Reply To: Generalizations vs. Statistics

Hello, I’m your friendly local statistician, please let me help you all out today!

Firstly, here are some online calculators make your life a lot easier. Statistical Computing for The Internet Savvy:

1. Input your data, SD (standard deviation) and at easycalculations’s calculator to obtain a z-score.

2. Plug in your Z-score at Fourmilab’s Z-score calculator to calculate the probability of your z-score, which will give you a sense of how rare your result was.

3. Or, use the normal distribution calculator by computerpsych research software, which takes your z-scores and places them on a bell curve which is conveniently drawn for you.

Another note: Chebyshev’s rule says for a standard distribution of data, 99.7% of results will fall within 3 standard deviations from the mean. So if you’re getting a z-score above 3, make sure that’s what you’re expecting.

What does this have to do with Popa’s coin flipping?

For Popa’s original scenario, a z-score of 3 or so sounds about right. 1 means typical, 2 means a little less than typical, 3 means very rare, 4 means nearly unheard of.

Popa, a Z score of 10 is about 1 in infinity. The chance of getting 1000 heads is not infinity.

We’re all familiar with IQs and SATs as probability distributions. In standardized testing, a perfect score is nearly always set to 3 SDs above the mean.

A Z score of 10 would be equivalent to an IQ of 250, which is considered a mathematically vacuous result. To put that into perspective, a Z score of 10 is the equivalent of an SAT score of 1500 PER SECTION, for a total of 3000 rather than 1600 which is the maximum SAT score. It is functionally meaningless.

I hope that was helpful. This is your friendly local statistician signing off for today!