抄録
Under an assumption that particle size x follows log-normal distribution in its parent population, any population mean particle size defined by
Γ= {∫∞0xa+bƒ(x)dx/∫∞0xbdx} 1/cor Γ=exp {∫∞0 (lnxe)ƒ(x)dx}
can be estimated by the following Hatch-Choate formula.
gH=exp (Aμ+Bô2)
μ=n∑i=1yi/n, ô2n∑i=1 (yi-μ) 2/ (n-1)
yi=lnxi
(A, B) = {a/c, a2+2ab/2c} or {e, 0}
This estimate gH, which may be called sample mean particle size by Hatch-Choate, follows asymptotically log-normal distribution, that is, h=ln gH follows asymptotically normal distribution N {Ω, D2 (h)}.
Ω=lnΓ
D2 (h) = (A2/n) σ2+ {2B2/ (n-1)} σ4
So far as sample size n is larger than about 100, the distribution of h= ln gH can be approximately expressed by N {Ω, D2 (h)}. Owing to this conclusion, some of statistical tests and inference about population mean particle sizes become possible with a numerical table of t-distribution.
