R

2.1: Density Curves, Normal Distributions, and Segmented Bar Charts

Normal Curves

  • Some data can be graphed on a relatively smooth curve
    • Detail may be lost about some specific data, but this is outweighed by the simplicity the curve allows us to analyze with
    • Mathematical model: an idealized description of a situation
  • When a curve is used to approximate a distribution, the scale is adjusted to make it into a density curve
  • Density curve: a distribution which is always above the x-axis and always has an area of 1
    • The area in any part of the curve is equal to the percent of data present in that part of the curve
  • Normal curve: a distribution which is symmetric, bell-shaped, and has a single peak
    • Defined by their means (μ) and standard deviations (σ)
  • Normal curves may be analyzed further using z-scores, which can be found and interpreted in relation to probabilities using a z-chart
    • This chart is provided on the AP exam, along with many relevant formulas
  • When solving z-score problems, it is always necessary to:
    • Show work for getting the z-score
    • Write a probability statement
    • Give the final probability
  • Data may be represented as N(μ, σ)
    • μ: mean
    • σ: standard deviation

Case Study: Male vs. Female Height

  • Who is taller relative to their gender: a 76.5-inch-tall woman or a 80-inch-tall man?
    • Female heights are N(64, 2.5) and male heights are N(69.1, 2.8)
  • Female: 76.5 inches
    • That means that this value is 5 standard deviations above the mean
    • This is where the term “off the charts” comes from → if you reference what 5 standard deviations above the mean would look like on the normal curve above in the notes, 5 standard deviations to either side is virtually unheard of and statistically very rare
  • Male: 80 inches
    • The height of an 80-inch-tall man is 3.89 standard deviations above the mean
  • This means that statistically, the woman is taller proportionate to her gender than the man is proportionate to his.

Segmented Bar Charts

  • Segmented bar chart: a display of categorical data showing such data as a percentage of the whole
    • Like a pie chart in bar form
  • Mosaic plot: segmented bar charts for comparing groups where the widths of the bars are proportional to the size of the groups