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Statistics LibreTexts

6.7: Chapter 6 Key Items

  • Page ID
    4579
  • Normal Distribution
    a continuous random variable \((RV)\) with pdf \(f(x) =\)

    \[\frac{1}{\sigma \sqrt{2 \pi}} \mathrm{e}^{\frac{-(x-\mu)^{2}}{2 \sigma^{2}}}\nonumber\]

    , where \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation; notation: \(X \sim N(\mu, \sigma)\). If \(\mu = 0\) and \(\sigma = 1\), the \(RV\), \(Z\), is called the standard normal distribution.
    Standard Normal Distribution
    a continuous random variable \((RV) X \sim N(0, 1)\); when \(X\) follows the standard normal distribution, it is often noted as \(Z \sim N(0, 1)\).
    z-score
    the linear transformation of the form \(z=\frac{x-\mu}{\sigma}\) or written as \(z=\frac{|x-\mu|}{\sigma}\); if this transformation is applied to any normal distribution \(X \sim N(\mu, \sigma)\) the result is the standard normal distribution \(Z \sim N(0,1)\). If this transformation is applied to any specific value \(x\) of the \(RV\) with mean \(\mu\) and standard deviation \(\sigma\), the result is called the z-score of \(x\). The z-score allows us to compare data that are normally distributed but scaled differently. A z-score is the number of standard deviations a particular \(x\) is away from its mean value.