Data Transformations

• Measures of center include the mean, median, quartiles, minimum, and maximum of a set of data

• Measures of spread include the IQR, range, and standard deviation

• When a constant is added to every number in a list,

  • The measures of center increase by that amount and the measures of spread remain the same

• When every number in a list is multiplied by a constant,

  • The measures of center are multiplied by that amount and the measures of spread also get multiplied by that amount

Comparing Unalike Figures

• Suppose we have two values that we want to compare, but they don’t come from the same distribution

  • Eg. getting a 610 on SAT math vs. a 24 on ACT math

  • Eg. being a 80” guy vs. a 76.5” girl

• We can compare these figures by getting a standardized score called a z-score

  • z = (x-μ)/σ

    • z: z-score

    • x: value

    • μ: population mean

    • σ: population standard deviation

• A z-score shows exactly how many standard deviations above or below the mean a value is

• Even if two data points come from different distributions, they can be directly compared if converted to z-scores

• When comparing two z-scores, the higher z-score is the better relative score

Z-Charts

• A chart which can be used to convert proportions to z-scores and vise-versa

  • Available on the “cheat sheet” during the AP exam

Example Problem

• Find percent of values less than 54

  • μ = 65

  • σ = 9

• X~N (μ,σ) → Z~N (65,9)

• Finding P(x<54) → x = 54, put into equation z = (x-μ)/σ

  

• Locate z = -1.2222 on z-chart

  • 

• P(x < 54) = P(z < -1.22) = 0.1112 (11.12%)