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)

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• 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

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  • 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

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  • 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