Chapter 2: Modeling Distributions of Quantitative Data

2.1 Describing Location in a Distribution

• The pth percentile is the percent of values in a distribution that are less than or equal to the individual’s data value.
  • Ex: 4 students of 25 earned a higher test score than Emily. 21/25 ppl have scores less than or equal to her, so she’s in the 84th percentile.
  • Ex: The table reveals that 20.0% of U.S. presidents took office by the time they turned 50. In other words, the 20th percentile of the distribution of inauguration age is 50.000 years.
• A Cumulative Relative Frequency Graph plots a point corresponding to the cumulative relative frequency
  • Make one by
  • Useful to measure percentiles in a distribution
• The standardized score (z-score) tells us how many standard deviations a value is from the mean
  • z = (value - mean) / (standard deviation)
  • Values larger than mean have positive z-scores
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• Transformations (a & b are constants):
  • Neither transformations will change the shape.
  • If data is being added/subtracted by a constant (a), all the values in a data set, measures of center and location—mean, five-number summary, percentiles—increase (decrease) by 𝑎
  • Measures of variability—range, IQR, SD—do not change.
  • When you multiply (divide) all the values in a data set by a positive constant 𝑏 everything changes: measures of center, location, and variability are multiplied (divided) by 𝑏.
  • It’s possible to have multiple transformations of a and b in one distribution

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2.2 Density Curves and Normal Distributions:

• Density Curves always add up to 1 or 100%

  • Its basically a dot plot with a shit ton of calculus (area under the dot plot)

• If a density curve is symmetric both the mean and the median will lie at the same point of symmetry.

• If a curve is skewed, however, the mean is closer to tail, median is farther from tail

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• Normal Density Curve is drawn over a histogram (calculus shit again)

  • Shape: All Normal distributions have the same overall shape: symmetric, single-peaked (unimodal), and bell-shaped.
  • Center: The mean μ is located at the midpoint of the symmetric density curve and is the same as the median.
  • Variability: The standard deviation σ measures the variability (width) of a Normal distribution.
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• The Empirical rule states that: 68% for [-1, 1] SD, 95% for [-2,2] SD, 99.7 for [-3,3] SD

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• Reminder: You need to draw the normal graph & the scaled graph:

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• Make sure to write N=(Mean, SD) for left graph, and N=(0,1) for right normalized graph

• To convert a percentile to a z score do backwards stuff:

  • Look in the body of Table A for the value closest to 0.20. A 𝑧-score of –0.84 gives the closest value (0.2005).
  • −0.84 = (x-5.3) / 0.9 → x = 4.544mins

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