# How to assess the statistical significance of a single data point?

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.

• You'd need a model for the data, a test statistic and a null and alternative hypothesis. Jul 12, 2013 at 2:43
• 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.
– whuber
Jul 12, 2013 at 14:39