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This question already has an answer here:

Let X = [11.17, 9.52, 10.69, 9.84, 10.84, 9.88, 10.28, 12.23, 8.49, 10.79] be a normally distributed data with mean = 10.37 and standard deviation = 1.02. How to determine the corresponding value to +1.02 and -1.02 standard deviations, which covers 69% area of the probability distribution.

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marked as duplicate by Glen_b, Xi'an, Andy, gung, Scortchi Nov 25 '14 at 9:49

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ The question is unclear as stated: do you ask what is the mean+standard deviation value? 10.37+1.02=11.39... A random variable does not take an actual value along the x-axis, but any value when realised. Please rephrase title and text. $\endgroup$ – Xi'an Nov 25 '14 at 5:10
  • $\begingroup$ @Xi'an No, I didn't asked for mean + standard deviation (i.e., 10.37 + 1.02 = 11.39). The min = 8.49 and the max = 12.23. How to determine the exact lower and upper values which covers 0.69% (derived from 1.02 from left and right side of the mean) area. $\endgroup$ – mani Nov 25 '14 at 5:59
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    $\begingroup$ Please rephrase your title to How to derive quantiles of a non-standard distribution. The use of the same value 1.02 both for the empirical standard deviation and for the 84% quantile of the N(0,1) distribution is quite confusing. $\endgroup$ – Xi'an Nov 25 '14 at 6:08
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    $\begingroup$ The highly detailed and informative answer provided by Glen_b to your earlier question already makes this point that you seem confused between the standard deviation of your sample and the normal quantiles. $\endgroup$ – Xi'an Nov 25 '14 at 6:11
  • $\begingroup$ what do you mean by 84% quantile? $\endgroup$ – mani Nov 25 '14 at 6:17
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This derivation follows from the linearity properties of the normal distribution.

Since for a standard normal $X\sim N(0,1)$, $-1.02\le X\le 1.02$ with probability $0.69$, for a normal $Y\sim N(10.37,1.02)$, $$9.33=-1.02\cdot1.02+10.37\le Y\sim1.02\cdot X+10.37\le 1.02\cdot 1.02+10.37=11.41$$

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  • $\begingroup$ How did you get 9.33 on left hand side by 1.02*1.02 +10.37? $\endgroup$ – mani Nov 25 '14 at 6:34
  • $\begingroup$ I corrected the typo, there was a - in front. $\endgroup$ – Xi'an Nov 25 '14 at 6:40

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