"Generate a circle with a choosen R (=0.75) centered at (h,k), and export the data as data_circle_unknown_center.dat"
SetDirectory[NotebookDirectory[]];
h=1.40;
k=0.9;
T=2Pi;
R=0.75;deltaT=0.003T;
x[t_,R_]:=R*Cos[t]+h;y[t_,R_]:=R*Sin[t]+k;
pp[R_]:=ParametricPlot[{x[t,R],y[t,R]},{t,0,2Pi}];
Show[{pp[R]}];
txy=Table[{x[t,R],y[t,R]},{t,0,T,deltaT}];
lptxy=ListPlot[txy,AspectRatio->1];
lphk=ListPlot[{{h,k}},PlotRange->All];
Show[{lphk,pp[R]}]
Export["data_circle_unknown_center.dat",txy];

"Import and listplot the data file, data_circle_unknown_center.dat"
SetDirectory[NotebookDirectory[]];
rawdata=Import["data_circle_unknown_center.dat"];
lprawdata=ListPlot[rawdata,AspectRatio->1,PlotStyle->{Red},PlotRange->All];

"Try drawing a circle with trial values of (h,k), R=0.75 and overlap it on the rawdata circle"
k=0.6;h=1.5;
R=0.75;
x[t_,R_]:=R*Cos[t]+h;y[t_,R_]:=R*Sin[t]+k;
pp[R_]:=ParametricPlot[{x[t,R],y[t,R]},{t,0,2Pi},PlotStyle->{Blue},PlotRange->All];
lphk=ListPlot[{{h,k}},PlotStyle->Blue];
Show[{pp[R],lprawdata,lphk}]

"Quantify the error of the trial values of (h,k) using R^2-((xrawdata[i]-h)^2+(yrawdata[i]-k)^2);"
R=0.75;
xrawdata[i_]:=rawdata[[i]][[1]];yrawdata[i_]:=rawdata[[i]][[2]];
Deltasq[i_,h_,k_]:=(R^2-((xrawdata[i]-h)^2+(yrawdata[i]-k)^2))^2;
variance[h_,k_]:=Sum[Deltasq[i,h,k],{i,1,Length[rawdata]}]/(Length[rawdata]);
stderror[h_,k_]:=Sqrt[variance[h,k]];
hini=1.0;hlast=2.0;
kini=0.0;klast=2.0;
NDeltah=75;(* NDeltah must be interger *)
NDeltak=75;(* NDeltak must be interger *)
Deltak=(klast-kini)/NDeltak;
Deltah=(hlast-hini)/NDeltah;
(*
For[ih=0,ih≤NDeltah,ih++,
h=hini+ih*Deltah;
For[ik=0,ik≤NDeltak,ik++,
k=kini+ik*Deltak;
Print[{h,k,Deltay[h,k]}]
];(*end for ih*)
];(*end for ik*)
*)
thkstderror=Table[{h,k,stderror[h,k]},{h,hini,hlast,Deltah},{k,kini,klast,Deltak}];
thkstderror=Flatten[thkstderror,1];
ListPointPlot3D[thkstderror]

stderrory[i_]:=thkstderror[[i,3]]
tstderrory=Table[stderrory[i],{i,1,Length[thkstderror]}];
stderrorymin=Sort[tstderrory][[1]];
minposition=Ordering[tstderrory][[1]];
Print[{stderrorymin,minposition,thkstderror[[minposition]]}]