5.4: The Normal Distribution
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Normal Distribution is one of the most important and widely used probability distributions in statistics. Also known as the bell curve, it shows up naturally in many real-world processes from test scores to measurement errors, heights to IQs, and much more.
Continuous Distributions: A Broader View
Before we zoom in on the Normal Distribution, it’s helpful to remember that it’s just one of many types of continuous probability distributions used in statistics.
Unlike discrete distributions (which list clear, separate outcomes like rolling a die), continuous distributions model variables that can take on any value in an interval or range. We do not assign probabilities to individual values instead, we talk about the probability of falling within an interval.
- A continuous probability distribution is represented by a density curve.
- The total area under the curve equals 1.
- The probability of landing exactly on one number (like \( P(X = 5) \)) is always zero. We only deal with ranges, like \( P(4 \leq X \leq 6) \).
There are many different continuous distributions beyond the Normal! Think of these models as tools for different situations:
- Normal distribution — symmetric, bell-shaped
- Student t-distribution — symmetric, bell-shaped
- Uniform distribution — all intervals of equal width are equally likely
- Exponential distribution — often used for waiting times
- Beta & Gamma distributions — useful in Bayesian statistics and modeling skew
Now let’s dive deeper into the most famous of these: the Normal Distribution.
Definition: Normal Distribution
The Normal Distribution is a symmetric, bell-shaped, continuous probability distribution defined by two parameters:
- μ (mu): the mean or center of the distribution
- σ (sigma): the standard deviation, which controls the spread
We write this as:
\( X \sim N(\mu, \sigma) \)
The probability density function (PDF) for a normal distribution is:
\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{ -\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2 } \]
And for the standard normal distribution, where μ = 0 and σ = 1:
\[ f(z) = \frac{1}{\sqrt{2\pi}} e^{ -\frac{z^2}{2} } \]
Note: You won’t be computing probabilities directly using this formula. Finding the area under this curve (which gives us probability) requires technology, integration, or a Z-table. But knowing the form of the function helps us understand how the curve behaves, especially how it centers around μ and falls off with distance from μ.
Key Features of the Normal Distribution:
- It is symmetric around the mean.
- The mean = median = mode.
- It follows a predictable shape centered on μ with most of the area near the center and less in the tails.
- The curve never touches the x-axis (it is asymptotic).
- About 68% of the data fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 (we'll formalize this soon as the Empirical Rule).
Think About It
Have you ever heard of "grading on a curve"? Or taken a standardized test? The logic behind those systems often assumes that results follow a normal distribution.
Can you think of a situation in which a bell-shaped curve might not be a good fit for data? What might make a distribution not normal?