Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

6: The Normal Distribution

Last updated

Mar 17, 2025
Save as PDF
- 5.10: Solutions
- 6.0: Introduction to Normal Distribution **

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

6.0: Introduction to Normal Distribution **
This page discusses the normal probability density function, a significant continuous distribution characterized by a bell-shaped curve, relevant in fields like psychology, economics, and mathematics. It highlights key parameters such as mean and standard deviation, noting the curve's symmetry and the equality of mean and median. Changes in these parameters alter the curve's position and shape. The standard normal distribution is mentioned as an important variant.
6.1: The Standard Normal Distribution
This page explains the standard normal distribution, defined by z-scores that indicate a value's distance from the mean (0) in terms of standard deviations (1). Z-scores can be positive or negative, and the Empirical Rule highlights that approximately 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations from the mean. Examples provided illustrate these concepts with specific normal distributions.
6.2: Using the Normal Distribution
This page explains probability calculations using the normal distribution, focusing on its symmetry and the Standard Normal Distribution for simplifying these calculations using z-scores. It includes practical examples, such as household computer usage, where the probability of usage between 1.8 and 2.75 hours is calculated as 0.5886, and quartile assessments.
6.3: Estimating the Binomial with the Normal Distribution
This page discusses estimating binomial processes using various probability distributions, particularly the normal, Poisson, and hypergeometric distributions. It notes that the normal distribution effectively approximates binomial outcomes when both np and n(1-p) exceed five. An example involving Australian Shepherd puppies illustrates these calculations for binomial probabilities, highlighting the utility of the normal distribution in simplifying computations.
6.4: Normal Distribution - Pinkie Length (Worksheet) **
A statistics Worksheet: The student will compare empirical data and a theoretical distribution to determine if data from the experiment follow a continuous distribution.
6.5: Key Terms
This page defines normal distribution in statistics, detailing its probability density function, mean ( $\mu$ ), and standard deviation ( $\sigma$ ). It highlights the standard normal distribution ( $Z \sim N(0, 1)$ with $\mu = 0$ and $\sigma = 1$ ) and explains the z-score, which standardizes normal variables for comparison by indicating their distance in standard deviations from the mean.
6.6: Chapter Review
This page describes the standard normal distribution, which has a mean of zero and a standard deviation of one, noted as Z ~ N(0, 1). It explains the concept of the z-score, measuring distance from the mean in standard deviations. Additionally, it highlights the normal distribution's importance in probability theory, characterized by its continuous, bell-shaped form with parameters mean (µ) and standard deviation (σ), particularly focusing on the relevance of z-scores across various fields.
6.7: Formula Review
This page offers a concise overview of the normal distribution, defining $X$ as a variable with mean $\mu$ and standard deviation $\sigma$ . It discusses the standard normal distribution, where $Z$ has a mean of 0 and a standard deviation of 1, and explains how to convert between observed values and $z$ -scores using specified formulas. Furthermore, it highlights the use of normal distribution to approximate binomial distributions.
6.8: Practice
This page discusses problems related to the standard normal distribution, including random variables, means, medians, standard deviations, and z-scores. It examines normal distribution probabilities, z-score interpretations, and connections to binomial distributions using real-world examples like CD player lifespans.
6.9: Homework
This page provides exercises on normal distribution covering patient recovery times, SAT scores, and physical measurements, emphasizing concepts like z-scores and probability estimations. It includes multiple-choice questions for calculations and interpretations of statistical data.
6.10: References
This page discusses various online resources regarding the Standard Normal Distribution and its applications across health, education, and technology. It includes studies on blood pressure, educational statistics, ACT scores, demographic data, and tools for epidemiological calculations. Additionally, it covers analyses related to stadium capacities, lottery ticket strategies, and statistics on smartphone and Facebook usage, with specific URLs provided for access.
6.11: Solutions
This page covers statistical calculations focused on z-scores, probabilities, and normal distributions, showcasing mean and standard deviations for datasets like NBA player heights and SAT/ACT scores. It provides examples of computing probabilities from normal distributions and discusses the likelihood of events based on random variables. The summaries present specific statistical outcomes and interpretations, often linked to education probabilities and criminal trial durations.

Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN

Support Center

How can we help?