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I am not very well knowledgeable in statistics as I have yet to take a formal class in it (but have signed up for one next year) and yet find myself in need of finding out whether or not a single data point in a list of values is statistically significant. My data is a list of 4000 values (a power spectrum to be exact) ranging from 130 to near zero (on the order of $10^{-7}$)in which almost all values are less than 0.2. Because of this I want to know if a few of these data points which stand out are statistically significant at very low values of 0.1 because they are surrounded by values of 0.05 in a noise region caused by the fact that the data was generated from samplings in nature. Everything which I have looked up on statistical significance has mentioned only the significance of a whole data set through calculation of t value from the standard deviation and whatnot, and yet have never mentioned just checking one point to see if it is reasonably out of the noise and something is worth noticing, or a random anomaly.

This may be just a simple easy question that my lack of knowledge is stopping me from getting and if so feel free to berate me about it.

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    $\begingroup$ You'd need a model for the data, a test statistic and a null and alternative hypothesis. $\endgroup$
    – Glen_b
    Commented Jul 12, 2013 at 2:43
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    $\begingroup$ You are asking about outlier identification. This is usually not conducted as a formal statistical test, in part because applying any such test is questionable: what, after all, brought a value to your attention in the first place? You certainly didn't randomly select it from your 4000 values! There are many other subtle issues involved, depending on whether you know beforehand how many outliers you want to detect, whether they come in clusters, whether they can be both high and low, etc. Entire books have been written on the subject. $\endgroup$
    – whuber ♦
    Commented Jul 12, 2013 at 14:39

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There are a few reasons why it doesn't make sense to talk about the "statistical significance" of a single data point.

First of all statistical significance is a property of differences. Wen we say something is statistically significant, what we really mean is that some value A is (statistically) significantly different from some other value B. So it doesn't make sense to talk about a single value being "significant," unless you have something to compare it to. We do sometimes refer to specific quantities like regression coefficients as being statistically significant, but that is a sort of short hand for "significantly different from zero".

Second, statistical significance is a way of measuring the uncertainty associated with estimating a value from a random sample, and then generalizing that result to a full population. We might be comparing a single estimate from a sample to some fixed value like zero (a t or z value), or comparing two estimates to each other (that's a t test). But there is always an estimate from a sample involved. If a difference is statistically significant, then what that really means is something like "if in reality the true difference we're estimating was zero and we repeated this study a zillion times, drawing a different random sample each time, it would be really really unlikely for us to estimates that the difference is as large as we actually estimated it." (just what "really really unlikely" means depends on your chosen threshold of significance, e.g. 95% or 99% confidence).

So that's why this idea doesn't really make sense in your example. If your data of 4,000 points is a full population, than the idea of statistical significance doesn't come up, because you are not trying to generalize to some larger population. And if you are just talking about a single point, then there is no difference to apply the concept to.

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