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STATISTICAL ACCURACY



intro: TOC for Knowledge Concepts, Exercises, and Solutions



Let's say you hear someone declare, "70 percent of all Americans 
like astronomy."  The first question to ask is, how did we get
that number, 70 percent?  Perhaps the person took a poll.
What if another pollster, however, said, "80% of all Americans
like astronomy?"  Then which number is right, 70% or 80%?  

If you found out that Pollster One asked 50 people and 35 of 
them said they like astronomy, whereas Pollster Two asked 400
people and 320 of them said they like astronomy, your common
sense would suggest that Pollster Two's result is more likely
to be accurate.  Why?  Well, you think, she asked a lot more
people, so her result is more believable.

The quantitative way to express this "gut feeling" is with
statistics.  If you're counting up data points, you can use
something called "counting statistics" or "Poisson statistics,"
named after a French guy named Poisson.  In this case, you're
counting up answers to a question; you could also count up
marbles in a bag, or photons (light "particles") collected
by your telescope - anything you're trying to get a sum total
or "count" of.  In this case, your accuracy is determined by the
number of counts you have.

With Poisson statistics, the ERROR in your measurement is the
square root of the total number of counts.  Thus Pollster One's
error is sqrt(50), or about 7, and Pollster Two's error is
sqrt(400), or 20.  The MARGIN OF ERROR in your measurement would
be the fraction or percentage of the total count number that the
error is.  So Pollster One has a margin of error of

  sqrt(50)
  -------- = margin of error = 0.14, or 14%
     50

whereas Pollster Two has a margin of error of

  sqrt(400)
  --------- = margin of error = 0.05, or 5%
     400

So Pollster One's value of 70% of Americans liking astronomy is
really "70% plus or minus 14%"; similarly, Pollster Two's result
is "80% plus or minus 5%" - much more accurate.