g., who tends to be taller, year 11 boys or year 11 girls?) but as well as this, we want them to be alert for anything else that the data may be trying to tell them, anything else which is interesting, unusual or unexpected. The way to do this is, in the early days, require them to look at and comment on specific properties like shape and spread and after a while it will become second nature to them, they will just do it automatically, almost reflex-like. It all boils down to a matter of good statistical practice.
2. A paper we wrote in 2010 including the appendix. See: Pfannkuch, M., Regan, M., Wild, C.J., & Horton, N. (2010). Telling data stories: essential dialogues for comparative reasoning. Journal of Statistics Education, 18(1).
Note what is said on p. 14:
Starting at ages 9 and 10 years in the new New Zealand curriculum (Ministry of Education, 2007), students will describe features of dot plots, and at ages 11 to 13 students will make descriptive statements about two samples through visually comparing dot plots. Therefore, the reasoning elements described in the Analysis section of this document (Overall visual comparisons; Shift and overlap; Summary; Spread; Shape, Individual, and Gaps/Clusters) should already be embedded into students’ thinking to some extent. By stressing all the reasoning components involved in comparison we wish to emphasize that point estimates and zeroing in on immediately making a call are insufficient and only a very small part of the underlying rich conceptual repertoire and data-dialogue.
3. Some activities for teaching to students to think about shape.
