$X\sim N(\mu ,{{\sigma }^{2}})$ • Mean = Mode = Median = $\mu $ • \(\text{P}\hspace{0.1em}\left(

Question

$X\sim N(\mu ,{{\sigma }^{2}})$ • Mean = Mode = Median = $\mu $ • $\text{P}\hspace{0.1em}\left( \left| X-\mu \right|<\sigma \right)\approx 0.68$ • $\text{P}\hspace{0.1em}\left( \left| X-\mu \right|<2\sigma \right)\approx 0.95$ • $\text{P}\hspace{0.1em}\left( \left| X-\mu \right|<3\sigma \right)\approx 0.997$

Accepted Answer

For $X\sim N(\mu,\sigma^2)$, individual points have probability 0, so endpoint inclusion does not affect interval probabilities; about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of $\mu$, respectively.

Introduction to Normal Distribution

Introduction to Normal Distribution

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