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Given a simple data set, the two standard deviation is calculated as a negative number. Shouldn't the standard deviations (min or max) be within the range of the data set?

My data set: 2 31 32 22 10 43 12 46 48 51 51 52 82 88 10 01 07 12 31 40 15 41 30 Average: 66.2 Standard Deviation: 45.02 Min second Standard deviation: -23.8 (average - stddev x 2) Max second standard deviation: 156.20 (average + stddev x 2)

https://simple.wikipedia.org/wiki/Standard_deviation#/media/File:Standard_deviation_diagram.svg

picolo
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1 Answers1

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Assuming your calculations are correct you have a mean $\mu=66.2$ and standard deviation $\sigma=45.02$ so $\mu-2\sigma=-23.84,\,\mu+2\sigma=156.24$. I've never hard anyone call $\mu-2\sigma$ the "second standard deviation", and it has no theoretical reason to be $\ge 0$.

J.G.
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  • Thanks for your comment. I am trying to calculate −2σ based on the link I provided in the question. – picolo Mar 08 '19 at 14:59
  • @picolo Well, what that diagram calls $-2\sigma$ really means $\mu-2\sigma$. Since you can slide the data along the line by a constant, $\mu$ might as well be $0$ in that context. Statisticians like to use IQ as an example, because its mean and standard deviation are easy to memorise ($100$ and $15$), so $\mu-2\sigma=70$ (but we'd often just label it $-2\sigma$ instead). – J.G. Mar 08 '19 at 15:09