6.20: Normal Random Variables (2 of 6)

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Learning Objectives

At these points, the curve changes the direction of its bend and goes from bending upward to bending downward, or vice versa. A point like this on a curve is called an inflection point. Every normal curve has inflection points at exactly 1 standard deviation on each side of the mean.

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https://assessments.lumenlearning.co...sessments/3564
Now we extend this idea to look at the probability of a value falling within 2 standard deviations of the mean or 3 standard deviations of the mean.
If X is a normal random variable with mean $\mathrm{μ}$ and standard deviation $\mathrm{σ}$ , then
1. To summarize using probability notation:
  $\begin{array}{l}\mathrm{1.}P(\mu-\sigma<X<\mu+\sigma)=0.68\\ \mathrm{2.}P(\mu-2\sigma<X<\mu+2\sigma)=0.95\\ \mathrm{3.}P(\mu-3\sigma<X<\mu+3\sigma)=0.997\end{array}$
  Comment
  Let’s take a moment to look a bit deeper at what the empirical rule tells us.
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