I have 116 Bessel-corrected sample variances (average of squared distances from sample mean), each from a sample of three measurements. All measurements were done using the same method.
I had predicted that the empirical cumulative distribution would match that of a scaled chi-squared curve with 2 degrees of freedom, so I tried plotting them on the same graph.
For the chi-squared distribution's scaling, I scaled according to the degree of freedom and my prediction for the population variance (0.01192), in the form (for y-values) =CHISQDIST(2*E2/0.01192,2,1) , where E2 corresponding to an x-value point. For the PDF later, the form was =CHISQDIST(2*F2/0.01192,2,0) * (2/0.01192) .
For the sample variances curve, I sorted the sample variances by size and for the y-values used 1/116, 2/116, ..., 116/116.
My intention was that both curves would represent distributions with the mean at the population variance (of the original, normal distribution).
In Image 1 below, the yellow line represents my uncorrected sample variances CDF, the blue line represents my 2-degrees-of-freedom scaled chi-squared distribution CDF, and the red line represents a 1-degree-of-freedom scaled chi-squared distribution CDF after I began trying to rule out possibilities.
To my eyes, the yellow line seems to have a mostly-consistent shape of its own, distinct from both the blue and red lines, but I have no idea how to identify what CDF that is. My knowledge of CDFs is shallow and so I could also be imagining something to hold import that holds none.
In Image 2 below, remembering the PDF to be the slope of the CDF, I have approximated the PDF of the samples' distribution by dy/dx using adjacent points (so for point 2, using $\frac{y_3 - y_1}{x_3 - x_1}). The y-axis is logarithmic to better include all points shown. I again am concerned by that the shape does not seem to match the 2-degrees-of-freedom curve and not knowing why.
[I have somewhat rewritten/redone the above question before reading welcome offered information in more detail!]
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