9.3 Normal distribution

The normal distribution is a crucial concept in probability theory, shaping our understanding of data spread. It's characterized by its symmetrical bell curve, with two key parameters: the mean (μ) and standard deviation (σ). These determine the curve's center and spread, respectively.

Standardizing normal variables transforms them into a standard normal distribution with a mean of 0 and standard deviation of 1. This process, along with the z-table, allows us to calculate probabilities for various scenarios, making the normal distribution a powerful tool in statistical analysis.

Normal distribution properties

Characteristics and parameters

• Normal distribution, also known as Gaussian distribution, exhibits symmetry around its mean
• Two parameters characterize the normal distribution
  • Mean (μ) determines the center of the distribution
  • Standard deviation (σ) measures the spread of the distribution
• Probability density function (PDF) for a normal distribution follows the equation: $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
• Bell-shaped curve represents the normal distribution
  • Highest point occurs at the mean
  • Probability decreases symmetrically as values move away from the mean

Distribution properties

• Empirical rule (68-95-99.7 rule) describes data distribution
  • 68% of data falls within one standard deviation of the mean
  • 95% of data falls within two standard deviations
  • 99.7% of data falls within three standard deviations
• Unimodal distribution with mode, median, and mean equal and centered
• Total area under the normal distribution curve always equals 1
  • Represents the sum of probabilities for all possible outcomes

Standardizing normal variables

Standardization process

• Standardization converts a normal random variable X to a standard normal variable Z
  • Resulting Z has a mean of 0 and standard deviation of 1
• Formula for standardization: $Z = \frac{X - \mu}{\sigma}$
  • X represents the original value
  • μ represents the mean
  • σ represents the standard deviation
• Standard normal distribution (z-distribution) results from standardization
  • Special case of normal distribution with μ = 0 and σ = 1

Using the standard normal table

• Z-table (standard normal table) provides cumulative probabilities for the standard normal distribution
• Steps to find probabilities using the z-table:
  1. Standardize the given value
  2. Locate the corresponding probability in the table
• Z-table typically gives area to the left of a given z-score
  • Can be used to find areas to the right or between two z-scores through calculations
• Interpolation may be necessary for z-scores falling between provided table values
  • Example: For z-score 1.234, interpolate between values for 1.23 and 1.24

Probabilities for normal distributions

Calculating probabilities

• Determine probabilities for general normal distributions by standardizing values and using the z-table
• Cumulative distribution function (CDF) gives probability that X ≤ x for a random variable X
• Find probabilities between two values by calculating the difference between their CDFs
• Use symmetric intervals around the mean with the formula: $P(\mu - k\sigma < X < \mu + k\sigma) = 2\Phi(k) - 1$
  • Φ represents the standard normal CDF
  • Example: Probability of X falling within 2σ of the mean is 2Φ(2) - 1 ≈ 0.9545

Determining quantiles

• Calculate quantiles (percentiles, quartiles) using inverse standardization and the z-table
• Formula for finding a quantile: $X = \mu + (Z * \sigma)$
  • Z represents the z-score corresponding to the desired percentile
• Interquartile range (IQR) for a normal distribution approximately equals 1.34σ
  • Useful for identifying potential outliers
  • Example: In a normal distribution with σ = 10, IQR ≈ 13.4

Normal approximation of binomial distributions

Conditions for approximation

• Normal distribution approximates binomial distribution when:
  • Sample size n is large
  • Probability p is not too close to 0 or 1
• Rule of thumb for using normal approximation
  • Both np and n(1-p) should be ≥ 5 or 10, depending on desired accuracy
  • Example: For n = 100 and p = 0.3, np = 30 and n(1-p) = 70, satisfying the condition

Applying the approximation

• Approximating normal distribution parameters:
  • Mean = np
  • Standard deviation = √(np(1-p))
• Apply continuity correction when using normal approximation
  • Add or subtract 0.5 to the value of interest
  • Depends on calculating "less than" or "greater than" probability
• Accuracy improves as n increases and p approaches 0.5
• Useful for large n values where direct binomial calculation becomes computationally intensive
• Recognize limitations and use exact binomial probabilities when high precision is required
  • Example: Medical studies often require exact probabilities rather than approximations