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Consider a data

data = {{x1,y1},{x2,y2},...}

with n rows. I want it to be organized into $n$ bins for $x$ and $m$ bins for y, with the positions $x_{i}, x_{i+1}$ and $y_{j}, y_{j+1}$ of the bins chosen in the way such that the number of events in each bin will be the same. And the resulting data should be

dataBinned = {{(x1+x2)/2, (y1+y2}/2, N/((x2 - x1)(y2-y1))},{(x1+x2)/2, (y2+y3}/2, N/((x2 - x1)(y3-y2))}, …, {(x2+x3)/2, (y1+y2}/2, N/((x3 - x2)(y2-y1))},{(x2+x3)/2, (y2+y3}/2, N/((x3 - x2)(y3-y2))}, …}

Here, $N$ denotes the number of events in each bin (it should be provided "automatically" by the procedure which finds $x_{i}, y_{j}$).

The method which makes the same for one-dimensional data data1 (see also a question) is the following:

(*Organizing data into bins*)
data1Bins = Partition[Sort[#], Round[Length[#]/#2]] &[data1, n];
(*Construction the binned data*)
data1Binned = 
  Table[{(data1Bins[[i]][[1]] + data1Bins[[i]][[Length[data1Bins[[i]]]]])/2 //. {x_} :> x, Length[data1Bins[[i]]], Length[data1Bins[[i]]]/(data1Bins[[i]][[
Length[data1Bins[[i]]]]] - data1Bins[[i]][[1]]) //. {x_} :> x}, {i, 1, n, 1}];

But I do not imagine how to construct the same for 2-dimensional data. Could you please help me with this?

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I don't think what you want is possible for all cases. As a simple example, consider the following data set {{x1,y1},{x2,y2},...} = {{1,1,},{2,2},...}. Let the gridlines represent the bins. There is no way to bin it out such that the off diagonal bins can have any content.

enter image description here

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