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STATISTICS OF DISTRIBUTIONS
If you're not counting photons or marbles or poll responses and the like, but rather are getting a distribution of answers to a question, you can't use Poisson statistics! For example, you want to know the height of an American adult. So you ask 89 adults how tall they are. You might get a range of answers like this: Ht. (inches) Number of people ------------ ---------------- 60 ooo 61 oo 62 oooo 63 oooo 64 ooooo ooooo oooo 65 ooooo oo 66 ooooo ooooo ooo 67 ooooo o 68 ooooo ooooo o 69 ooooo ooo 70 oo 71 oo 72 oo 73 oooo 74 o 75 ooo 76 oo 77 78 o You could just take the average, that is the arithmetic mean; add up all the heights, then divided by the number of people in the survey. You'll get a number like 5'7" or so. How accurately does that number represent the height of an American adult? Well, it may be close to an average, but clearly there are a LOT of people who aren't 5'7" (or, 67" if we just use inches). What we realize here is that this distribution is more like a "bell curve" with a peak around 66" or so; and the width of the "bell curve" halfway between the base and the peak tells you how widely distributed the spread is. Let's use a more contrived example. I ask the students in a class to measure the brightness of the Moon. This is the distribution of answers: 12 oo oooo oooo 9 ooooo ooooooo ooooooooo 6 oooooooooo oooooooooooo ooooooooooooooo 3 o ooooooooooooooo o o ooooooooooooooooooo o oooooooooooooooooooooooooooo -|--------|---------|---------|----------|- 90 100 110 120 130 (flux units) It looks like there is a bell-shaped curve, with a maximum of 12 units. The width of the bell curve at half the maximum (in this case, 6 units) runs from about 106 to 117 units; and the center of the bell curve looks to be about 110.5 units. So in this case, the most likely value of the true brightness of the moon is 110.5 flux units, and the margin of error is plus or minus 5.5 flux units. This means that the FULL WIDTH AT HALF MAXIMUM of this distribution is 11 units (106 to 117), and the margin of error is half of that width in each direction. Notice there are irregularities and slight asymmetry in the bell curve. Data from real-world observations are always likely to be imperfect; it is up to us, the analyzers of the data, to spot the peak, the FWHM, and interpret the data effectively. |