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Back-of-the-Envelope Statistics

Back-of-the-Envelope Statistics. Jason Zimba Student Achievement Partners and Bennington College. Signal and Noise Significance vs. Size Regression Models. Call It:. I flip a coin a million times. Results: 51% heads, 49% tails. Do we think the coin is fair?.

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Back-of-the-Envelope Statistics

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  1. Back-of-the-Envelope Statistics Jason Zimba Student Achievement Partners and Bennington College

  2. Signal and Noise • Significance vs. Size • Regression Models

  3. Call It: • I flip a coin a million times. • Results: 51% heads, 49% tails. • Do we think the coin is fair?

  4. With N data points, expect random fluctuations to be of order in percentage terms. The Rule of Thumb: • Apply rule of thumb to coin problem: • N ~ 1,000,000  square root of N is 1000 • Expect fluctuations of about 1/1000 = 0.1% • Actual fluctuations ~1% • Ten times larger than one intuitively expects • Don’t bet on the coin being fair…!

  5. Implications of the Rule • If you want to cut random noise in half, expect to gather four times as much data. • Suppose an astronomer wants to image a star. To make the image twice as clear, it will take 4times as much telescope time. • Square root dependency makes good data expensive to collect.

  6. Implications of the Rule • The larger the N, the smaller the noise in percentage terms (one hopes….) • N = 100  expect ~ 10% noise • N = 10,000  expect ~ 1% noise

  7. Implications of the Rule When N is too small, noise swamps the signal H. Wainer, “The Most Dangerous Equation,” American Scientist, May-June 2007

  8. Signal and Noise • rule of thumb and its implications • Significance vs. Size • Regression Models

  9. “Significant” means… • The effect being studied is real. • You’re looking at signal, not noise. • The effect being studied is unlikely to be due to chance. • p value: “p < 0.05.”

  10. “Significant” does not mean that the effect is large. • If N is large enough, then the noise will be damped down, and very weak signals will emerge. • These weak signals are termed “significant” despite their small size (because they are real). • The p-value of an effect does not tell you how large (or important) the effect is. • For the coin flip problem, the 1% deviation is very significant (p < 0.000…001). The coin is almost certainly biased - just not very strongly.

  11. Describing Effect Sizes Sizes are always relative. • A year is a long time for a snowstorm to last, a short time for an empire to last. • A millimeter is large for a molecule, small for an engagement ring. • A ton is large for a hog, small for an asteroid. Comparisons must be like with like. • “A year is longer than an ounce.” (??)

  12. Computing effect size Example: How strongly does gender affect weight? • US avg. adult male weight = 190 lb  50 lb • US avg. adult female weight = 160 lb  50 lb • Diff. between avg. M and avg. F = 30 lb • Natural variability in adult weights = 50 lb • Effect size: • d = (diff. between averages)  (natural variability) • = (30 lbs)  (50 lbs) • = 30  50 • d = 0.6.

  13. “Packaging” Effect Sizes • Beware of authors describing effect sizes as being ‘large’ … ‘substantial’ … ‘small’ … ‘quite small’ … ‘slight’, …. • “There is evidence of slightlygreater male variability in scores, although the causes remain unexplained.”* • Meaningful assessments of effect size go beyond everyday language to compare the effect numerically to some meaningful standard measured in the same units. • e.g. ‘The top quartile-bottom quartile difference in value-add among third-year teachers was X times as large as the first-year gain to experience.’ *Hyde et al., “Gender Similarities Characterize Math Performance,” Science, 25 July 2008, Vol. 321, pp. 494, 495.

  14. Signal and Noise • rule of thumb and its implications • Significance vs. Size • Significant means real/not spurious • Effect could still be small and unimportant • Don’t let people “package” for you • Describe effect size numerically with like-to-like comparison • Regression Models

  15. Birth Wt = 0.067 Wt of Mother - (0.57 lb/wk)  Wks premature

  16. Signal and Noise • rule of thumb and its implications • Significance vs. Size • Significant means real/not spurious • Effect could still be small and unimportant • Don’t let people “package” for you • Describe effect size numerically with like-to-like comparison • Regression Models • Regression models tell you how much a change in the “inputs” affects the “output.” • What looks like noise might actually be an effect - an effect of an unmeasured variable. • But if you put in too many variables, you can explain anything.

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