Data set testing?

Question

If Y = X/Z where Y is the expected result of a measured X and Y. I have noted that if one sorts the data points in numerical ascending order for Y then plots Y against the data point number, it becomes obvious when there are deviations in the data set from expected results. But I don't quite understand why mathematically.
Serious help please.

Sorry, of course it is X and Z. Too much champagne!

Answer 1

If you have an a priori distribution in mind for your measurements, and you histogram enough measurements, then you expect the histogram to take the shape of the pdf (probability distribution function) of your expected distribution.

If you sort the measurements and plot them on the vertical axis against either 1 to n on the horizontal axis or perhaps 1/(n+1) to n/(n+1) on the horizontal axis, then you would expect to see the shape of the cdf (cumulative distribution function) of the expected distribution. Put the locations of the cdf-inverse(1/(n+1)) through cdf-inverse(n/(n+1)) and you expect a straight line (Q-Q plot), where it is even easier to see the deviations from your expected distribution.

If you consider what the Kolmogorov-Smirnov (KS) test is testing, the "cdf plot" of your sorted measurements is kind of a visual analog of the KS test.

So there is mathematical information in the plot you made.

Dan

Answer 2

It would help a lot if you were more specific about these anomalies. But if X and Z are independent variables, I would suppose either that the data point sequence is unrelated to values of X or Z (you didn't specify the nature of the sequence), or that X is changing with respect to Z in some unanticipated way.