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Formula

Normal probability distribution

X ~ N ( μ , σ )

μ = the mean σ = the standard deviation

Formula

Standard normal probability distribution

Z ~ N ( 0 , 1 )

z = a standardized value (z-score)

mean = 0 standard deviation = 1

Formula

Finding the kth percentile

To find the kth percentile when the z-score is known: k = μ + ( z ) σ

Formula

Z-score

z = x - μ σ

Formula

Finding the area to the left

The area to the left: P ( X x )

Formula

Finding the area to the right

The area to the right: P ( X x ) = 1 - P ( X x )

Definitions

    Normal distribution

  • A continuous Random Variable (RV) with Probability Density Function (PDF) f ( x ) = 1 σ 2 π e - 1 2 ( x - μ σ ) 2 , where μ is the mean of the distribution and σ is the standard deviation.
  • Notation: X ~ N(μ, σ). If μ=0 and σ=1, the RV is called the standard normal distribution.

    Standard normal distribution

  • A continuous random variable (RV) X~N(0,1). When X follows the standard normal distribution, it is often noted asZ~N(0,1).

    Z-score

  • The linear transformation of the form z = x - μ σ . If this transformation is applied to any normal distribution X~N(μ,σ), the result is the standard normal distribution Z~N(0,1). If this transformation is applied to any specific value x of the RV with mean μ and standard deviation σ , the result is called the z-score of x. Z-scores allow us to compare data that are normally distributed but scaled differently

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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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