The standard normal distribution is a powerful tool in statistics, helping us understand data patterns. It's a bell-shaped curve with a mean of 0 and standard deviation of 1, making it easy to compare different datasets.

Z-scores are key in using this distribution. They tell us how far a value is from the mean in terms of standard deviations. This lets us find probabilities and percentiles, which are super useful in real-world situations like test scores or quality control.

Standard Normal Distribution

Characteristics of standard normal distribution

• Represents a continuous probability distribution that follows a symmetrical bell-shaped curve
• Has a mean ($\mu$) equal to 0 and a standard deviation ($\sigma$) equal to 1
• Encompasses a total area under the curve that sums to 1, representing all possible outcomes
• Uses the variable $Z$ to distinguish it from other normal distributions
• Functions as a standardized version of any normal distribution for comparison purposes

Z-scores for value standardization

• Transform values from the original normal distribution to the standard normal distribution using z-scores
• Calculate z-scores using the formula: $Z = \frac{X - \mu}{\sigma}$, where $X$ represents the value, $\mu$ represents the mean, and $\sigma$ represents the standard deviation
• Express the number of standard deviations a value is from the mean using z-scores
  • Indicate values above the mean with positive z-scores (right side of curve)
  • Indicate values below the mean with negative z-scores (left side of curve)

Applying the Standard Normal Distribution

Probabilities using z-scores and tables

• Provide probabilities, percentiles, and areas under the curve using standard normal distribution tables (z-tables)
• Locate the z-score in the table and find the corresponding probability to determine the probability of a value being less than or equal to a given z-score
• Calculate the probability of a value falling between two z-scores:
  1. Use the table to find the areas to the left of each z-score
  2. Subtract the smaller area from the larger area to obtain the probability
• Locate the z-score in the table and multiply the corresponding area to the left by 100 to find the percentile for a given z-score

Empirical rule for normal distributions

• Estimate the percentage of data within specific standard deviations of the mean using the Empirical Rule (68-95-99.7 Rule)
  • Contains 68% of data within 1 standard deviation of the mean ($\mu \pm 1\sigma$)
  • Contains 95% of data within 2 standard deviations of the mean ($\mu \pm 2\sigma$)
  • Contains 99.7% of data within 3 standard deviations of the mean ($\mu \pm 3\sigma$)
• Estimate proportions quickly without using z-tables for normally distributed data

Applications in real-world scenarios

• Recognize that many real-world variables approximately follow a normal distribution (heights, weights, test scores)
• Solve problems involving normally distributed variables:
  1. Determine the mean and standard deviation of the distribution
  2. Standardize the values and locate their relative positions using z-scores
  3. Determine probabilities, percentiles, or areas under the curve using z-tables or the Empirical Rule
• Calculate the percentage of students scoring above a certain value on a standardized test (SAT, GRE)
• Assess the probability that a randomly selected product weighs less than a specified amount (quality control)
• Identify the minimum or maximum value corresponding to a given percentile in a population (income, IQ scores)