Chapter 3 Notes: The Normal Distributions

Chapter 3: The Normal Distributions

Overview of Chapter Topics

  • Density curves
  • Describing density curves
  • Normal distributions
  • The 68–95–99.7 rule
  • The standard Normal distribution
  • Finding Normal proportions
  • Using the standard Normal table
  • Finding a value given a proportion

Density Curves

  • Definition: A density curve models the overall pattern of a distribution, summarizing a large number of observations.
    • Always lies on or above the horizontal axis.
    • Total area under the curve equals 1.
  • Steps for Exploring a Distribution:
    • Always plot your data (e.g., histogram, stemplot).
    • Observe the overall pattern of shape, center, and variability (look for outliers).
    • Compute a numerical summary for center and variability.

Example of Density Curves

  • Vocabulary scores of 947 seventh-graders represented in a histogram.
    • A smooth curve fitting the histogram approximates the observed data pattern.
    • Example area under the curve for a score of 6.0 is 0.293, indicating the proportion of scores at or below this value.

Characteristics of Density Curves

  • Median: The point that divides the area into two equal halves under the curve.
  • Mean: The “balance point” of the curve; both mean and median coincide in symmetric distributions but may differ in skewed distributions.
  • Mean (μ) and standard deviation (σ) for actual observations vs. those represented by the density curve.

Normal Distributions

  • Definition: Normal distributions are a specific class of density curves that are symmetric, single-peaked, and bell-shaped.
  • A Normal distribution is fully described by its mean (μ) and standard deviation (σ).
    • Changing the mean shifts the curve along the horizontal axis.
    • The standard deviation affects the spread: larger σ results in a flatter curve.

68-95-99.7 Rule

  • Approximately:
    • 68% of observations lie within 1 standard deviation (σ) of the mean (μ)
    • 95% within 2σ
    • 99.7% within 3σ

Standardizing and z-Scores

  • Standardizing: Converting a value from an original Normal distribution to a standard Normal distribution (mean = 0, standard deviation = 1).
    • Formula: ( z = \frac{x - \mu}{\sigma} ) (where x = original value)
  • Example: For women’s height with a mean of 64.1 inches and a standard deviation of 3.7 inches, a height of 70 inches has a z-score of 1.59, indicating it is 1.59 standard deviations above the mean.

The Standard Normal Distribution

  • A standard Normal distribution has a mean of 0 and a standard deviation of 1.
  • Any Normal distribution can be standardized to this format.

Cumulative Proportions

  • Definition: The cumulative proportion for a value represents the proportion of observations less than or equal to that value.

Example Calculation

  • For SAT scores with a mean of 1059 and standard deviation of 210, to find the proportion of students scoring 980 or above, standardize the score:
    • ( z = \frac{980 - 1059}{210} = -0.38 )
    • Use standard normal tables or software to calculate the area under the curve.

Normal Calculations

  • Using Standard Normal Table:
    • Steps:
    1. Define observed variable x (set plot).
    2. Standardize x to a z-score.
    3. Refer to Table A for area calculations.
  • Finding Scores Given a Proportion:
    • To find a score for the top 10%, find the z-score corresponding to 0.90 cumulative proportion (1.28 from tables), then unstandardize:
    • Formula: ( x = \mu + z \cdot \sigma )
    • For SAT, ( x = 531 + 1.28 \cdot 104 \approx 664.12 )

Conclusion

  • Mastery of the normal distribution involves understanding density curves, z-scores, and how to manipulate the mean and standard deviation to calculate proportions and values within a dataset.