6.5 Normal distribution: summary of formulas

Normal probability distribution

$X$ ~ $N(\mu ,\sigma )$

$\mu$ = the mean $\sigma$ = the standard deviation

Standard normal probability distribution

$Z$ ~ $N(0,1)$

$z$ = a standardized value (z-score)

mean = 0 $$ standard deviation = 1

Finding the kth percentile

To find the kth percentile when the z-score is known: $k=\mu +\left(z\right)\sigma$

Z-score

$z=\frac{x-\mu }{\sigma }$

Finding the area to the left

The area to the left: $P(X< x)$

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) $f\left(x\right)=\frac{1}{\sigma \cdot \sqrt{2\cdot \pi }}\cdot e^{-\frac{1}{2}\cdot {\left(\frac{x-\mu }{\sigma }\right)}^2}$ , 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 $z=\frac{x-\mu }{\sigma }$ . 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