## Posts Tagged ‘Statistics’

### US Supreme Court calls almost one quarter of blacks retards

June 12, 2014

In the recent Hall v. Florida decision, the US Supreme Court (besides demonstrating a limited understanding of statistics; but what do you expect, they went to law school) has effectively raised the IQ threshold with regards to the level of mental handicap to bar execution to 75. However, when one consideres that the median black American IQ is 85, with standard deviation 13.5 (here), this cutoff has a z-score of (75-85)/13.5≈0.74, which corresponds to a percentile of 23%. Thus, SCOTUS has effectively called almost a quarter of American blacks retarded.

### Monday Math 31

August 4, 2008

The binomial distribution, the discrete probability distribution of obtaining n successes out of N Bernoulli trials, each with probability p of sucess, is
.
We see that the probabilities sum to unity due to the binomial theorem:
.
Now, consider the expected number of successes:

Now, note that the n=0 term is zero, and may be dropped:

Shifting the index by defining k=n-1 and M=N-1, we have
.

Now, suppose we rewrite it in terms of the expected number of successes  in place of p:
.

Let us consider what happens if we take the limit as the sample size , while holding λ fixed:


Now, in that first fraction, the numerator has n terms:
,
and so as N increases without bound, each of these terms approaches unity. The next fraction,  is independent of N.
In the last term, , the base of the exponent approaches unity, while the exponent is fixed, so that term in turn approaches unity.
Lastly, we recall that
,
and so we see that the penultimate term fits this form, and so
.

This distribution is known as the Poisson distribution, which is the distribution for a large number of trials of a Poisson process, and which models a number of real-world processes, such as radioactive decay, or shot noise.

### Lies, Damn Lies, andâ€¦

March 29, 2008

The New York Times has an article about high school graduation rates, and how many states, such as Mississippi, have been inflating their graduation rate statistics, particularly those they send to Washington D.C.