Normal Distribution
nothing is called 95% until it has been counted
Depth

Series — page 3

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The weight on the population, σ = 3. Each curve is one population spread τ. A group's estimate moves B = se²/(se² + τ²) of the way to the population mean, where se = σ/√n is what the group's own mean does not know. At τ = 1 a group of 9 observations sits halfway.

Shrinkage

  1. 2 The weight that decides
  2. 3 What the plug-in forgets
  3. 4 A group from the population's own tail
  4. 5 Estimates that are too alike
  5. 6 A league table of a hundred
5 essays · hierarchical Student's t on 5 degrees of freedom, against the normal. The two-sided 95% critical value is 2.571 for t(5) and 1.960 for the normal — 31% wider. Using the normal at this sample size makes every interval too short by that much.

Student

  1. 2 The correction for not knowing the spread
  2. 3 Where the two tails disagree
  3. 4 A degrees of freedom that is not a count
  4. 5 The skewness of a difference
  5. 6 The side a bound is read from
5 essays · intervals Four datasets, slope 0.50, R² 0.67. Every one of these fits reports the same slope to two decimals and the same R². Only the first is a linear relationship with noise: the second is a curve, the third is a line with one outlier, and the fourth has its slope set by a single point.

Summary

  1. 1 Four datasets, one summary
  2. 2 R² is not a measure of fit
  3. 3 R² is a property of the design
  4. 4 The t statistic wearing different clothes
  5. 5 The summary that was meant to work
5 essays · regression 8 exponential draws, standardised, against the normal. The source is one-sided and skewed. At n = 8 the standardised sum has skew 0.695, and the theory says 2/sqrt(n) = 0.707 — so the convergence is visible AND its rate is predicted.

Clt

  1. 1 Sums of almost anything
  2. 2 Where the derivative is zero
  3. 3 A ratio whose interval has to be the whole line
  4. 4 A flat point with more than one direction
4 essays · normal The variance touches p and never crosses it. d(x) = f(x)′M⁻¹f(x) along the diagonal of a square region, for the D-optimal measure. The line at 6 is the number of parameters in the model. Kiefer and Wolfowitz's theorem says a design is D-optimal exactly when the largest d anywhere in the region is p — not approximately, equals — so the optimal curve is tangent to that line at its support points and below it everywhere else. Here the largest value anywhere on a 41×41 grid is 6.000000000.

Equivalence

  1. 1 The theorem that says when to stop
  2. 2 How many places a design goes
  3. 3 Augmenting a design that has already run
  4. 4 The criterion with no derivative
4 essays · optimality The false-positive rate against the number of analyses, on pure noise. The data has no effect in it. Each analysis is correct and each p-value is honest. With 20 correlated outcomes available, something reaches p below 0.05 58% of the time.

Forking

  1. 3 Twenty analyses of nothing
  2. 4 How many analyses there really were
  3. 5 What naming it in advance costs
  4. 6 The correction that makes the estimate worse
4 essays · testing Twenty points and one more, at leverage 0.74. Without the distant point the slope is 0.495; with it the slope is -0.389. Its leverage is 0.737 and its Cook's distance is 24.1, against a conventional threshold of 1.

Leverage

  1. 1 The line that one point drew
  2. 2 Two points that hide each other
  3. 3 A robust loss and a far x
  4. 4 The start an efficient robust line inherits
4 essays · regression Coverage against sample size, true proportion 0.15. Coverage does not improve monotonically. n = 19 covers 93.8% while the larger n = 20 covers 81.9%. The sample space is discrete, so the endpoints jump as n changes.

Oscillation

  1. 2 More data is not monotonically better
  2. 3 A hole no sample size fills
  3. 4 What a guaranteed minimum costs
  4. 5 The coin that makes it exact
4 essays · intervals Where the normal approximation converges, and where it does not. Relative error against the exact binomial. At n = 1280 the error at the median is 0.96% and three sigma out it is 25.7% — a factor of 27. The tail is where the approximation is used.

Rate

  1. 2 The tail converges last
  2. 3 A bound written for a coin
  3. 4 A correction that goes below zero
  4. 5 An approximation built at the threshold
4 essays · normal Twenty 95% intervals for a proportion that really is 0.35. 2 of the twenty miss the true value. The 95% is a property of the procedure across repetitions — no single interval has a 95% chance of anything, because it either contains 0.35 or it does not.

Repetition

  1. 1 Twenty intervals and one expected miss
  2. 2 Five times in six
  3. 3 Two intervals that overlap
  4. 4 Significant in one, not in the other
4 essays · intervals Which allocations reverse the overall comparison. The per-group success rates are held fixed; only the split of each group between treatment and control changes. 32% of the allocations reverse, and the worst reverses by 13.1 percentage points.

Simpson

  1. 1 Simpson's reversal is a region, not a table
  2. 2 The reversal a coin cannot prevent
  3. 3 The change that is not confounding
  4. 4 Conditioning on what the treatment caused
4 essays · paradox 20,000 p-values from a true null, n = 12. Flat, as it must be: under the null a p-value is uniform on (0,1). The Kolmogorov–Smirnov distance from uniform is 0.0090 (p = 0.81). That flatness is the check that catches an error a single rejection rate would miss.

Uniformity

  1. 1 A p-value that is not flat is not a p-value
  2. 2 The p-value a replication gets
  3. 3 Two ways to combine p-values
  4. 4 The smallest of three combinations
4 essays · testing weakly informative — Beta(2, 2), updated by 5 of 20. The prior is worth 4 observations. With 20 observations the posterior mean is 0.292, against a data proportion of 0.250 and a prior mean of 0.500.

Prior

  1. 1 What a prior is worth
  2. 2 The prior the data estimates
  3. 3 A prior on the spread
3 essays · bayes Coverage of four nominal 95% intervals, n = 30. Computed exactly by summing over all 31 possible counts, not simulated. The Wald interval drops to 26.0% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Routes

  1. 2 Two routes to every number
  2. 3 The arcsine that closes it, and the error that was overstated
  3. 4 The check worth more than the check
3 essays · method Coverage of four nominal 95% intervals, n = 20. Computed exactly by summing over all 21 possible counts, not simulated. The Wald interval drops to 18.2% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Coverage

  1. 1 What the 95% refers to
  2. 2 An interval that covers and says nothing
2 essays · intervals 12,000 studies of a real effect of 0.3, n = 16. Power is 21%. The studies that reached significance report a mean effect of 0.621 — 2.07 times the truth. Every one of them is honest; the selection did the inflating.

Curse

  1. 3 The winner's curse
  2. 4 The estimate after the choice
2 essays · testing

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