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I am revising undergrad statistics course via this course, where i am learning technique to pull out sample from population.

While ensuring that sample is decent representative of population, i am left with this couple of question.

  • Should we care about identification and rectification of outliers prior to taking sample from population ?

Here is my work from where i came up with this question

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  • $\begingroup$ How would you be able to identify outliers without looking at a sample? $\endgroup$ – Michael R. Chernick Aug 6 '18 at 7:11
  • $\begingroup$ If you can look at the outliers of the whole population, you should. $\endgroup$ – Ingolifs Aug 6 '18 at 7:20
  • $\begingroup$ @MichaelChernick I meant to say, should I figure out outlier by exploring population itself. $\endgroup$ – Gaurav Gupta Aug 6 '18 at 7:34
  • $\begingroup$ @Ingolifs From the exercise point of view, yes i can look for outliers on population, but practically, i believe that won't be possible as dataset can be huge. So, consider the case, if we can't find outliers on population, then how should we progress in such scenarios. $\endgroup$ – Gaurav Gupta Aug 6 '18 at 7:37
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This depends hugely of what you even want to do about outliers, through what mechanism they arise and whether they even affect what you want to do.

If outliers are perfectly good values that just happen to be (a bit) extreme, then it may be more a matter of making sure it does not affect what you intend to do too much. Some statistical procedures/summary measures are inherently more robust in this respect, while other have more problems.

  1. E.g. if you want to find the population median of something, outliers do not really matter much.
  2. On the other hand, a mean might be much more heavily affected, e.g. if you are trying to find the mean of a log-normal distribution, directly taking the sample mean may be a bad idea, if you have just a few values. On the other hand, estimating the parameters of the distribution and then obtaining the theoretical mean from that would likely work a lot better.

If outliers arise through a corrupted sampling process (e.g. people deliberately give nonsense answers to a questionnaire, some biological sample was accidentally contaminated with some other substance etc.), you probably do indeed want to do something about those values. However, what you do depends on what you assume the process that creates these corrupted values to be (following the terminology of missing completely at random/at random/not at random for missing data):

  • Is it "completely at random" (it might be a reasonable assumption that whether or not a lab technician spoilt a biological sample had nothing to do with what you would have measured on this sample)?
  • Is it "at random" (perhaps young people are more likely to give nonsense answers to questions, but their true response would otherwise have been distributed like for other young people)?
  • Or is it "not at random" (e.g. people are reluctant to given answer A and rather give a nonsense answer instead, or the lab technician accidentally spoilt the biological sample, because she was shocked by how many bacteria she saw etc.)?
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  • $\begingroup$ This is a good discussion of the issues with outlier. But it can't be an answer to a meaningless question. If you know the population that doesn't tell you anything about outliers. If the OP means assume the distribution is say normal than you could test for outliers in a sample. The test just tells you that it would be very unusual to get this value if the data were normal. The OP has not made that kind of clarification and he speaks about identifying outliers without collecting a sample which makes no sense. $\endgroup$ – Michael R. Chernick Aug 6 '18 at 13:32
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In general, you won't have the population. So, if you, in class exercises, remove the outliers from the population before taking the sample, your exercises will be poor examples for the students when they do future work.

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  • $\begingroup$ I think I see what you are trying to say but Peter the statement :"remove the outlier from the population before taking the sample" makes no sense. $\endgroup$ – Michael R. Chernick Aug 6 '18 at 13:34
  • $\begingroup$ Sure it does, if you have the whole population. $\endgroup$ – Peter Flom - Reinstate Monica Aug 7 '18 at 0:13
  • $\begingroup$ I am not clear on your response. But if you are saying that you can remove outliers when you have the whole population I disagree. The population provides the distribution you are interested in and why would you change the distribution? $\endgroup$ – Michael R. Chernick Aug 7 '18 at 0:21
  • $\begingroup$ "Outlier" just means "surprising point". Some points in a population are surprising (it could also be data entry error. I read that, in one US Census (1960 IIRC) they reported that there were thousands of 12 year old widows. Turned out some card (back in the punch card days) had slipped. $\endgroup$ – Peter Flom - Reinstate Monica Aug 7 '18 at 0:45
  • $\begingroup$ Suppose you have a Cauchy distribution. Then a point far from the mean is less surprising for the Cauchy than it would be for a normal. So if you have a theoretical reason to assume the Cauchy then you may not want to call that observation an outlier. During my career I have done some research on outlier. What you define as an outlier is wrong. An outlier is an observation is surprising relative to an assumed population distribution or family of distributions. Without that how can you test for outliers? Also to test you need data. $\endgroup$ – Michael R. Chernick Aug 7 '18 at 1:00

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