Keep this page open while you read the notes. The single most important distinction in the course runs down the first three tables: a parameter is a fixed unknown, a statistic / estimator is a random function of the sample, and an estimate is one realized number. Confusing them is the root of most inferential mistakes, so the symbols are chosen to keep them apart. All numeric values shown are from the synthetic recurring reading-fluency study and are verified: false (the math gate is blocked pending sign-off).
Parameters (fixed unknowns)
A parameter is a fixed feature of the population or process. It does not have a sampling distribution and never takes a “hat.”
| \(\theta\) |
a generic parameter (in this course, usually the true pass rate) |
| \(p\) |
a success probability / proportion parameter |
| \(\mu\) |
a population mean |
| \(\sigma\), \(\sigma^2\) |
a population standard deviation, variance |
Statistics and estimators (random — functions of the sample)
A statistic is computed from the random sample, so it is itself random and has a sampling distribution. An estimator is a statistic used to estimate a parameter.
| \(X_1,\dots,X_n\), \(X\) |
the sample (random variables); lowercase \(x\) are observed values |
| \(\bar X\) |
the sample mean (estimator of \(\mu\)) |
| \(\hat p = X/n\) |
the sample proportion (estimator of \(p\) or \(\theta\)) |
| \(S\) |
the sample standard deviation (estimator of \(\sigma\)) |
| \(\hat\theta\) |
a generic estimator of \(\theta\) |
Estimates (one realized number)
| \(\hat p = 0.65\) |
the observed sample proportion (\(26\) of \(40\)) |
| \(\bar x = 8.0\) |
the observed sample mean gain |
| \(s = 6.0\) |
the observed sample SD of the gains |
The hat does double duty — \(\hat\theta\) can mean the estimator (a random variable) or its realized value (a number). Say which you mean.
Variability, bias, and error
| \(\operatorname{SE}(\hat\theta)\) |
standard error — the (estimated) SD of the estimator’s sampling distribution (not the SD of the data) |
| \(\operatorname{Bias}(\hat\theta) = E[\hat\theta] - \theta\) |
bias; the estimator is unbiased when this is \(0\) |
| \(\operatorname{Var}(\hat\theta)\) |
the variance of the estimator |
| \(\operatorname{MSE}(\hat\theta) = \operatorname{Var}(\hat\theta) + \operatorname{Bias}(\hat\theta)^2\) |
mean squared error; the bias–variance decomposition |
Likelihood and maximum likelihood
| \(L(\theta)\) |
the likelihood — a function of \(\theta\) given the data; not a distribution over \(\theta\) (it need not integrate to 1) |
| \(\ell(\theta) = \log L(\theta)\) |
the log-likelihood; maximized at the same \(\theta\) as \(L\) |
| \(\hat\theta_{\text{MLE}}\) |
the maximum likelihood estimate; for the study, \(26/40 = 0.65\) |
Intervals and testing
| CI |
confidence interval — a procedure with long-run coverage; never a probability statement about a fixed \(\theta\) |
| \(z^{*}\), \(t^{*}\) |
the critical multiplier (\(1.96\) for a 95% normal interval; \(t_{n-1,\,0.975}\) for a mean) |
| \(H_0\), \(H_a\) |
null and alternative hypotheses |
| \(T\), \(Z\) |
a test statistic |
| \(p\)-value |
\(P(T \text{ as or more extreme} \mid H_0)\) — a tail probability under \(H_0\), not \(P(H_0 \text{ true})\) |
| \(\alpha\) |
significance level; the Type I error rate (rejecting a true \(H_0\)) |
| \(\beta\) |
the Type II error rate (failing to reject a false \(H_0\)) |
| power |
\(1 - \beta\) |
Simulation-based inference
| \(\hat\theta^{*}\) |
a bootstrap resampled estimate (resample the sample with replacement) |
| percentile interval |
the middle 95% of the bootstrap distribution |
| permutation null |
a randomization / reference distribution built by relabeling under the null |
Bayesian inference
| \(\pi(\theta)\) |
the prior distribution for \(\theta\) (an assumption, not data) |
| \(p(x \mid \theta)\) |
the likelihood of the data given \(\theta\) |
| \(\pi(\theta \mid x)\) |
the posterior: \(\pi(\theta \mid x) \propto p(x \mid \theta)\,\pi(\theta)\) |
| \(p(x)\) |
the evidence / marginal likelihood — the normalizing constant dropped by \(\propto\) |
| \(\propto\) |
“proportional to”; always know which constant (\(p(x)\)) it dropped |
| \(\text{Beta}(a,b)\) |
the Beta prior/posterior (shape–shape); mean \(a/(a+b)\) |
| credible interval |
a Bayesian posterior interval — a genuine probability statement about \(\theta\) (the opposite of a CI) |
| posterior predictive |
the distribution of a future observation, averaging the likelihood over the posterior |
Decisions
| \(a\) |
an action / decision |
| \(\operatorname{Loss}(\theta, a)\) |
a decision loss — written out, never \(L\) (which is the likelihood) |
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