5.1 Normal Distribution and its Properties

Normal distributions are the backbone of many statistical analyses. They're characterized by their symmetric bell shape and defined by two key parameters: the mean and standard deviation. These distributions are incredibly useful for modeling real-world phenomena and making predictions.

Understanding normal distributions is crucial for probability calculations. The empirical rule helps estimate probabilities within certain ranges, while z-scores standardize values for easier comparison. Probability density and cumulative distribution functions further aid in calculating specific probabilities for various scenarios.

Properties and Characteristics of Normal Distributions

Properties of normal distribution

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• Symmetric bell-shaped curve represents the shape of the distribution
  • Mean, median, and mode are equal located at the center (peak) of the distribution
• Continuous probability distribution represents the data
  • Probability represented by the area under the curve (total area always equals 1)
• Extends infinitely in both directions (left and right)
  • Asymptotically approaches the x-axis but never touches it (tails extend indefinitely)
• Defined by two parameters: mean ($\mu$) and standard deviation ($\sigma$)
• 68-95-99.7 rule (empirical rule) applies to normal distributions
  • Specifies the percentage of data within 1, 2, and 3 standard deviations of the mean

Parameters of normal distribution

• Mean ($\mu$) determines the location (center) of the distribution
  • Shifting the mean shifts the entire distribution left (negative shift) or right (positive shift)
• Standard deviation ($\sigma$) determines the spread (width) of the distribution
  • Larger standard deviation results in a wider, flatter distribution (more variability)
  • Smaller standard deviation results in a narrower, taller distribution (less variability)

Probability Calculations and Functions

Probabilities using empirical rule

• Empirical rule (68-95-99.7 rule) specifies data percentages within standard deviations
  • Approximately 68% of data falls within one standard deviation of the mean ($\mu \pm \sigma$)
  • Approximately 95% of data falls within two standard deviations of the mean ($\mu \pm 2\sigma$)
  • Approximately 99.7% of data falls within three standard deviations of the mean ($\mu \pm 3\sigma$)
• Z-scores standardize values to represent the number of standard deviations an observation is from the mean
  • Formula: $z = \frac{x - \mu}{\sigma}$ (observation minus mean, divided by standard deviation)
  • Can be used to calculate probabilities using a standard normal distribution ($\mu = 0, \sigma = 1$)

Density vs cumulative distribution functions

• Probability density function (PDF) denoted as $f(x)$
  • Describes the relative likelihood of a random variable taking on a specific value
  • Area under the curve between two points represents the probability of the random variable falling within that range (not the height of the curve)
• Cumulative distribution function (CDF) denoted as $F(x)$
  • Describes the probability that a random variable is less than or equal to a specific value
  • Monotonically increasing function, ranging from 0 to 1 (never decreases)
  • $F(x) = P(X \leq x)$, where $X$ is the random variable
  • Can be used to calculate probabilities for intervals by subtracting CDF values (upper bound minus lower bound)