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Equation 13 · A Showdown Ladder Rating Measures the Ladder, Not the Player

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EE

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EE

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Classical Test Theory gives psychometrics its founding decomposition: an observed score X on any single administration of a test equals a true score T — the score a person would get on average across infinitely many parallel administrations — plus measurement error E , so X = T + E , with E assumed to average to zero and to be uncorrelated with T across a population of test-takers [ 4 ] . From that single axiom the whole apparatus of reliability follows: reliability is the proportion of observed-score variance attributable to true-score variance, and the standard error of measurement, SEM\mathrm{SEM} = SDX\mathrm{SD}_X1−reliability\sqrt{1 - \text{reliability}} , is the quantity that turns a single observed…
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Classical Test Theory gives psychometrics its founding decomposition: an observed score X on any single administration of a test equals a true score T — the score a person would get on average across infinitely many parallel administrations — plus measurement error E , so X = T + E , with E assumed to average to zero and to be uncorrelated with T across a population of test-takers [ 4 ] . From that single axiom the whole apparatus of reliability follows: reliability is the proportion of observed-score variance attributable to true-score variance, and the standard error of measurement, SEM\mathrm{SEM} = SDX\mathrm{SD}_X1−reliability\sqrt{1 - \text{reliability}} , is the quantity that turns a single observed score into a defensible interval rather than a point claim — a test-taker’s true score is estimated to lie within roughly X ±\pm 1.96 ⋅\cdot SEM\mathrm{SEM} at 95% confidence [ 4 ] .

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