You need to read up on BLUE (best linear unbiased estimator). See for example http://en.wikipedia.org/wiki/Least_squares the bit on weighted least squares. You can search for the "Aitken" on that website.

Also, see

http://reference.wolfram.com/applications/eda/FittingDataToLinearModelsByLeast-SquaresTechniques.html

Cheers, Marco

Ok, for the sake of completeness, if you want to use the information of the errors for the fit this will do the trick:

fit = NonlinearModelFit[data[[All, {1, 2}]], a*Exp[b x], {a, b}, x, Weights -> 1/data[[All, 3]]^2]

which gives $a=241.238$ and $b=-0.138946$.If you then plot:

Show[Plot[{Normal[fit], fit["MeanPredictionBands", ConfidenceLevel -> .99]}, {x, 0, 13}, PlotStyle -> Red, Filling -> {2 -> {1}}], ErrorListPlot[Table[{data[[i, {1, 2}]], ErrorBar[10*data[[i, 3]]]}, {i, 1, Length[data]}]]]

Cheers, Marco

Thank you very much Marco, but I have another question: How can I use the function Y(A,B,x)=Aexp(B*x) as a fit-function and determine the parameteres A and B?

Dear Sarmed,

<< ErrorBarPlots`

data = {
  {1, 200.0, 1.5},
  {2, 180.0, 1.2},
  {3, 150.0, 1.1},
  {4, 140.0, 1.0},
  {5, 120.0, 0.9},
  {6, 110.0, 0.9},
  {7, 100.0, 0.8},
  {8, 90.0, 0.7},
  {9, 80.0, 0.7},
  {10, 60.0, 0.6},
  {11, 50.0, 0.6},
  {12, 40.0, 0.5},
  {13, 30.0, 0.5}}

ErrorListPlot[
 Table[{data[[i, {1, 2}]], ErrorBar[10*data[[i, 3]]]}, {i, 1, 
   Length[data]}]]

gives

Note that I have multiplied the errors by 10, because they are so small that do not see the bars on the plot if you do not do this.

You can also use functional programming:

ErrorListPlot[Transpose[{data[[All, {1, 2}]], ErrorBar /@ (10*data[[All, 3]])}]]

Cheers, M.