Monday, November 18, 2013

Normal Snare Distribution

Here is the head of a snare drum (thanks Sean) showing the two dimensional, joint distribution of his drumstick hits. The maker, Evans, produces drum heads with two plys of plastic bonded together. In this image the pattern of wear and use reveal themselves through contour lines as closed curves indicating regions of similar frequency of use. The greatest wear is in the lightest colored central region, having seen the greatest frequency of hits. This central region has worn through the first ply showing the remaining plastic support underneath. Sean seems to have a very stable left hand, consistently hitting this small central region in nearly a circular pattern. In this region, the horizontal location of his hits seems independent of the vertical, producing this near circular pattern of the joint distribution. Surrounding this is the darker region of the top ply of plastic. This layer retains more dirt and grime than the underlying supporting plastic. Again the pattern of these hits is nearly circular. It turns out that with simple assumptions, like radial symmetry and independence, the pattern can be shown to be that of a bivariate normal distribution. A result that was thought first published (p. 398) by John Herschel in 1850, but actually discovered much earlier by the American mathematician Robert Adrain in 1808. More details on that in a future post.

But here, perhaps we see slight deviations from normality. The darker ring seems to show slightly  more variability extending vertically and a greater clustering of hits on the bottom of this image. This indicates a bit of skewness towards the top of the picture. Less use and wear is finally shown in the cream colored outer region that has seem very few hits. Thanks again Sean.

No comments: