Put error bars in a plot from a non-linear fit together with confidence level

Question

I want to ask you a problem that I have. I make a non linear fit with some data, and I want to know how can I put the error bars in the plot together with the confidence level of the fit that I perform. I have tried this but doesn't work:

data = {{0, 1}, {1, 0}, {3, 2}, {5, 4}, {6, 4}, {7, 5}};
errors = {0.1, 0.1, 0.2, 0.05, 0.2, 0.15};
nlm = NonlinearModelFit[data, Log[a + b x^2], {a, b}, x, 
   Weights -> 1/errors^2, VarianceEstimatorFunction -> (1 &)];


{bands80[x_], bands90[x_], bands95[x_], bands99[x_]} = 
  Table[nlm["SinglePredictionBands", 
    ConfidenceLevel -> cl], {cl, {0.6, .9, .95, 0.99}}];
Show[Plot[{nlm[x], bands80[x], bands90[x], bands95[x], 
   bands99[x]}, {x, 0, 7}, 
  Filling -> {2 -> {1}, 3 -> {2}, 4 -> {3}, 5 -> {4}}], 
 Graphics[{PointSize[Large], Blue, Point[data]}]]

I have though that maybe using ErrorListPlot[withError, PlotStyle -> Red], could work, but it doesn't...Do you know how to do it? Thanks

Answer 1

Your code works:

You can add error bars showing, for example, the range of the 90% confidence level of the fit. (It's up to you to decide if it makes sense to do that...)

bars90 = Table[bands90[d[[1]]] + -d[[2]], {d, data}]
(* {{-2.69685,1.58472},{-0.928799,3.33932},{-1.26757,2.99974},{-2.30463,1.96317},{-1.95044,2.31746},{-2.6485,1.61947}} *)

Needs["ErrorBarPlots`"]
Show[Plot[{nlm[x], bands80[x], bands90[x], bands95[x], bands99[x]}, {x, 0, 7}, 
  Filling -> {2 -> {1}, 3 -> {2}, 4 -> {3}, 5 -> {4}}, AxesOrigin -> {-.1, 0}], 
 ErrorListPlot[
  Table[{data[[n]], ErrorBar[bars90[[n]]]}, {n, Length[data]}], PlotMarkers -> {Automatic, Medium}, PlotStyle -> Blue]]

Or, per JimB's suggestion, add error bars corresponding to the values in the list errors (this makes more sense to me):

Show[Plot[{nlm[x], bands80[x]}, {x, 0, 7}, Filling -> {2 -> {1}}, AxesOrigin -> {-.1, 0}], 
 ErrorListPlot[
  Table[{data[[n]], ErrorBar[errors[[n]] {-1, 1}]}, {n, Length[data]}], PlotMarkers -> {Automatic, Tiny}, PlotStyle -> Blue]]

(The implementation of error bars has changed in v.12 - this is for earlier versions.)