Standard Deviation Distribution

Author

Eric W. Weisstein
October 17, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Statistics/StandardDeviationDistribution.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/StandardDeviationDistribution.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Code

P[n_,σ_,s_]:=((2((((n/(2(σ^2)))))^((n-1)/2)))/(Gamma[(n-1)/2]))Exp[-n s^2/(2(σ^2))] (s^(n-2))

Plot

<<Graphics`Colors`

Plot[Evaluate[Table[P[n,1,s],{n,2,12,2}]],{s,0,2},PlotStyle->{Red,Orange,Yellow,Green,Blue,Violet},
PlotRange->All,AxesLabel->TraditionalForm/@{s,(Subscript[f, N])[s]}]

Distribution

Normalization

Integrate[P[s],{s,0,∞},Assumptions->n>1&&σ>0]

Mean

Integrate[s P[s],{s,0,∞},Assumptions->n>1&&σ>0]

Raw Moments

FullSimplify[Integrate[s^r P[s],{s,0,∞},Assumptions->n>1&&σ>0&&r>0]]

Variance

(Overscript[μ, ′])[r_]:=(((2^(r/2)) (n^(-r/2)) (σ^r) Gamma[(1/2) (-1+n+r)])/(Gamma[(1/2) (-1+n)]))

FullSimplify[(Overscript[μ, ′])[2]-(Overscript[μ, ′])[1]^2]

Series Expansion of b(N)

b[n_]:=(((Sqrt[2])  Gamma[(n/2)])/((Sqrt[n]) Gamma[(1/2) (-1+n)]))

Limit[b[n],n->∞]

<<Graphics`Colors`

Plot[b[n],{n,0,10},PlotStyle->Red,PlotRange->{-1,1}]

s=Simplify[Series[b[n],{n,Infinity,8}]]

c=CoefficientList[Normal[s],1/n]

b(N) Table

{#,N[1/b[#],4]}&/@Join[Range[2,10],{20,30,50,100}]//TableForm

Plot

Plot[1/b[n],{n,1,20},PlotStyle->Red]