### Something close to normal (for a change) in the backyard

In yesterday's post I presented some shell measurements I had obtained with the land snail *Vertigo pygmaea* that lives around my house. To construct a histogram of shell heights, I combined all 3 sets of data†. The resulting histogram is below. I haven't done any statistical tests, but visually, the distribution of the shell heights of *V. pygmaea* seems close to being normal.

The blue curve is my combined measurements of

*Vertigo pygmaea*; the red curve is a normal curve with the same mean and standard deviation. I constructed the normal curve using a normal probability calculator. The total number of specimens for each curve is 132.

The graph below shows the area under a theoretical normal distribution. For our purposes, this graph tells us that in a sample of snail shells, 99.7% of the measured heights are expected to be within plus or minus 3 standard deviations (SD) of the mean. In more practical terms, one could expect to find on the average 3 shells outside the (mean±3xSD) range out of about every 1000 shells collected. However, this is not a hard rule; it only tells us what things will be like on the average in the long run. In fact, on a few occasions, I have found very small or very big shells in samples much smaller than 1000 specimens. It is, however, good to keep in mind these theoretical limitations expected from a normal distribution. For example, if I happened to find, in a sample of 100 shells, say, 3 shells whose heights were <(mean-3xSD), I'd strongly suspect that something was going on. For example, the measurements might be wrong, the sample might be biased (the small shells might have been selectively taken from a larger sample), the distribution might not be normal, the shells might have been broken and repaired‡, the small shells might belong to another species, etc.

Area under a normal curve. From Brase & Brase,

*Understanding Basic Statistics*, Houghton Mifflin, 1997.

My combined sample of 132 shells of

*V. pygmaea*has a mean and SD of 1.89 mm and 0.085 mm, respectively, giving a (mean±3xSD) range of 1.64-2.15 mm. The shortest (1.65 mm) and the longest (2.12 mm) shells are within the range and quite close to the "limits".

I have written about miniaturization in land snails (here and here) and the smallest land snails. I don't know how small

*V. pygmaea*can get. I will collect another sample for measurement about 2 years from now or sooner, if I chance upon large numbers of them. It follows from the argument above that in a large enough sample, I should find a shell smaller than what I have so far measured.

†Although this is in general a questionable practice, I am justifying it in this case, because all the specimens were from more or less the same locality (I don’t have a large yard), the time span was relatively short and the mean and the standard deviation values of the samples were close enough to each other.

‡Repaired shells sometimes end up being smaller than they were before. I have an example here.

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