Sandbox
Forsooth
"There are 33 percent more such women in their 20s than men. To help us see what a big difference 33 percent is, Birger invites us to imagine a late-night dorm room hangout that’s drawing to an end, and everyone wants to hook up. 'Now imagine,' he writes, that in this dorm room, 'there are three women and two men.'"
Imagine 33 percent at Jordan Ellenberg's Quodmodocumque blog (20 November 2015).
Submitted by Priscilla Bremser
Tests for gerrymandering
Let math save our democracy
by Sam Wang, New York Times, 5 December 2015
How gerrymandered is your Congressional district?
by Christopher Ingraham, Washington Post,15 May 2014
appeal to isoperimetric inequality
How to rig an election
Economist, 25 April 2002
"Worst of all is the state's extraordinary 17th District, which is a crab (see chart). Though most of it lies in the western part of the state, two claws stretch out towards the eastern part to grab Democratic cities in order to make the surrounding 18th and 19th districts more reliably Republican."
"as used to be said of the old Texas 6th (which was a road from Houston to Dallas), that you could kill most of the constituents by driving down the road with the car doors open."
Diet science
Are fats unhealthy? The battle over dietary guidelines
by Aaron E. Carroll, “Upshot” blog, New York Times, 12 October 2015.
Related “Upshot”: Behind new dietary guidelines, better science, February 23, 2015
Earthquake prediction
Why a 99.9% earthquake prediction is 100% controversial
by Rong-Gong Lin II, Los Angeles Times, 23 October 2015
"This report
Donnellan added that the USGS' 85% probability and her 99.9% chance still favored a big earthquake in the next three years.
"If an earthquake happens in three years, we're both right," Donnellan said.
USGS https://www.facebook.com/USGeologicalSurvey/posts/955479124498071:0 responded]
Some math doodles
<math>P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)</math>
<math>\hat{p}(H|H)</math>
<math>\hat{p}(H|HH)</math>
Accidental insights
My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end. I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics. But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.
While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.
As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.” Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.
| Piece | Sq. Inches | % of Total |
|---|---|---|
| 1 | 43.25 | 31.9% |
| 2 | 35.25 | 26.0% |
| 3 | 23.25 | 17.2% |
| 4 | 14.10 | 10.4% |
| 5 | 7.10 | 5.2% |
| 6 | 4.70 | 3.5% |
| 7 | 3.60 | 2.7% |
| 8 | 3.03 | 2.2% |
| 9 | 0.66 | 0.5% |
| 10 | 0.61 | 0.5% |
The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line. I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect? What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head: “On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”
My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from Nature News on research by Hans Herrmann, et. al. Shattered eggs reveal secrets of explosions. As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions. Bill also provided a link to a vignette from CRAN describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.
Submitted by William Montante