Plot[PDF[NormalDistribution[0,1],x],{x,-6,6},Filling->Axis,PlotLabel->"Normal Distribution\nμ=0, σ=1"]

Plot[Evaluate@Table[PDF[NormalDistribution[0,σ],x],{σ,{.75,1,2}}],{x,-6,6},Filling->Axis]

normDIST[]:=DynamicModule[{},

Manipulate[

Plot[PDF[NormalDistribution[μ,σ],x],{x,-10,10},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ], PlotRange->Full,AxesOrigin->{0,0}],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},0.01,10}]
]

normDIST[]

m4[]:=DynamicModule[{g1,g2},

Manipulate[

g1=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,7},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z<"<>ToString[z]<>")="<>ToString[N@CDF[NormalDistribution[μ,σ],z]], PlotRange->Full,AxesOrigin->{0,0}];
g2=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,z},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z<"<>ToString[z]<>")", PlotRange->Full,AxesOrigin->{0,0},FillingStyle->{Red}];
Show[g1,g2],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},0.01,10},{{z,1,"Z"},-7,7}]
]

m4[]

m6[]:=DynamicModule[{g1,g2,x,sol,z},

Manipulate[

sol=NSolve[CDF[NormalDistribution[μ,σ],x]== pz,{x}, Reals];
z=x/.sol[[1]];
g1=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,7},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z<"<>ToString["z"]<>")="<>ToString[pz]<>"\nz="<>ToString[z], PlotRange->Full,AxesOrigin->{0,0}];
g2=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,z},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z<"<>ToString[z]<>")", PlotRange->Full,AxesOrigin->{0,0},FillingStyle->{Red}];
Show[g1,g2],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},0.01,10},{{pz,0.5,"P(Z<z)"},0,1}]
]

m6[]

m7[]:=DynamicModule[{g1,g2,x,sol,z},

Manipulate[

sol=NSolve[CDF[NormalDistribution[μ,σ],x]== 1-pz,{x}, Reals];
z=x/.sol[[1]];
g1=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,7},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z<"<>ToString["z"]<>")="<>ToString[Chop[pz,6]]<>"\nz="<>ToString[z], PlotRange->Full,AxesOrigin->{0,0}];
g2=Plot[PDF[NormalDistribution[μ,σ],x],{x,z,7},Filling->Axis, PlotRange->Full,AxesOrigin->{0,0},FillingStyle->{Red}];
Show[g1,g2],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},0.01,10},{{pz,0.5,"P(Z>z)"},0.0000000001,0.9999999999}]
]

m7[]

zCasioRIGHT[]:=DynamicModule[{g1,g2,x,sol,z},

Manipulate[

sol=NSolve[CDF[NormalDistribution[μ,σ],x]== 1-pz,{x}, Reals];
z=x/.sol[[1]];
g1=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,7},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z⩾"<>ToString["z"]<>")="<>ToString[Chop[pz,10^-6]]<>"\nz="<>ToString[Chop[z,10^-6]], PlotRange->Full,AxesOrigin->{0,0}];
g2=Plot[PDF[NormalDistribution[μ,σ],x],{x,z,7},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z⩾"<>ToString[z]<>")", PlotRange->Full,AxesOrigin->{0,0},FillingStyle->{Red}];
Column[{
Show[g1,g2,ImageSize->300],
"\nCasio Instructions:",
"1. Select STAT F5 (DIST) and then F1(NORM). Now select F3(InvN) and the following options:",
"2. Data: F2 (Variable)",
"3. Tail: F1 (Right)",
"4. Area: "<>ToString[pz]<>" EXE",
"5. σ: "<>ToString[σ]<>" EXE",
"6. μ: "<>ToString[μ]<>" EXE",
"7. Save Res: If you do not want to save results in a list press F1 (None). If you want to save results in a list press F2 and type the number of list where you want to save results.",
"8. Execute: key EXE or F1 (Calc).",
"You will get xInv = "<>ToString[Chop[z,10^-6]]<>". This means P(Z⩾"<>ToString[Chop[z,10^-6]]<>")="<>ToString[Chop[pz,10^-6]]
}],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},0.01,10},{{pz,0.5,"P(Z>z)"},0.0000000001,0.9999999999}]
]

zCasioRIGHT[]

zCasioLEFT[]:=DynamicModule[{g1,g2,x,sol,z},

Manipulate[

sol=NSolve[CDF[NormalDistribution[μ,σ],x]== pz,{x}, Reals];
z=x/.sol[[1]];
g1=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,7},Filling->Axis,PlotLabel->"Normal Distribution\n"<>"μ="<>ToString[μ]<>", σ="<>ToString[σ]<>"\nP(Z⩽"<>ToString["z"]<>")="<>ToString[Chop[pz,10^-6]]<>"\nz="<>ToString[Chop[z,10^-6]], PlotRange->Full,AxesOrigin->{0,0}];
g2=Plot[PDF[NormalDistribution[μ,σ],x],{x,-7,z},Filling->Axis, PlotRange->Full,AxesOrigin->{0,0},FillingStyle->{Red}];
Column[{
Show[g1,g2,ImageSize->300],
"\nCasio Instructions:",
"1. Select STAT F5 (DIST) and then F1(NORM). Now select F3(InvN) and the following options:",
"2. Data: F2 (Variable)",
"3. Tail: F1 (Left)",
"4. Area: "<>ToString[pz]<>" EXE",
"5. σ: "<>ToString[σ]<>" EXE",
"6. μ: "<>ToString[μ]<>" EXE",
"7. Save Res: If you do not want to save results in a list press F1 (None). If you want to save results in a list press F2 and type the number of list where you want to save results.",
"8. Execute: key EXE or F1 (Calc).",
"You will get xInv = "<>ToString[Chop[z,10^-6]]<>". This means P(Z⩽"<>ToString[Chop[z,10^-6]]<>")="<>ToString[Chop[pz,10^-6]]
}],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},0.01,10},{{pz,0.5,"P(Z⩽z)"},0.0000000001,0.9999999999}]
]

zCasioLEFT[]

m0[]:=DynamicModule[{data},

Manipulate[


Column[{

ListPlot[RandomVariate[NormalDistribution[μ,σ],100],Axes->True,PlotLabel->"Normally distributed points in 1D",AxesOrigin->{0,0},Filling->Axis,ImageSize->300],
ListLinePlot[RandomVariate[NormalDistribution[μ,σ],100],Axes->True,PlotLabel->"Normally distributed points in 1D",AxesOrigin->{0,0},PlotLabel->"Normally distributed points in 1D",Filling->Axis,ImageSize->300]
}],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},1/2,3}]
]

m0[]

m2[]:=DynamicModule[{},

Manipulate[

Graphics[{AbsolutePointSize[1],Point[RandomVariate[NormalDistribution[μ,σ],{10^4,2}]]},Axes->True,PlotLabel->"Normally distributed points in 2D",AxesOrigin->{0,0}],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},1/2,3}]
]

m2[]

m3[]:=DynamicModule[{},

Manipulate[

Graphics3D[{AbsolutePointSize[1],Point[RandomVariate[NormalDistribution[μ,σ],{10^4,3}]]},Axes->True,PlotLabel->"Normally distributed points in 3D",BoxStyle->Directive[Dashed]],

{{μ,0,"μ"},-3,3},{{σ,1,"σ"},0.1,10}]
]

m3[]

m5[]:=DynamicModule[{changeColor,ts,ub},

Manipulate[
Module[{axisLevel,zZero,zPos1,zNeg1,zPos2,zNeg2,zPos3,zNeg3,arrowPos1,arrowNeg1,arrowPos2,arrowNeg2,arrowPos3,arrowNeg3,arrow1sd,arrow2sd,arrow3sd,r1,r2,r3,r4,r4b,r5,r6,r7,r8,r8b,totalArea,oneSD,twoSD,threeSD,approximations,oneSDarrows,twoSDarrows,threeSDarrows,approximationArrows,empRules,empValues,pdf,dens},


axisLevel=-0.01pdf[μ];
pdf[x_]:=PDF[NormalDistribution[μ,σ],x];


zZero=zPos1=Graphics[{Dashed,Thickness[0.005],GrayLevel[0.5],Line[{{μ,axisLevel},{μ,pdf[μ]}}]}];
If[MemberQ[whatToShow,1],
zPos1=Graphics[{Dashed,Thickness[0.005],Darker[Red],Line[{{μ+σ,axisLevel},{μ+σ,pdf[μ+σ]}}]}];
zNeg1=Graphics[{Dashed,Thickness[0.005],Darker[Red],Line[{{μ-σ,axisLevel},{μ-σ,pdf[μ-σ]}}]}],

zPos1=Graphics[{Dashed,Thickness[0.005],Darker[Red],Line[{{μ+σ,axisLevel},{μ+σ,pdf[μ+σ]}}]}];
zNeg1=Graphics[{Dashed,Thickness[0.005],Darker[Red],Line[{{μ-σ,axisLevel},{μ-σ,pdf[μ-σ]}}]}]          ];

If[MemberQ[whatToShow,2],
zPos2=Graphics[{Dashed,Thickness[0.005],Blue,Line[{{μ+2σ,axisLevel},{μ+2σ,2.5pdf[μ+2σ]}}]}];
zNeg2=Graphics[{Dashed,Thickness[0.005],Blue,Line[{{μ-2σ,axisLevel},{μ-2σ,2.5pdf[μ-2σ]}}]}],
zPos2=Graphics[{Dashed,Thickness[0.005],Blue,Line[{{μ+2σ,axisLevel},{μ+2σ,pdf[μ+2σ]}}]}];
zNeg2=Graphics[{Dashed,Thickness[0.005],Blue,Line[{{μ-2σ,axisLevel},{μ-2σ,pdf[μ-2σ]}}]}]                 ];

If[MemberQ[whatToShow,3],
zPos3=Graphics[{Dashed,Thickness[0.005],GrayLevel[0.2],Line[{{μ+3σ,axisLevel},{μ+3σ,pdf[μ+2σ]}}]}];
zNeg3=Graphics[{Dashed,Thickness[0.005],GrayLevel[0.2],Line[{{μ-3σ,axisLevel},{μ-3σ,pdf[μ-2σ]}}]}],

zPos3=Graphics[{Dashed,Thickness[0.005],GrayLevel[0.2],Line[{{μ+3σ,axisLevel},{μ+3σ,pdf[μ+3σ]}}]}];
zNeg3=Graphics[{Dashed,Thickness[0.005],GrayLevel[0.2],Line[{{μ-3σ,axisLevel},{μ-3σ,pdf[μ-3σ]}}]}]          ];





arrowPos1=Graphics[{Darker[Red],Arrowheads[{0,.035}],Arrow[{{μ+σ,-.2pdf[μ+σ]},{μ+σ,axisLevel}}],Style[Text[Row[{"μ + σ = ",μ+σ}],{μ+σ,-.3pdf[μ+σ]}],ts]       }];

arrowNeg1=Graphics[{Darker[Red],Arrowheads[{0,.035}],Arrow[{{μ-σ,-.2pdf[μ-σ]},{μ-σ,axisLevel}}],Style[Text[Row[{"μ - σ = ",μ-σ}],{μ-σ,-.3pdf[μ-σ]}],ts]                 }];

arrowPos2=Graphics[{Blue,Arrowheads[{0,.035}],Arrow[{{μ+2σ,-.4pdf[μ+σ]},{μ+2σ,axisLevel}}],Style[Text[Row[{"μ + 2σ = ",μ+2σ}],{μ+2σ,-.5pdf[μ+σ]}],ts]       }];

arrowNeg2=Graphics[{Blue,Arrowheads[{0,.035}],Arrow[{{μ-2σ,-.4pdf[μ-σ]},{μ-2σ,axisLevel}}],Style[Text[Row[{"μ - 2σ = ",μ-2σ}],{μ-2σ,-.5pdf[μ-σ]}],ts]                 }];

arrowPos3=Graphics[{GrayLevel[0.2],Arrowheads[{0,.035}],Arrow[{{μ+3σ,-.6pdf[μ+σ]},{μ+3σ,axisLevel}}],Style[Text[Row[{"μ + 3σ = ",μ+3σ}],{μ+2.5σ,-.7pdf[μ+σ]}],ts]       }];

arrowNeg3=Graphics[{GrayLevel[0.2],Arrowheads[{0,.035}],Arrow[{{μ-3σ,-.6pdf[μ-σ]},{μ-3σ,axisLevel}}],Style[Text[Row[{"μ - 3σ = ",μ-3σ}],{μ-2.5σ,-.7pdf[μ-σ]}],ts]                 }];

arrow1sd=Graphics[{Darker[Red],Arrowheads[{-.035,.035}],Arrow[{{μ-σ,.5pdf[μ]},{μ+σ,.5pdf[μ]}}],Style[Text["68%",{μ,.55pdf[μ]}],18]       }];

arrow2sd=Graphics[{Blue,Arrowheads[{-.035,.035}],Arrow[{{μ-2σ,.3pdf[μ]},{μ+2σ,.3pdf[μ]}}],Style[Text["95%",{μ,.35pdf[μ]}],18]       }];

arrow3sd=Graphics[{GrayLevel[0.2],Arrowheads[{-.035,.035}],Arrow[{{μ-3σ,.1pdf[μ]},{μ+3σ,.1pdf[μ]}}],Style[Text["99.7%",{μ,.15pdf[μ]}],18]       }];


r1=Graphics[{GrayLevel[gL1],Button[Polygon[{
Join[{{μ,axisLevel},{μ+σ,axisLevel},{μ+σ,pdf[μ+σ]}},
Reverse[Table[{μ+0.05k σ,pdf[μ+0.05k σ]},{k,0,20}]] ,
{{μ,pdf[μ]},{μ,axisLevel}}
]
}]  ,gL1=1.8-gL1;area1=.34-area1 ]   } ];

r2=Graphics[{GrayLevel[gL2],Button[Polygon[{
Join[{{μ+σ,axisLevel},{μ+2σ,axisLevel},{μ+2σ,pdf[μ+2σ]}},
Reverse[Table[{μ+σ+0.05k σ,pdf[μ+σ+0.05k σ]},{k,0,20}]] ,
{{μ+σ,pdf[μ+σ]},{μ+σ,axisLevel}}
]
}]  ,gL2=changeColor[gL2] ;area2=.135-area2  ]   } ];


r3=Graphics[{GrayLevel[gL3],Button[Polygon[{
Join[{{μ+2σ,axisLevel},{μ+3σ,axisLevel},{μ+3σ,pdf[μ+3σ]}},
Reverse[Table[{μ+2σ+0.05k σ,pdf[μ+2σ+0.05k σ]},{k,0,20}]] ,
{{μ+2σ,pdf[μ+2σ]},{μ+2σ,axisLevel}}
]
}]  ,gL3=changeColor[gL3]; area3=.0235-area3    ]   } ];

r4=Graphics[{GrayLevel[1],Button[Polygon[{
{μ+3σ,axisLevel},{μ+4σ,axisLevel},{μ+4σ,pdf[μ+2σ]},
{μ+3σ,pdf[μ+2σ]},{μ+3σ,axisLevel}}
],gL4=changeColor[gL4]  ;area4=.0015-area4  ]   } ];


r4b=Graphics[{GrayLevel[gL4],Button[Polygon[{
Join[{{μ+3σ,axisLevel},{μ+4σ,axisLevel},{μ+4σ,pdf[μ+4σ]}},
Reverse[Table[{μ+3σ+0.05k σ,pdf[μ+3σ+0.05k σ]},{k,0,20}]] ,
{{μ+3σ,pdf[μ+3σ]},{μ+3σ,axisLevel}}
]
}] ,gL4=changeColor[gL4]  ;area4=.0015-area4  ]   } ];


r5=Graphics[{GrayLevel[gL5],Button[Polygon[{
Join[{{μ,axisLevel},{μ-σ,axisLevel},{μ-σ,pdf[μ-σ]}},
Reverse[Table[{μ-0.05k σ,pdf[μ-0.05k σ]},{k,0,20}]] ,
{{μ,pdf[μ]},{μ,axisLevel}}
]
}]  ,gL5=changeColor[gL5];area5=.34-area5     ]   } ];

r6=Graphics[{GrayLevel[gL6],Button[Polygon[{
Join[{{μ-σ,axisLevel},{μ-2σ,axisLevel},{μ-2σ,pdf[μ-2σ]}},
Reverse[Table[{μ-σ-0.05k σ,pdf[μ-σ-0.05k σ]},{k,0,20}]] ,
{{μ-σ,pdf[μ-σ]},{μ-σ,axisLevel}}
]
}]  ,gL6=changeColor[gL6]; area6=.135-area6    ]   } ];


r7=Graphics[{GrayLevel[gL7],Button[Polygon[{
Join[{{μ-2σ,axisLevel},{μ-3σ,axisLevel},{μ-3σ,pdf[μ-3σ]}},
Reverse[Table[{μ-2σ-0.05k σ,pdf[μ-2σ-0.05k σ]},{k,0,20}]] ,
{{μ-2σ,pdf[μ-2σ]},{μ-2σ,axisLevel}}
]
}]  ,gL7=changeColor[gL7] ; area7 = .0235-area7   ]   } ];

r8=Graphics[{GrayLevel[1],Button[Polygon[{
{μ-3σ,axisLevel},{μ-4σ,axisLevel},{μ-4σ,pdf[μ-2σ]},
{μ-3σ,pdf[μ-2σ]},{μ-3σ,axisLevel}}
],gL8=changeColor[gL8] ;area8=.0015-area8  ]   } ];

r8b=Graphics[{GrayLevel[gL8],Button[Polygon[{
Join[{{μ-3σ,axisLevel},{μ-4σ,axisLevel},{μ-4σ,pdf[μ-4σ]}},
Reverse[Table[{μ-3σ-0.05k σ,pdf[μ-3σ-0.05k σ]},{k,0,20}]] ,
{{μ-3σ,pdf[μ-3σ]},{μ-3σ,axisLevel}}
]
}] ,gL8=changeColor[gL8]  ;area8=.0015-area8  ] } ];


totalArea=area1+area2+area3+area4+area5+area6+area7+area8;

If[MemberQ[whatToShow,1],oneSD={zPos1,zNeg1,arrow1sd},oneSD={zPos1,zNeg1}];
If[MemberQ[whatToShow,2],twoSD={zPos2,zNeg2,arrow2sd},twoSD={zPos2,zNeg2}];
If[MemberQ[whatToShow,3],threeSD={zPos3,zNeg3,arrow3sd},threeSD={zPos3,zNeg3}];

approximations={oneSD,twoSD,threeSD};

oneSDarrows={arrowPos1,arrowNeg1};
twoSDarrows={arrowPos2,arrowNeg2};
threeSDarrows={arrowPos3,arrowNeg3};

approximationArrows={oneSDarrows,twoSDarrows,threeSDarrows};


dens=Plot[pdf[x],{x,μ-4σ,μ+4σ},PlotStyle->{Black},Axes->{True,False},Ticks->{Automatic,None},AxesOrigin->{μ-3.5σ,axisLevel}];

If[totalArea==0.,theText=Invisible[Row[{"The shaded region has area ",totalArea,"."}]],theText=Row[{"The shaded region has approximate area ",totalArea,"."}]];
empRules=approximations;
empValues=approximationArrows;

Pane[
Text@Column[{Style[Row[{"N(",μ,", ",σ,")"}],16],

Style[theText,18]  ,
If[sv,

Show[dens,r1,r2,r3,r4b,r5,r6,r7,r8b,dens,zZero,empRules,empValues,ImagePadding->{{20,20},{20,20}},PlotRange->{{μ-4σ,μ+4σ},{-.45pdf[μ],1.1pdf[μ]}},AspectRatio->0.8,ImageSize->450],

Show[dens,r1,r2,r3,r4b,r5,r6,r7,r8b,dens,zZero,empRules,dens,ImagePadding->{{20,20},{20,20}},PlotRange->{{μ-4σ,μ+4σ},{-.45pdf[μ],1.1pdf[μ]}},AspectRatio->0.8,ImageSize->450]],


},Alignment->Center],ImageSize->{460,420},Alignment->Center]
](*end of module*),

{{μ,100},0,999,ControlType->Slider,ImageSize->80},
{{σ,25},0.1,500,0.1,ControlType->Slider,ImageSize->80},
Delimiter,
Button["N(0, 1)",μ=0;σ=1,ImageSize->120],
Delimiter,
"Click under the graph to shade.",
Button["clear shading",
gL1=1;
gL2=1;
gL3=1;
gL4=1;
gL5=1;
gL6=1;
gL7=1;
gL8=1;

area1=0;
area2=0;
area3=0;
area4=0;
area5=0;
area6=0;
area7=0;
area8=0,ImageSize->120],
Delimiter,
Column[{"empirical rule",
TogglerBar[Dynamic[whatToShow],{1->"σ",2->"2σ",3->"3σ"}]}],
Delimiter,
{{sv,False,"show values"},{False,True},Enabled->(whatToShow≠{}||ub)},
{{whatToShow,{}},ControlType->None},
{{gL1,1},ControlType->None},
{{gL2,1},ControlType->None},
{{gL3,1},ControlType->None},
{{gL4,1},ControlType->None},
{{gL5,1},ControlType->None},
{{gL6,1},ControlType->None},
{{gL7,1},ControlType->None},
{{gL8,1},ControlType->None},
{{area1,0},ControlType->None},
{{area2,0},ControlType->None},
{{area3,0},ControlType->None},
{{area4,0},ControlType->None},
{{area5,0},ControlType->None},
{{area6,0},ControlType->None},
{{area7,0},ControlType->None},
{{area8,0},ControlType->None},
{{totalArea,0},ControlType->None},
{{theText,""},ControlType->None},

{{axisLevel,-0.01pdf[μ]},ControlType->None},
ControlPlacement->Left,
TrackedSymbols->True,
AutorunSequencing->{1,2},
Initialization:>(changeColor[x_]:=1.8-x;
ts=16)
]

]

m5[]