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I understand how to calculate the probability that a value falls above or below a certain Z score. How about calculating the probability that a value is equal to a particular Z?

In my homework I see that probability that P(X=m) (the mean) is 0 since 50 percent of the data lies above and below the point. Can I use the same logic to say that the probability of observing any single point is 0? If so, this would seem to present a logical contradiction.

whuber
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Info5ek
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  • This sure seems like homework, in which case, it should have the "self-study" tag. – Peter Flom Feb 02 '14 at 21:16
  • This question is already covered by a number of answers on this site, for example [here](http://stats.stackexchange.com/questions/17000/probability-of-continuous-random-variable), and there's discussion that is directly relevant to the issue in many other less-obviously-related questions, such as [this one](http://stats.stackexchange.com/questions/4220/a-probability-distribution-value-exceeding-1-is-ok) – Glen_b Feb 02 '14 at 22:02
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    It is amusing that in adding the [tag:continuous-data] tag, its wiki excerpt provided the answer ("the chance that it takes any particular value is zero ... for every real number x"). Now if we could only make *all* wiki excerpts that intelligent and useful! – whuber Feb 04 '14 at 15:25
  • The relations between this question and the "duplicates" are not obvious to a beginner. It seems safe to assume that those asking fundamental questions don't also have the ability to deduce and search for all linguistically and statistically equivalent permutations of their question in order to avoid conceptual duplicates. This question makes it more likely that another person looking to learn about this concept will be able to find what they're looking for if they don't know all such possible permutations. – Info5ek Feb 04 '14 at 19:01

1 Answers1

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This seems like homework so I will just give a hint:

Is there anything special about the mean for the purposes of this question? If there is, what? If not, then you have the answer.

EDIT: Given that you've said it is not homework: There is nothing special about the mean. The likelihood of any exact value from a continuous distribution is 0.

However, in real life (as opposed to the world of math or homework problems!) data get reported rounded; e.g. if you collect data on people's weights, the likelihood of someone weighing 150.000000... pounds is 0. But the likelihood of a reported weight of 150 is far from 0.

Peter Flom
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  • This is a non-graded question from an already graded homework assignment. To your question, the mean is just a data point as any other, so I can't see why it should be treated differently but I could be wrong. – Info5ek Feb 02 '14 at 21:37
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    user1800340 - a non-graded question from an already graded homework assignment certainly counts as `self-study` (it's the kind of question that makes it that, not just whether you get a grade for it). Would you mind adding the tag (and since self-study questions are treated specially, could you also perhaps take a look through the [tag wiki info](http://stats.stackexchange.com/tags/self-study/info))? – Glen_b Feb 02 '14 at 21:58