PP

Chapter 6: The Normal Distribution

6.1 The Standard Normal Distribution

  • Normal distribution: A bell curve or a Gaussian distribution curve. If a random variable has a probability distribution whose graph is continuous, symmetric, and bell-shaped.

  • Approximately normally distributed variables: Many continuous variables (e.g. weights, heights) have distributions that are bell-shaped

  • Standard normal distribution: a normal distribution with a mean of µ = 0 and a standard deviation of σ = 1.

  • Z-score formula: X − 𝝁 / σ

    Same means but different standard deviations

    Different means but the same standard deviations

    Different means and different standard deviations

Summary of the Properties of the Theoretical Normal Distribution

  • Mean = Median = Mode

  • A normal distribution curve is unimodal.

  • The curve never touches the x-axis.

  • The total area under a normal distribution curve is equal to 1.00

6.2 Using the Normal Distribution

  • Normal Distribution: X ~ N(µ, σ) where µ is the mean and σ is the standard deviation.
  • (P → X): X = 𝝁 + 𝝈Z
  • Number of units (individual/items) satisfying a condition→ Total number of units given in the problem × are calculated based on condition
  • Calculator function for probability: normalcdf (lower x value of the area, upper x value of the area, mean, standard deviation)
  • Calculator function for the kth percentile: k = invNorm (area to the left of k, mean, standard deviation)

6.3 The Central Limit Theorem

  • Sampling distribution of sample means: a distribution using the means computed from all possible random samples of a specific size taken from a population.
  • Sampling error: the difference between the sample measure and the corresponding population measure because the sample is not a perfect representation of the population.

Properties of the Distribution of Sample Means

  • The mean of the sample means = population mean.
  • The standard deviation of the sample means < standard deviation of the population
  • The standard deviation of the sample means will be equal to the population standard deviation divided by the square root of the sample size.
  • The Central Limit Theorem: sample size n, mean µ, and standard deviation σ are given
  • Standard error of the sample mean: 𝝈/ √n

Examples