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So I was thinking about this question: Maximum minus average?

This question is on hold while work calls me. When I get a chance I am going to substantially update it.

Background:

At my previous job, I was able to find a metric that started triggering (onset of sensitivity) at 1.15 standard deviations from the mean, and reliably triggered above 1.54 standard deviations. The metric of Mean-to-max has onset at about 2.84 standard deviations, and the metric min-to-max has onset at about 2.69 standard deviations. This was applied to pre-reflow solder-ball coplanarity as a predictor of solder-joint reliability after reflow. The idea was that a pre-reflow outlier would not behave as it should and would cause itself (if too small) or its neighbors (if too large) to not reflow properly.

The Question/s:

  • What are some real-world use cases where "this thing isn't like its normally-distributed neighbors" detection is useful? (outside of solder-balls)
  • Are there metrics that have sensitivity on 1200 samples of a normally distributed phenomena that can detect outlierhood at thresholds below 1.15 standard deviations from the mean? If so, what are they? How do they work? What is their onset of sensitivity? At what level of variation does the signal unilaterally emerge from noise?
  • How do I determine whether something like this might be valuable to "the community of (statistics-oriented) scholars"?
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    $\begingroup$ Starting with a statistic seems kind of backwards. It usually is more fruitful to begin with a problem and use statistical principles to derive appropriate decision procedures, rather than to begin with a statistic and ask what it might be good for. This one is particularly suspect as a useful statistic for two reasons: (1) the variability of the maximum will be large and therefore the difference between the max and average is likely to lead to inadmissible tests; and (2) a single high outlier will totally screw up the value. $\endgroup$
    – whuber ♦
    Commented Oct 8, 2014 at 14:47
  • $\begingroup$ After 1.54 standard deviations from the mean, the false-positive rate went to zero. The outlier is what was trying to be detected, selected, and removed. I would want a test to flag on it. $\endgroup$
    – EngrStudent
    Commented Oct 8, 2014 at 17:35
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    $\begingroup$ Are you asking about how to detect outliers? If so, we need to have much more context, because what constitutes an outlier depends on the form of the analysis. An additional point of confusion is that it will almost never be the case that rejecting data more than 1.54 SDs from their mean will adequately detect outliers in any normal sense of the word: there must be something very special indeed about the situation you refer to. $\endgroup$
    – whuber ♦
    Commented Oct 8, 2014 at 17:38
  • $\begingroup$ I can go into details on coplanarity and solder-balls if it helps. I suspect that a review of some of JEDEC might give physical context, but insufficient technical context. How do I determine the level of "special" of a situation? I'm still working on the nomenclature. $\endgroup$
    – EngrStudent
    Commented Oct 8, 2014 at 17:41
  • $\begingroup$ @whuber - If I have a standard normal distribution, and I take 1200 iid samples from it, then take one of those samples and replace it with a constant value. At what point can you detect that I have done this? If the magnitude of the constant is larger than about 2.8, then a metric like range of the data, or distance between mean and maximum tail, can indicate the presence of a replacement not from the standard normal distribution. Is it relevant if there is a metric that can reliably detect this sort of outlier at 1.54 standard deviations from the mean? $\endgroup$
    – EngrStudent
    Commented Oct 10, 2014 at 21:28

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