µgeo or µ = true geometric mean (wahrer geometrischer mittelwert)

µgeo or µ = µari/e^(0,5*(ln σd)^2) ≠ µari for phi factor and phi density normal distribution

true arithmetic mean (µari) and true geometric mean (µgeo or µ)

µari = true arithmetic mean (wahrer arithmetischer mittelwert)

µgeo = true geometric mean (wahrer geometrischer mittelwert)

µari = µgeo*e^(0,5*ln(σd)^2) ≠ µgeo

µgeo = µari/e^(0,5*ln(σd)^2) ≠ µari

estimated arithmetic mean (xari) and estimated geometric mean (xgeo)

xari = estimated arithmetic mean (geschätzter arithmetischer mittelwert)

xgeo = estimeted geometric mean (geschätzter geometrischer mittelwert)

xgeo(x_i;n) = e^(1/n*((ln(x₁)+ln(x₂)+.....+ln(x_n)))

xgeo(sd;xari) = xari/e^(0,5*ln(sd)^2)

xari(x_i;n) = 1/n*(x₁+x₂+ ....+x_n)

xari(sd;xgeo) = xgeo*e^(0,5*ln(sd)^2)

true standard scattering probability limit factor sigma deisenroth σd and estimated standard scattering probability limit factor sigma deisenroth sd

σd = true standard scattering probability limit factor sigma deisenroth (wahrer standardstreufaktor deisenroth)

sd = estimated standard scattering probability limit factor sigma deisenroth (geschätzter standardstreufaktor deisenroth)

σd = e(2*ln(µari/µgeo))^0,5

sd = e(2*ln(xari/xgeo))^0,5

scattering probability limit factor normal distribution sigma zeta phi deisenroth

σd(ϕd;n)= σd^(±√((-ln(ϕd)/n))) = x(ϕd;n)/µgeo

σd(ϕd;n) = σd^(±ζd(ϕd;n) = x(ϕd;n)/µgeo

ϕd(x;n) = e^(-n*((ln(x/µgeo))/(ln(σd))²)

x(ϕd;n) = µgeo∗σd^(±√((-ln(ϕd)/n)))

ϕd(σd;n) = e^(-n*((ln(x(ϕd;n)/µgeo))/(ln(σd))²)

ζd(ϕd) = ±√(-ln(ϕd)/n)

factor normal distribution (Faktor Normalverteilung)

probability limit factor normal distribution sigma zeta phi deisenroth = σd(ϕd;n) = σd^±ζd(ϕd;n) = σd^±√(-ln(ϕd)/n) = x(ϕd;n)/µgeo ; σd=1,01

probability limit factor normal distribution sigma zeta phi deisenroth = σd(ϕd;n) = σd^±ζd(ϕd;n) = σd^±√(-ln(ϕd)/n) = x(ϕd;n)/µgeo ; σd=2