### Introduction

*X* ∼ *N*(*μ*, *σ*)

*μ* = the mean;
*σ* = the standard deviation

### 6.1 The Standard Normal Distribution

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

mean = 0; standard deviation = 1

To find the observed value, *x*, when the *z*-scores is known:*x* = *μ* + (*z*)*σ*

*z*-score: *z* = $\frac{x\text{\u2013}\mu}{\sigma}$

*Z* = the random variable for *z*-scores

### 6.2 Using the Normal Distribution

Normal Distribution: *X* ~ *N*(*µ*, *σ*) where *µ* is the mean and *σ* is the standard deviation.

Standard Normal Distribution: *Z* ~ *N*(0, 1).

Calculator function for probability: normalcdf (lower *x* value of the area, upper *x* value of the area, mean, standard deviation)

Calculator function for the *k*^{th} percentile: *k* = invNorm (area to the left of *k*, mean, standard deviation)