Formula
Normal probability distribution
$X$ ~
$N(\mu ,\sigma )$
$\mu $ = the mean
$\phantom{\rule{20pt}{0ex}}\sigma $ = the standard deviation
Formula
Standard normal probability distribution
$Z$ ~
$N(0,1)$
$z$ = a standardized value (z-score)
mean = 0
$\phantom{\rule{20pt}{0ex}}$ standard deviation = 1
Formula
Finding the kth percentile
To find the
kth percentile when the z-score is known:
$k=\mu +\left(z\right)\sigma $
Formula
Z-score
$z=\frac{x-\mu}{\sigma}$
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)
$f\left(x\right)=\frac{1}{\sigma \cdot \sqrt{2\cdot \pi}}\cdot {e}^{-\frac{1}{2}\cdot {\left(\frac{x-\mu}{\sigma}\right)}^{2}}$ , 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