Calculating the dimensions of an icosahedron from a sample of cross-sectional widths

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Hi all,
I'm a biologist with a mathematical and statistical question. I have a dataset of measurements of the maximal cross-sectional width of icosahedral bodies inside bacteria. Since they are icosahedral, the cross-sectional width (if precisely through the middle of the object) is a measure of the diameter of the circumscribed sphere of the icosahedron, from which I should be able to calculate dimensional data. The problem is that, of course, I essentially have a dataset of values whose mean will be less than the average circumscribed sphere diameter!

Consider, as an example, determining the average volume of non-uniform (but similarly sized) balls in a box where the only known data is from a set of cross-sectional measurements of a sample of the balls. The measurements will be a range of diameters as the cross-section will sometimes be through the middle, sometimes near the top and sometimes near the bottom of the ball, and other values in between. The maximum measured diameter will be representative of the actual mean diameter of all the balls, but it will be one of very few measurements as outliers of a normal distribution. In addition, given that the diameter of balls in the box vary, the large size outliers may also represent a small cross-sectional measurement of a very large ball! How can I calculate a value close to the mean diameter of all balls in the box given that the value I need to use is poorly represented in the dataset?? Any help appreciated. Any further questions, please post.

Thanks and cheers,

botbml

An example dataset is attached. If anyone has any clues.
 

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We could assume that the measured mean equals the mean radius ##R_m##. The mean radius in case of balls is ##R_m=2 \displaystyle{\int_0^r}\sqrt{r^2-x^2}\,dx = \dfrac{\pi}{2}\,r^2## where ##r## is the true radius. The check of the distance between ##r## and the measured maximum is at the same time a measure for the sample size, i.e. if we actually hit a cross section through a center.
 
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