Introduction to Probability

Multivariable Calculus

Algorithms: Part II

Algorithms: Part I

Introduction to Software Design and Architecture

Calculus Two: Sequences and Series

LAFF Linear Algebra

Stanford Machine Learning

Calculus One

Computational Thinking

Effective Thinking Through Mathematics

CS50 Introduction to Computer Science

Others

Lecture 13

Statistical Abuses

Anscombe’s Quartet

  • Summary statistics for groups identical
    • Mean x = 9.0
    • Mean y = 7.5
    • Variance of x = 10.0
    • Variance of y = 3.75
    • Linear regression model: y = 0.5x + 3
  • Are four data sets really similar?
  • Moral:
    • Sometimes, Statistics about the data is not the same as the data
    • Use visualization tools to look at the data itself

Lying with Pictures

  • Telling the Truth with Pictures
  • Moral: Look carefully at the axes labels and scales

GIGO (Garbage In, Garbage Out)

  • Moral: Analysis of bad data can lead to dangerous conclusions.

Non-representative Sampling

  • “Convenience sampling” not usually random, e.g.,
    • Survivor bias, e.g., course evaluations at end of course or grading final exam in 6.00.2x on a curve
    • Non-response bias, e.g., opinion polls conducted by mail or online
  • Moral: Understand how data was collected, and whether assumptions used in the analysis are satisfied. If not, be wary.

A Comforting Statistic?

  • 99.8% of the firearms in the U.S. will not be used to commit a violent crime in any given year
    • How many privately owned firearms in U.S.?
    • 300,000,000
    • 300,000,000*0.002 = 600,000
  • Moral: Context matters. A number means little without context.

Relative to What?

  • Consider drugs X and Y for treating acne
    • X cures acne twice as well as Y
    • X kills twice as many acne patients as Y
  • Do you want to take X or Y?
    • Suppose Y kills 0.00001% of cases, and cures 50% of them
  • Moral: Beware of percentages when you don’t know the baseline

Lurking Variable

  • Does going to school contribute to the spread of flu?
  • Moral:
    • Establishing Causation
      • Attempt to control for all variables other than the variables of interest
        • Rarely possible
      • Randomized control studies the gold standard
        • Start with a population
        • Randomly assign members to either
          • Control group
          • Treatment group
        • Deal with two groups identically except with respect to the one thing being evaluated
        • Very hard to do