example 1:  sampling from a normal distribution

Print["The parent population is the normal distribution. "]
Plot[PDF[NormalDistribution[0,1],x],{x,-3,3}]

Clear[SampleSets,SampleMeans,SampleStDevs,SampleMaxs,SampleMedians,nSize,i,numSets]
nSize=10;
numSets=200;
SampleMeans={};
SampleStDevs={};
SampleMaxs={};
SampleMedians={};
For[i=1,i≤ numSets,i++,
SampleSets[i] = RandomReal[NormalDistribution[0,1],nSize];
AppendTo[SampleMeans, Mean[SampleSets[i]] ];
AppendTo[SampleStDevs, StandardDeviation[SampleSets[i]]  ];
AppendTo[SampleMaxs, Max[SampleSets[i]]   ];
AppendTo[SampleMedians, Median[SampleSets[i]] ]
]
Print["One of the samples of values from this distribution, along with its mean, standard deviation, max, and median"]
SampleSets[3]
SampleMeans[[3]]
SampleStDevs[[3]]
SampleMaxs[[3]]
SampleMedians[[3]]

Print["Two histograms of one of the samples, shown with absolute frequencies."]
Histogram[SampleSets[3]]
Histogram[SampleSets[3],4]
Print["A sample should resemble its parent population."]

Print["Samples vary, especially for a small sample size!"]
For[i=1,i≤4,i++,
Print[GraphicsArray[Table[Histogram[SampleSets[5i+j],4],{j,1,5}]]
]]

sample means - Central Limit Theorem applies

SampleMeans

Clear[ListOfSampleStatistics];
ListOfSampleStatistics = SampleMeans;
Clear[xmin,xmax,a,b];
a=Min[ListOfSampleStatistics];
b=Max[ListOfSampleStatistics];
xmin=a-.1*(b-a);
xmax=b+.1*(b-a);
Clear[Model1];
Model1[x_]:= PDF[NormalDistribution[0,1/Sqrt[nSize]],x];   (*   CLT model for samples from N[0,1]  *)
Print["Histogram of the sample means from our collection of samples, shown with absolute frequencies"]
Histogram[ListOfSampleStatistics]
Print["This collection of sample means is itself a sample, and "]
Print["it should resemble the population from which the sample means are drawn, "]
Print["which is the sampling distribution of sample means."]

Print["Histogram of this collection of sample means, shown with relative frequencies"]
Histogram[ListOfSampleStatistics,Automatic,"Probability"]

Print["Histogram of this collection of sample means, shown with heights of the absolute frequencies divided by the bin widths"]
Show[{Histogram[ListOfSampleStatistics,Automatic,"ProbabilityDensity"],
Plot[Model1[x],{x,xmin,xmax},PlotStyle->{Red}]          }]
Print["Red is CLT Model that applies to the population of sample means."]

sample maxima - NO Central Limit Theorem

Clear[ListOfSampleStatistics];
ListOfSampleStatistics = SampleMaxs;
Clear[xmin3,xmax3];
a=Min[ListOfSampleStatistics];
b=Max[ListOfSampleStatistics];
xmin3=a-.1*(b-a);
xmax3=b+.1*(b-a);
Clear[Model3];
Model3[x_]:= PDF[NormalDistribution[Mean[ListOfSampleStatistics],StandardDeviation[ListOfSampleStatistics]],x]
Print["Red is CLT Model.  Histogram and a normal comparison for the collection of sample maximums"]
Show[{Histogram[ListOfSampleStatistics,Automatic,"ProbabilityDensity",ChartStyle->Green],
Plot[Model1[x],{x,xmin,xmax},PlotStyle->{Red}],
Plot[Model3[x],{x,xmin3,xmax3},PlotStyle->{Green,Dashed,Thick}]
}]
Print["The collection of sample maxs should resemble the population from which the sample maxs are drawn, "]
Print["which is the sampling distribution of sample means."]

sample standard deviations - NO Central Limit Theorem

Clear[ListOfSampleStatistics];
ListOfSampleStatistics = SampleStDevs;
Clear[xmin3,xmax3];
a=Min[ListOfSampleStatistics];
b=Max[ListOfSampleStatistics];
xmin3=a-.1*(b-a);
xmax3=b+.1*(b-a);
Clear[Model3];
Model3[x_]:= PDF[NormalDistribution[Mean[ListOfSampleStatistics],StandardDeviation[ListOfSampleStatistics]],x]
Print["Red is CLT Model.  Histogram and a normal comparison for the collection of sample standard deviations"]
Show[{Histogram[ListOfSampleStatistics,Automatic,"ProbabilityDensity",ChartStyle->Orange],
Plot[Model1[x],{x,xmin,xmax},PlotStyle->{Red}],
Plot[Model3[x],{x,xmin3,xmax3},PlotStyle->{Orange,Dashed,Thick}]
} ]

sample medians - NO Central Limit Theorem

Clear[ListOfSampleStatistics];
ListOfSampleStatistics = SampleMedians;
Clear[xmin3,xmax3];
a=Min[ListOfSampleStatistics];
b=Max[ListOfSampleStatistics];
xmin3=a-.1*(b-a);
xmax3=b+.1*(b-a);
Clear[Model3];
Model3[x_]:= PDF[NormalDistribution[Mean[ListOfSampleStatistics],StandardDeviation[ListOfSampleStatistics]],x]
Print["Red is CLT Model.  Histogram and a normal comparison for the collection of sample medians"]
Show[{Histogram[ListOfSampleStatistics,Automatic,"ProbabilityDensity",ChartStyle->Gray],
Plot[Model1[x],{x,xmin,xmax},PlotStyle->{Red}],
Plot[Model3[x],{x,xmin3,xmax3},PlotStyle->{Green,Dashed,Thick}]
} ]

see the random variation in multiple data sets

Do[
Clear[SampleSets,SampleMeans,SampleStDevs,SampleMaxs,i];
SampleMeans={};
SampleStDevs={};
SampleMaxs={};
For[i=1,i≤ numSets,i++,
SampleSets[i] = RandomReal[NormalDistribution[0,1],nSize];
AppendTo[SampleMeans, Mean[SampleSets[i]] ];
AppendTo[SampleStDevs, StandardDeviation[SampleSets[i]]  ];
AppendTo[SampleMaxs, Max[SampleSets[i]] ]     ];
Print[GraphicsArray[{
Show[{Histogram[SampleMeans,Automatic,"ProbabilityDensity"],
Plot[Model1[x],{x,-1.5,2}]},
PlotRange->{{-2,4},{0,1}},PlotLabel->"Means"],
Show[{Histogram[SampleMaxs,Automatic,"ProbabilityDensity",ChartStyle->Green],
Plot[Model1[x],{x,-1.5,2}]},
PlotRange->{{-2,4},{0,1}},PlotLabel->"Maxs"],
Show[{Histogram[SampleStDevs,Automatic,"ProbabilityDensity",ChartStyle->Orange],
Plot[Model1[x],{x,-1.5,2}]},
PlotRange->{{-2,4},{0,1}},PlotLabel->"StdDevs"], 
Show[{Histogram[SampleMedians,Automatic,"ProbabilityDensity",ChartStyle->Gray],
Plot[Model1[x],{x,-1.5,2}]},
PlotRange->{{-2,4},{0,1}},PlotLabel->"Medians"]                     }]                 ],            {j,1,3}]

example 2 - sampling from a uniform distribution

Print["The parent population is a uniform distribution. "]
Plot[PDF[UniformDistribution[{250,500}],x],{x,200,600}]
Clear[SampleSets2,SampleMeans2,SampleStDevs2,SampleMaxs2,nSize2,i,numSets2]
nSize2=10;
numSets2=200;
SampleMeans2={};
SampleStDevs2={};
SampleMaxs2={};
For[i=1,i≤ numSets2,i++,
SampleSets2[i] = RandomReal[UniformDistribution[{250,500}],nSize2];
AppendTo[SampleMeans2, Mean[SampleSets2[i]] ];
AppendTo[SampleStDevs2, StandardDeviation[SampleSets2[i]]  ];
AppendTo[SampleMaxs2, Max[SampleSets2[i]] ]
]
SampleSets2[3]
SampleMeans2[[3]]
SampleStDevs2[[3]]
SampleMaxs2[[3]]

Clear[mu2,sigma2,sig2DivRootn];
mu2 = Mean[UniformDistribution[{250,500}]]
sigma2 = Sqrt[ExpectedValue[(x-mu2)^2,UniformDistribution[{250,500}],x] ]

sig2DivRootn = sigma2/Sqrt[nSize2]

N[%]

Clear[pxmin2,pxmax2];
pxmin2 = mu2 - 3sig2DivRootn;
pxmax2 = mu2 + 3sig2DivRootn;

Clear[p2];
p2=Plot[PDF[NormalDistribution[mu2,sig2DivRootn],x],{x,pxmin2,pxmax2},PlotStyle->{Red}];
Print["Red is CLT Model.  Histogram for the collection of sample means"]
Show[{Histogram[SampleMeans2,Automatic,"ProbabilityDensity"],
p2}]

Print["Red is CLT Model.  Histogram for the collection of sample maximums"]
Show[{Histogram[SampleMaxs2,Automatic,"ProbabilityDensity",ChartStyle->Green],
p2}]

Print["Red is CLT Model.  Histogram for the collection of sample standard deviations"]
Show[{Histogram[SampleStDevs2,Automatic,"ProbabilityDensity",ChartStyle->Orange],
p2}]

example 3 -- sampling from an exponential distribution

Print["The parent population is the exponential distribution. "]
Plot[PDF[ExponentialDistribution[1],x],{x,0,5}]
Clear[SampleSets3,SampleMeans3,SampleStDevs3,SampleMaxs3,nSize3,i,numSets3,mu3,sigma3,sig3DivRootn,pxmin3,pxmax3]
nSize3=10;
numSets3=200;
SampleMeans3={};
SampleStDevs3={};
SampleMaxs3={};
For[i=1,i≤ numSets3,i++,
SampleSets3[i] = RandomReal[ExponentialDistribution[1],nSize2];
AppendTo[SampleMeans3, Mean[SampleSets3[i]] ];
AppendTo[SampleStDevs3, StandardDeviation[SampleSets3[i]]  ];
AppendTo[SampleMaxs3, Max[SampleSets3[i]] ]
]
mu3 = Mean[ExponentialDistribution[1]];
sigma3 = Sqrt[ExpectedValue[(x-mu3)^2,ExponentialDistribution[1],x] ];
sig3DivRootn = sigma3/Sqrt[nSize3];
pxmin3 = mu3 - 3sig3DivRootn;
pxmax3 = mu3+6sig3DivRootn;
Clear[p3];
p3=Plot[PDF[NormalDistribution[mu3,sig3DivRootn],x],{x,pxmin3,pxmax3},PlotStyle->{Red}];
Print["Red is CLT Model.  Histogram for the collection of sample means"]
Show[{Histogram[SampleMeans3,Automatic,"ProbabilityDensity"],p3}]

Print["Red is CLT Model.  Histogram for the collection of sample maximums"]
Show[{Histogram[SampleMaxs3,Automatic,"ProbabilityDensity",ChartStyle->Green],
Plot[PDF[NormalDistribution[mu3,sig3DivRootn],x],{x,pxmin3,pxmax3}]}]

Print["Red is CLT Model.  Histogram for the collection of sample standard deviations"]
Show[{Histogram[SampleStDevs3,Automatic,"ProbabilityDensity",ChartStyle->Orange],
Plot[PDF[NormalDistribution[mu3,sig3DivRootn],x],{x,pxmin3,pxmax3}]}]