The Variance of the Squared Deviation From the Mean
The missing character is an x bar (mean of X).
What is the variance of the squared deviation from the mean within the χ2 distribution? What this question involves is if the variance is normally distributed what is the variance of the variance of a normally distributed variance? This is a situation where some samples are taken and from them a mean and a variance is gathered. The variance of these distributions are thought to be normally distributed as the sample measurements are normally distributed.
The measurements taken from the samples have a mean, however, the mean is not necessarily the population mean (μ). Thereby, what we know is that the squared deviation from the mean is:
S2 = Σ (xi - )2 = Σ(xi - μ)2 - n(n( - μ)2.
The variance of the squared deviation from the mean is found by the formula for the variance:
Var(S2) = E(S2) - E2(S2).
The first term is E(ΣΣ(xi - μ)2 (xi - μ)2 - 2 Σ(xi - μ)2 n( - μ)2 + n( - μ)2 n( - μ)2), and the second term is E2 ( nσ2 - σ2) since E(Σ(xi - μ)2) = nσ2 and E(n( - μ)2) = σ2. The result is that the second term is n2σ4 - 2nσ4 + σ4 and the first term is n2σ4 - 2nσ4 + nσ4. Var(S2) = n2σ4 - 2nσ4 + σ4 - [n2σ4 - 2nσ4 + nσ4 ] = nσ4 - σ4 = σ4(n - 1).
If the distribution of the variance is normal then the variance of the distribution is 2 times the variance and the variance of S2 is 2(n-1)σ4 which is the answer usually given to this problem.
August 29th, 2006
Boca Raton, FL, USA