THE BOOK--Playing The Percentages In Baseball

Friday, July 21, 2006

Gory Details of the Gory Details

Andy expands on the calculations of pages 366-367 in The Book:

Let’s start with the variance of the variance, which is what is actually directly calculated.  To measure this, you need to compute the mean of the variance from any one observation (( x- )^2), relative to the expectation value of the variance (-^2).  Thus the variance of the variance equals:
< ( ( x- )^2 - ( - ^2) )^2 >

Expanding the terms, this becomes:
< ( x^2 - 2x - + 2^2 )^2 >
= < x^4 + 4x^2^2 + ^2 + 4^4 - 4x^3 - 2x^2 + 4x^2^2 + 4 x - 8x^3 - 4^2 >
= - 4 + 8^2 - ^2 - 4^4

The expectation values here for a Gaussian centered at with standard deviation sigma equal:
= x0
= sigma^2 + x0^2
= 3 x0 sigma^2 + x0^3
= 3 sigma^4 + 6 x0^2 sigma^2 + x0^4

Substituting these into the above equation, one gets that the variance of the variance equals
2 sigma^4

Naturally, this means the standard deviation of the variance, which is the square root of the variance of the variance, is sqrt(2) sigma^2.

To get to the standard deviation of the standard deviation, you use that if y is an arbitrary function of x, f(x), then the standard deviation of y equals the derivative of f(x) times the standard deviation of x.

In this case, abbreviating standard deviation and variance as SD and VAR, respectively, we have:
SD = sqrt(VAR)

so the standard deviation of the standard deviation is given by
sd(SD) = sd(VAR)/(2 sqrt(VAR))

since sd(VAR) is sqrt(2) sigma^2, and sqrt(VAR) is sigma, this becomes:
sd(SD) = sigma/sqrt(2).

Finally, the “N” comes into play since the uncertainty from repeated measurements equals 1/sqrt(N) times the standard deviation.  So the uncertainty in VAR equals sqrt(2/N)sigma^2, and the uncertainty in SD equals sigma/sqrt(2N).

-- Andy