Dear Bob,
Using your average & standard deviation this is what I get:

(21-15.5)/1.33= 4.135... & (15.5-10)/1.33= 4.135...

Thus, shells with 10 or 21 labial teeth are just beyond 4 standard
deviations away from the mean. In a normally distributed population,
0.0000317% of the specimens will be below & above the 4 SD mark. So, on the
average expect one shell with 10 & another with 21 teeth per about 3 million
shells! Then again, the next shell you pick up may turn out to have 10 or 21
teeth. Good luck.

I have looked at your results on the Internet. To me your Fig. 8 looks as
good as any other normal distribution I have seen. I wouldn't worry about
the results of chi-square too much. Sewall Wright has a section on
distributions of meristic variables (variables that are not continuous, such
as teeth counts) in vol. 1 (pp. 127-139) of his Evolution & the Genetics of
Populations. Most of that book is mathematically beyond my (and I suspect
most biologist's) understanding, but he does discuss in non-mathematical
terms many examples from the literature. I recommend the relevant section.
He states that the distributions of many meristic variables are "near
normal". That may be what you are seeing.

Have you considered publishing your results?

Aydin

>-----Original Message-----
>From: Bob Dayle [mailto:[log in to unmask]]
>Sent: Friday, May 03, 2002 8:41 PM
>To: [log in to unmask]
>Subject: Re: yet more tooth stats...
>
>
>Aydin (and any others),
>
>In an attempt to keep the "blurb" of data short,
>some things were held back.
>
>Yes, the distribution is normal (it is viewable
>at http://www.cowrys.org/research/hlvteeth.htm ).
>The C. helvola average labial tooth count is 15.5 and
>its standard deviation is 1.33 teeth. On a line, it
>looks like this (with std. dev. above the line and
>labial tooth count below the line):
>
>  -4    -3    -2    -1     M    +1    +2    +3    +4
>---^-----^-----^-----^-----^-----^-----^-----^-----^---
>^12     ^13     ^14     ^15 16^     17^     18^     19^
>
>The estimate of four std. dev. out is close for not
>knowing the mean and std. dev.! The shells with 11 and 20
>labial teeth are beyond the 5th std. dev. point, so there
>may _not_ be a possibillity of 10 or 21 labial teeth (in
>Hawaii, anyway. Who knows what is possible elsewhere?
>
>Although I think my sample of C. helvola is fairly
>representative of Hawaii, my samples of other species
>from Hawaii are much smaller and cannot be expected to
>contain ALL of the possibile labial tooth counts, so I
>"inflated" some populations to see what might be likely.
>
>A few ideas are yet to be fleshed out concerning C. helvola
>but most of the tooth count data are posted. A nagging
>mystery is why the difference between the labial & col-
>umellar counts on these specimens fits the normal curve
>best of all yet the columellar tooth count fails (badly)
>in matching a Gaussian distribution.
>
>Aloha,
>
>makuabob (a.k.a. Bob Dayle)
>
>_________________________________________________________________
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>