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Mean and Standard deviation

H.Tahsiri

Method 1 (using equations)

x={50.36,50.35,50.41,50.37,50.36,50.32,50.39,50.38,50.36,50.38}

     {50.36, 50.35, 50.41, 50.37, 50.36, 50.32, 50.39, 50.38, 50.36, 50.38}

n=Length[x]

     10

meanx=1/n Sum[x[[i]],{i,1,n}]

     50.368

sigmax=Sqrt[1/(n-1) Sum[(x[[i]]-meanx)^2,{i,1,n}]]

     0.024404

sdomx=sigmax/Sqrt[n]//N

     0.00771722

Method 2 (using Mathematica command)

<<DescriptiveStatistics.m

Clear[x,n,meanx,sigmax,sdomx]

x={50.36,50.35,50.41,50.37,50.36,50.32,50.39,50.38,50.36,50.38}

     {50.36, 50.35, 50.41, 50.37, 50.36, 50.32, 50.39, 50.38, 50.36, 50.38}

n=Length[x]

     10

meanx=Mean[x]

     50.368

sigmax=StandardDeviation[x]

     0.024404

sdomx=sigmax/Sqrt[n]//N

     0.00771722

StandardErrorOfSampleMean[x](*this uses Sqrt[n-1]*)

     0.00813467

Linear Least Squares Fits

Method 1 (using equations)

Clear[x,y,n]

x={1,2,3,4,5}

     {1, 2, 3, 4, 5}

y={7.57,11.97,16.58,21,25.49}

     {7.57, 11.97, 16.58, 21, 25.49}

n=Length[x Or y]

     5

s1= Sum[x[[i]] y[[i]],{i,n}];

s2=Sum[x[[i]],{i,n}];

s3=Sum[y[[i]],{i,n}];

s4= Sum[x[[i]]^2,{i,n}];

s5=(Sum[x[[i]],{i,n}])^2;

s7=(Sum[x[[i]]^2,{i,n}]);

s8=(Sum[y[[i]]^2,{i,n}]);

corrcoeff=(n s1-s2*s3)/((Sqrt[n s4-s5])*(Sqrt[n s8-s9]))//N

     0.99998

m=(n*s1-s2*s3)/(n*s4-s5)

     4.487

b=(s3*s4-s1*s2)/(n*s4-s5)

     3.061

Y=m X + b

     3.061 + 4.487 X

Clear[t,v0,v]

v[t_]:=3.061 + 4.487 t

fitplot=Plot[v[t],{t,0,5},DisplayFunction->Identity];

data={{1,7.57},{2,11.97},{3,16.58},{4,21},{5,25.49}}

     {{1, 7.57}, {2, 11.97}, {3, 16.58}, {4, 21}, {5, 25.49}}

dataplot=ListPlot[data,PlotRange->{{0,5},{0,30}},
PlotStyle->PointSize[0.02],DisplayFunction->Identity];
together=Show[{dataplot,fitplot},
DisplayFunction->$DisplayFunction];

Method 2 (using Mathematica command)

data={{1, 7.57}, {2, 11.97}, {3, 16.58}, {4, 21}, {5, 25.49}}
     {{1, 7.57}, {2, 11.97}, {3, 16.58}, {4, 21}, {5, 25.49}}

squarefit=Fit[data,{1,t},t]

     3.061 + 4.487 t

plotsquarefit=Plot[squarefit,{t,0,5},DisplayFunction->Identity];
dataplot=ListPlot[data,PlotStyle->PointSize[0.02],
DisplayFunction->Identity];
pp=Show[dataplot,plotsquarefit,DisplayFunction->$DisplayFunction];

General method (Mathematica command)

example

<<DescriptiveStatistics.m

x={50.36,50.35,50.41,50.37,50.36,50.32,50.39,50.38,50.36,50.38}

     {50.36, 50.35, 50.41, 50.37, 50.36, 50.32, 50.39, 50.38, 50.36, 50.38}

y={24.25,24.26,24.22,24.28,24.24,24.25,24.22,24.26,24.23,24.24}

     {24.25, 24.26, 24.22, 24.28, 24.24, 24.25, 24.22, 24.26, 24.23, 24.24}

n=Length[x Or y]

     10

xmean=Mean[x]

     50.368

ymean=Mean[y]

     24.245

sigmax=StandardDeviation[x]

     0.024404

sigmay=StandardDeviation[y]

     0.0190029

sdomx=sigmax/Sqrt[n]//N

     0.00771722

sdomy=sigmay/Sqrt[n]//N

     0.00600925

AreaAverage=N[Sqrt[(ymean^2*sdomx^2+xmean^2*sdomy^2)],1]

     0.4

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