Day 12

Math 216: Statistical Thinking

Bastola

The Normal Distribution

The Normal Distribution is a continuous probability distribution that is symmetrical around its mean, represented by \(\mu\). This distribution is crucial in statistics and is often used to represent real-world variables.

\[ f(x)=\frac{1}{\sigma \sqrt{2 \pi}} \exp \left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right) \]

\(\mu=\) Mean (center of the bell)
\(\sigma=\) Standard deviation (width of the bell)
\(\pi \approx 3.14159\), \(e \approx 2.71828\)

Key Insight: The formula looks complex, but it just describes a perfectly balanced bell curve!

Visualizing the Bell Curve

Real-World Applications: Where Normality Appears

Everyday Examples:

Human heights - cluster around average with few very tall/short people
Test scores - most students score near the middle
Measurement errors - equally likely to be positive or negative
Daily temperatures - vary around seasonal averages

The Standard Normal Distribution: Our Universal Tool

A special case where \(\mu=0\) and \(\sigma=1\). This is our universal translator for all normal distributions!

\[ f(z)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{1}{2} z^2\right) \]

Why it’s amazing:

Every normal problem becomes a standard normal problem
We can use Z-tables or software functions
It’s like having one key that opens every lock!

Area = Probability: A Visual Journey

The Empirical Rule

Converting Normal Variable to Standard Normal

To utilize the standard normal distribution effectively in statistical calculations, we convert a normal random variable \(x\) with any mean \(\mu\) and standard deviation \(\sigma\) to a standard normal variable \(z\).

Z-score

\[ Z = \frac{x - \mu}{\sigma} \]

Example:

To find the probability of \(x\) being less than a particular value \(x_0\), use: \[ P(x \leq x_0) = P\left(Z \leq \frac{x_0 - \mu}{\sigma}\right) \]

College Admissions Analysis

Phone Battery Life Analysis