Assessing approximate distribution of data based on a histogram Suppose I want to see whether my data is exponential based on a histogram (i.e. skewed to the right). 
Depending on how I group or bin the data, I can get wildly different histograms. 
One set of histograms will make is seem that the data is exponential. Another set will make it seem that data are not exponential. How do I make determining distributions from histograms well defined?
 A: A kernel density or logspline plot may be a better option compared to a histogram.  There are still some options that can be set with these methods, but they are less fickle than histograms.  There are qqplots as well.  A nice tool for seeing if data is close enough to a theoretical distribution is detailed in:

 Buja, A., Cook, D. Hofmann, H., Lawrence, M. Lee, E.-K., Swayne,
 D.F and Wickham, H. (2009) Statistical Inference for exploratory
 data analysis and model diagnostics Phil. Trans. R. Soc. A 2009
 367, 4361-4383 doi: 10.1098/rsta.2009.0120


The short version of the idea (still read the paper for details) is that you generate data from the null distribution and create several plots one of which is the original/real data and the rest are simulated from the theoretical distribution.  You then present the plots to someone (possibly yourself) that has not seen the original data and see if they can pick out the real data.  If they cannot identify the real data then you don't have evidence against the null.
The vis.test function in the TeachingDemos package for R help implement a form of this test.
Here is a quick example.  One of the plots below is 25 points generated from a t distribution with 10 degrees of freedom, the other 8 are generated from a normal distribution with the same mean and variance.

The vis.test function created this plot and then prompts the user to choose which of the plots they think is different, then repeats the process 2 more times (3 total).  
A: Cumulative distribution plots [MATLAB, R] – where you plot the fraction of data values less than or equal to a range of values – are by far the best way to look at distributions of empirical data. Here, for example, are the ECDFs of this data, produced in R:

This can be generated with the following R input (with the above data):
plot(ecdf(Annie),xlim=c(min(Zoe),max(Annie)),col="red",main="ECDFs")
lines(ecdf(Brian),col="blue")
lines(ecdf(Chris),col="green")
lines(ecdf(Zoe),col="orange")

As you can see, it's visually obvious that these four distributions are simply translations of each other. In general, the benefits of ECDFs for visualizing empirical distributions of data are:


*

*They simply present the data as it actually occurs with no transformation other than accumulation, so there's no possibility of accidentally deceiving yourself, as there is with histograms and kernel density estimates, because of how you're processing the data.

*They give a clear visual sense of the distribution of the data since each point is buffered by all the data before and after it. Compare this with non-cumulative density visualizations, where the accuracy of each density is naturally unbuffered, and thus must be estimated either by binning (histograms) or smoothing (KDEs).

*They work equally well regardless of whether the data follows a nice parametric distribution, some mixture, or a messy non-parametric distribution.


The only trick is learning how to read ECDFs properly: shallow sloped areas mean sparse distribution, steep sloped areas mean dense distribution. Once you get the hang of reading them, however, they're a wonderful tool for looking at distributions of empirical data.
A: 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
 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. 
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.  
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. 
Here we see Annie's data as a bounded kernel density using Gaussian smoothings of 0.1, 0.2, and 0.4. 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. So, this is no longer a histogram, but a PDF, and it serves the same role as a histogram without some of the warts.

