In[1]:=
<<Statistics`DescriptiveStatistics`
<<Statistics`MultiDiscreteDistributions`
<<Statistics`ContinuousDistributions`

NCSDist[k_,d_]:=NoncentralChiSquareDistribution[k,d]
ndist=NormalDistribution[0,1]
NTail[x_]:=1-CDF[ndist,x]
NapproxPt[n_,k_,q_,a_]:=
  Sqrt[n]*( n^(-1) Quantile[ChiSquareDistribution[k],1-a] - q)/(3/4)
CSTPow[k_,d_,a_]:=1-CDF[NCSDist[k,d],Quantile[ChiSquareDistribution[k],1-a]]

MultN[n_,p_]:=MultinomialDistribution[n,p]
p={1/4,1/2,1/4}
p0={1/3,1/3,1/3}
q=Sum[(p[[j]]-p0[[j]])^2/p0[[j]],{j,1,Length[p]}]


CSStatist[n_Integer,p0_,p_]:={
    x=Random[MultN[n,p]];
    Q=Sum[(x[[j]]-n*p0[[j]])^2/(n*p0[[j]]),{j,1,Length[p0]}]}

CS=Flatten[Table[CSStatist[100,p0,p],{k,1,5000}]];


nn=5000;
zz=Sort[CS];
hh=Table[k,{k,1,nn}]*(1/nn);//N
J1=Table[{zz[[i]],hh[[i]]},{i,1,nn}];
ListPlot[J1]
Delta[x_]:=0/;x<0
Delta[x_]:=1/;x³0
DE[x_]:=Table[{Delta[zz[[i]]-x]},{i,1,nn}]
EDF[nn_,x_]:=1.-Apply[Plus,DE[x]]*(1/nn)//N
P10=Plot[EDF[nn,x],{x,0,40},
    PlotStyle\[Rule]{Thickness[1/100],RGBColor[0.000,1.000,0.196],
        Dashing[{}]}]

NCSDist[k_,d_]:=NoncentralChiSquareDistribution[k,d]
ndist=NormalDistribution[0,1]
NTail[x_]:=1-CDF[ndist,x]
NapproxPt[n_,k_,q_,a_]:=
  Sqrt[n]*( n^(-1) Quantile[ChiSquareDistribution[k],1-a] - q)/(3/4)
CSTPow[k_,d_,x_]:=CDF[NCSDist[k,d],x]
P2=Plot[CSTPow[2,100/8,x],{x,0,40}]



Show[P10,P2]
1-EDF[nn,5.991]
1-EDF[nn,9.210]
Mean[CS]
Variance[CS]
LocationReport[CS]
DispersionReport[CS]

Out[5]=
NormalDistribution[0,1]
Out[10]=
\!\({1\/4, 1\/2, 1\/4}\)
Out[11]=
\!\({1\/3, 1\/3, 1\/3}\)
Out[12]=
\!\(1\/8\)
From In[1]:=
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\[SkeletonIndicator]Graphics\[SkeletonIndicator]
Out[32]=
{0.892}
Out[33]=
{0.736}
Out[34]=
\!\(18163\/1250\)
Out[35]=
\!\(352512873\/6248750\)
Out[36]=
\!\({Mean \[Rule] 18163\/1250, 
    HarmonicMean \[Rule] 
      176029623232064831269708213003286722326895612667132795832547171650418928\
158441402235110110250430998134469939831964371449905917251998521371860645990120\
997997272438316622383605228369825855018880296242406365359693334925696282302260\
782777151284501587264211134125247876749012108681543978190961911955079800120820\
151248000000/
        2122996263060170029031959355971984139498618089793654797632511136451863\
624474483674054877576507577682520416459819250187492887328557876935221090575647\
948825572845655280200401256477506186476363887486433217107978242372154280772678\
697643569387121729631358720698883474719938174928814002094578136955330409633895\
1678933474831, Median \[Rule] 14}\)
Out[37]=
\!\({Variance \[Rule] 352512873\/6248750, 
    StandardDeviation \[Rule] \(3\ \@\(39168097\/9998\)\)\/25, 
    SampleRange \[Rule] 2751\/50, MeanDeviation \[Rule] 18655713\/3125000, 
    MedianDeviation \[Rule] 261\/50, QuartileDeviation \[Rule] 261\/50}\)
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