Lecture Notes on Normal Distribution

Introduction

Understanding the normal distribution is crucial for many statistical analyses.
Key characteristics:
- Mean = Median = Mode
- Symmetric distribution
- Asymptotic: continues indefinitely without reaching zero

Models many real-world phenomena effectively.
Beyond three standard deviations from the mean, the curve captures nearly all data (99.7%).
Normal distribution is frequently used due to its properties.

Different distributions have different means and standard deviations.
The mean (population mean) represents the central average.
Standard deviation indicates the spread or variability in data.
- 1 standard deviation (±σ) covers approximately 68% of data.
- 2 standard deviations (±2σ) cover about 95%.
- 3 standard deviations (±3σ) cover about 99.7%.

Given:
- Population consumes an average of 100 pounds of gummy bears per day.
- Standard deviation is 10 pounds.
Question: What's the probability someone eats more than 140 pounds?

Identify Mean and Standard Deviation:
- Mean (µ) = 100 pounds
- Standard Deviation (σ) = 10 pounds
Calculate Z-Score:
- Formula: [ z = \frac{(X - \mu)}{\sigma} ]
- For 140 pounds: [ z = \frac{(140 - 100)}{10} = 4 ]
- Z-score of 4 indicates 4 standard deviations above the mean.
Interpret the Z-Score:
- Use z-score to find the corresponding probability from a z-table.
- For high z-scores (e.g., z = 4), probabilities are very low (e.g., less than 0.001).

A z-score represents how many standard deviations a data point is from the mean.
Positive z-score: above the mean
Negative z-score: below the mean
Useful for standardizing different normal distributions to utilize a common z-table.