Balls and Urns

Statisticians love balls and urns. A typical Stats 101 midterm, for example, usually includes a question along these lines:

“You take a simple random sample of 1000 balls from an urn containing 120,000,000 red and blue balls, and your sample shows 450 red balls and 550 blue balls. Construct a 95% confidence interval for the true proportion of blue balls in the urn.”

After choking back a giggle about “blue balls,” you whip out your calculator and text your frat brother who has a copy of last semester’s midterm. He instantly recognizes the correct formula is

95% confidence interval for P = +/- 1.96 * sqrt( p*(1-p) / n) * FPC   where P = the real, true, actual, honest-to-god proportion of blue balls in that great big f’ing urn 
p = the sample proportion of blue balls, or  0.55
n = the sample size = 1000
FPC = the “finite population correction” = sqrt((Nn)/(N-1))  where N=120,000,000
and the 1.96 has something to do with the 95% probability area under a standard normal distribution

That second part, after the “+/-“, is what you know as the “margin of error.” Your frat brother texts you back and reminds you that since the population is very large, the FPC is very close to 1 and can be dropped. He also reminds you to uses the conservative estimate of p= 0.5 in the margin of error calculation, since you don’t know the true value of p, only the sample estimate. So the whole formula simplifies to

p +/- 1.96 * sqrt( .25 / n)

=p +/-  0.98  / sqrt( n)
Assuming you still have juice in your calculator batteries and you’re not hung-over from the Sig Eps kegger last night, you should get

0.55 +/- 0.031
Now you could probably say you are 95% certain the real proportion of blue balls in that great big f’ing urn is 55%, plus or minus 3.1%. If you wanted to get extra credit points, you should probably say that “95% of all random samples of this size will have a computed confidence interval that contains the true population value.” But that’s just quibbling and brown-nosing the professor, who’s probably late for a faculty meeting anyway.

This is, for all intents and purposes, how political pollsters compute the mysterious “margin of error,” which has everything to do (and only to do) with pure mathematical sampling error. If you look at the formula above and round it just a smidge, you get a simple rule of thumb for the margin of error of a sampled probability:

Margin of Error = 1 / sqrt(n)
So if the sample size is 400, the margin of error is 1/20 = 5%; if the sample size is 625 the margin of error is 1/25 = 4%; if the sample size is 1000, it’s about 3%.

Works pretty well if you’re interested in hypothetical colored balls in hypothetical giant urns, or survival rates of plants in a controlled experiment, or defects in a batch of factory products. It may even work well if you’re interested in blind cola taste tests. But what if the thing you are studying doesn’t quite fit the balls & urns template?

  • What if 40% of the balls have personally chosen to live in an urn that you legally can’t stick your hand into?
  • What if 50% of the balls who live in the legal urn explicitly refuse to let you select them?
  • What if the balls inside the urn are constantly interacting and talking and arguing with each other, and can decide to change their color on a whim?
  • What if you have to rely on the balls to report their own color, and some unknown number are probably lying to you?
  • What if you’ve been hired to count balls by a company who has endorsed blue as their favorite color? 
  • What if you have outsourced the urn-ball counting to part-time temp balls, most of whom happen to be blue?
  • What if the balls inside the urn are listening to you counting out there, and it affects whether they want to be counted, and/or which color they want to be?

If one or more of the above statements are true, then the formula for margin of error simplifies to

Margin of Error = Who the hell knows?

Because, in this case, so-called scientific “sampling error” is completely meaningless, because it is utterly overwhelmed by immeasurable non-sampling error. Under these circumstances “margin of error” is a fantasy, a numeric fiction masquerading as a pseudo-scientific fact. If a poll reports it — even if it’s collected “scientifically” — the pollster is guilty of aggravated bullshit in the first degree.

The moral of this midterm for all would-be pollsters: if you are really interested in how many of us red and blue balls there are in this great big urn, sit back and relax until Tuesday, and let us show our true colors.

 –Author Unknown