Suggestion: Histograms usually only assign the x-axis data to have occurred at the midpoint of the bin and omit x-axis measures of location of greater accuracy. The effect this has on the derivatives of fit can be quite large. Let us take a trivial example. Suppose we take the classical derivation of a Dirac delta but modify it so that we start with a Cauchy distribution at some arbitrary median location with a finite scale (full width half-maximum). Then we take the limit as the scale goes to zero. If we use the classical definition of a histogram and do not change bin sizes we will capture neither the location or the scale. If however, we use a median location within bins of even of fixed width, we will always capture the location, if not the scale when the scale is small relative to the bin width.

For fitting values where the data is skewed, using fixed bin midpoints will x-axis shift the entire curve segment in that region, which I believe relates to the question above. 

STEP 1
[![funny histo][1]][1] Here is an almost solution. I used $n=8$ in each histogram category, and just displayed these as the mean x-axis value from each bin. Since each histogram bin has a value of 8, the distributions all look uniform, and I had to offset them vertically to show them. The display is not the correct answer, but it is not without information. It correctly tells us that there is an x-axis offset between groups. It also tells us that the actual distribution appears to be slightly U shaped. Why? Note that the distance between mean values is further apart in the centers, and closer at the edges. So, to make this a better representation, we should borrow whole samples and fractional amounts of each bin boundary sample to make all the mean bin values on the x-axis equidistant. Fixing this and displaying it properly would require a bit of programming. But, it may just be a way to make histograms so that they actually display the underlying data in some logical format. The shape will still change if we change the total number of bins covering the range of the data, but the idea is to resolve some of the problems created by binning arbitrarily.

STEP 2 So let's start borrowing between bins to try to make the means more evenly spaced.[![enter image description here][2]][2] 

Now, we can see the shape of the histograms beginning to emerge. But the difference between means is not perfect as we only have whole numbers of samples to swap between bins. To remove the restriction of integer values on the y-axis and complete the process of making equidistant x-axis mean values, we have to start sharing fractions of a sample between bins.

Step 3 The sharing of values and parts of values. [![histo3][3]][3] 

As one can see, the sharing of parts of a value at a bin boundry can improve the uniformity of distance between mean values. I managed to do this to three decimal places with the data given. However, one cannot, I do not think, make the distance between mean values exactly equal in general, as the coarseness of the data will not permit that. 

One can, however, do other things like use [kernel density estimation](https://en.wikipedia.org/wiki/Kernel_density_estimationestimation). 

Here we see Annie's data as a bounded kernel density using Gaussian smoothings of 0.1, 0.2, 0.4 and 0.8. The other subjects will have shifted functions of the same type, provided one does the same thing as I did, namely use the lower and upper bounds of each data set.

[![kernelsmooth][4]][4]


  [1]: https://i.sstatic.net/vKI3K.png
  [2]: https://i.sstatic.net/wiFcH.png
  [3]: https://i.sstatic.net/ZuDXb.png
  [4]: https://i.sstatic.net/O7v38.png