## Monday, December 10, 2012

### Not Normal Either and Why

I found this picture on Pinterest with the caption "Normal Distribution of cup lids." (Yes, I do troll for such things). I'm sorry, but yet again, this is not a normal distribution. Perhaps it resembles histogram bars from what could be a normal sample, piled high in the middle and lower on each side. But the problem is, as seen before, a normal distribution needs a measurement scale, a number line. Few small things on the left, few large things on the right, and many middle size things, of course, in the middle. This is the pattern shown by these lids, but there is no scale of measurement, no number line. It is not a frequency distribution of any obvious measurement. A similar stacking of items is shown below:
Scallop shells with different numbers of ribs (15 through 20) are stacked. This represents few small (scallops with 15 or 16 ribs), many middle-sized (with 17 or 18 ribs) and few large shells (with 19 or 20 ribs). This few-many-few pattern does resemble the cup lids above, but the lids have no measurement scale. The scallops are stacked into a real frequency distribution by their number of ribs. This is not a normal distribution since we are counting a discrete number of ribs. But it is a somewhat bell-shaped frequency distribution. The two scallops on the left have 15 and 20 ribs, respectively. More ribs seems to indicate an overall larger size or perhaps older scallop. A frequency distribution of a continuous measure of size, say mass or area, would be much more likely to be described a normal distribution.

Note: This is from the same book, Graphical Methods for Presenting Facts by Brinton (1914) that republished the earliest "living histogram" that I have found. This earliest one comes from Popular Science Monthly, September 1901 on page 447 of the article "The Statistical Study of Evolution" by C. B. Davenport.