Whoville Statistics: Regression, Probability, and Distributions Flashcards

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Correlation

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A numerical measure of the linear association between two variables. Values range from $-1$ to $1$, with sign indicating direction and magnitude indicating strength.

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Slope

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In regression, the estimated change in the response variable for a one-unit increase in the predictor. Computed as $b_1 = r \frac{s_{Y}}{s_{X}}$ for simple linear regression.

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Intercept

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The predicted value of the response when the predictor equals zero. Interpretation can be meaningless if $x=0$ is outside the observed range.

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Least-squares line

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The line that minimizes the sum of squared residuals between observed values and predicted values. It provides the best linear unbiased estimate under standard assumptions.

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Residual

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The difference between an observed value and its predicted value from a regression model. Residual = observed $-$ predicted.

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Coefficient of determination

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Denoted $R^2$, it measures the proportion of variance in the response explained by the predictor(s). In simple regression, $R^2 = r^2$.

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Prediction vs. Causation

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Regression can predict associations but does not prove causation unless the study design (e.g., randomized experiment) supports causal claims. Correlation alone is insufficient.

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Binomial model

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Models the number of successes in $n$ independent trials with constant success probability $p$. Denoted $Bin(n,p)$ with mean $np$ and variance $np(1-p)$.

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Normal approximation

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Approximating a Binomial $Bin(n,p)$ by a Normal when $n$ is large and $p$ not too close to 0 or 1. Use mean $np$ and variance $np(1-p)$ and apply continuity correction.

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Continuity correction

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Adjustment when approximating discrete distributions (like binomial) by a continuous distribution (normal). E.g., $P(X\ge k)$ approximated by $P(X>k-0.5)$.

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Expected value

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The weighted average of all possible values of a random variable, using their probabilities. For discrete $X$, $E[X]=\sum x p(x)$.

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Variance

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The expected squared deviation from the mean: $Var(X)=E[(X-\mu)^2]=E[X^2]-\mu^2$. It measures spread of the distribution.

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Standard deviation

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The square root of the variance. It is on the same scale as the variable and describes typical deviation from the mean.

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Bayes' theorem

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A formula to reverse conditional probabilities: $P(A|B)=\dfrac{P(B|A)P(A)}{P(B)}$. Useful when updating probabilities given new evidence.

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Finite population correction

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An adjustment when sampling without replacement from a finite population. When the sample is less than about 5% of the population, the correction is negligible.

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Z-score

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Standardized value representing how many standard deviations a data point is from the mean: $z=\dfrac{x-\mu}{\sigma}$. Used to find probabilities under the normal curve.

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Prediction interval

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An interval estimate for an individual future observation that accounts for both uncertainty in the regression parameters and residual variability. Wider than a confidence interval for the mean.

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Sample size rule

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For normal approximation of a binomial, ensure $np$ and $n(1-p)$ are both reasonably large (common rule: at least 5 or 10). This ensures approximation quality.