Notation glossary
Every symbol used across the course, collected in one place
What this page is
This page is the public mirror of the course’s private notation ledger, the internal document that first fixed every symbol, convention, and known confusion before any week note was drafted. That private ledger is build bookkeeping, not something you need to read. What you need from it is here: the symbols themselves, the convention this course chose for each one, and — because a symbol memorized without a worked number rarely sticks — a tie-back to the recurring MAC Study numbers wherever that helps. Keep this page open in a second tab whenever a week’s “Concept development” or “Worked examples” section uses a symbol without re-deriving it from scratch.
Two families of symbols recur across the whole term: parameters (unknown, about the population) and statistics (computed from a sample, used to learn about the parameters). This course is consistent about marking the difference with a hat: a hatted symbol (\(\hat\theta\), \(\hat p\)) is something you compute; an unhatted Greek letter (\(\theta\), \(\mu\), \(\pi\)) is something you are trying to learn about and never actually see.
Core symbols
| Symbol | Meaning | Convention chosen |
|---|---|---|
| \(\theta\) | a generic population parameter | unknown; the target of every estimation problem this term |
| \(\hat\theta\) | an estimator (before data) / an estimate (after data) | a random variable before data arrive; a plain number once computed |
| \(\mu, \sigma\) | population mean, population standard deviation | never assumed known outside a flagged teaching simplification |
| \(\pi\) | population proportion | context disambiguates from the constant \(\pi \approx 3.14159\); the two never appear needing disambiguation on the same page |
| \(\bar X, \bar x\) | sample mean — the statistic (random variable), then its computed value (a number) | capital for the random variable, lowercase for the realized number |
| \(s\) | sample standard deviation | \(s = \sqrt{\frac{1}{n-1}\sum(x_i - \bar x)^2}\) — the \(n-1\) divisor is derived, not asserted (Week 4) |
| \(\hat p\) | sample proportion | \(\hat p = k/n\) for \(k\) successes in \(n\) trials |
| \(n\) | sample size | — |
| \(\operatorname{SE}(\hat\theta)\) | standard error of an estimator | the standard deviation of \(\hat\theta\)’s own sampling distribution |
| \(\operatorname{Bias}(\hat\theta)\) | \(E[\hat\theta] - \theta\) | requires a stipulated true \(\theta\) — a teaching device only, flagged every time it is used |
| \(\operatorname{Var}(\hat\theta)\) | variance of the estimator’s sampling distribution | how much \(\hat\theta\) would bounce around across repeated samples |
| \(\operatorname{MSE}(\hat\theta)\) | \(\operatorname{Var}(\hat\theta) + \operatorname{Bias}(\hat\theta)^2\) | the bias-variance decomposition, derived in Week 4 |
| \(L(\theta)\) | the likelihood function | a function of \(\theta\) for fixed, already-observed data — never a probability distribution over \(\theta\) |
| \(\ell(\theta)\) | the log-likelihood | \(\ell(\theta) = \ln L(\theta)\) — same maximizer as \(L(\theta)\), easier arithmetic |
| \(\hat\theta_{\text{MLE}}\) | the maximum likelihood estimate | \(\arg\max_\theta L(\theta)\) |
| CI | confidence interval | a property of the repeated-sampling procedure — never “the probability \(\theta\) is in this interval” |
| \(H_0, H_a\) | null hypothesis, alternative hypothesis | fixed before the data are examined |
| \(\alpha\) | significance level | chosen in advance, before seeing the data |
| \(p\text{-value}\) | tail probability of the data under \(H_0\) | \(P(\text{data this extreme or more} \mid H_0 \text{ true})\) — never \(P(H_0 \text{ true})\) |
| \(\beta\) | Type II error probability | depends on which specific alternative is true — never known in practice, only stipulated for teaching |
| Power | \(1 - \beta\) | the procedure’s chance of correctly rejecting \(H_0\) against a specific stated alternative |
| \(\pi(\theta)\) | Bayesian prior density on \(\theta\) | context disambiguates from the proportion \(\pi\) above — the two are never both load-bearing on the same page |
| \(L(\theta \mid \text{data})\) | Bayesian likelihood | the same likelihood-function idea as \(L(\theta)\), written with the conditioning made explicit for the Bayesian update |
| \(\pi(\theta \mid \text{data})\) | Bayesian posterior density | prior updated by the likelihood; a genuine probability distribution over \(\theta\) |
| \(\text{Beta}(a,b)\) | the Beta distribution | mean \(a/(a+b)\); the conjugate prior family for a Binomial proportion |
| \(\Phi(\cdot)\) | the standard normal cumulative distribution function | used throughout to convert a \(z\)-score into a tail probability |
| \(z, t\) | standardized test statistics | \(z\) when \(\sigma\) is treated as known; \(t\) when \(\sigma\) is estimated by \(s\) — flagged explicitly whenever the distinction matters |
The pairs that get confused
Some of these symbols look, or sound, close enough to something else that mixing them up is the single most common inference mistake at this level. Each pair below gets its own subsection, with a MAC Study number attached so the distinction is not just verbal.
p-value vs. \(P(H_0 \text{ true})\)
The p-value is \(P(\text{data this extreme or more extreme} \mid H_0 \text{ true})\) — a statement about how surprising the data would be if \(H_0\) were true. It is not, and cannot be turned into, \(P(H_0 \text{ true})\), a statement about how likely the hypothesis itself is. The p-value never conditions on the hypothesis being questioned; it conditions on the hypothesis being assumed.
MAC Study tie-back (Week 8): testing \(H_0: \mu = 45\) against \(H_a: \mu \neq 45\), using the visit-duration sample (\(n = 36\), \(\bar x = 49.8\), known \(\sigma = 15\), so \(\operatorname{SE}(\bar X) = 2.5\)), gives \(z = (49.8 - 45)/2.5 = 1.92\) and a two-sided \(p \approx 2(1 - \Phi(1.92)) \approx 0.0548\). That number answers “if the true average visit duration really were 45 minutes, how often would a sample of 36 visits produce a mean at least this far from 45?” It does not answer “how likely is it that the true average visit duration is 45 minutes?” — that second question would require a prior over \(\mu\), which classical hypothesis testing never supplies. (Week 12’s Bayesian posterior is the tool that answers a question in that shape, and even then it is a posterior for the parameter, not a probability attached to \(H_0\) as classical testing frames it.)
A confidence interval vs. “95% probability \(\theta\) is in this interval”
A CI is a property of the procedure: build the interval \(\bar x \pm 1.96\,\operatorname{SE}(\bar X)\) the same way across many hypothetical samples, and 95% of the resulting intervals will contain the true \(\mu\). Once a single sample has been drawn and a single interval computed, \(\mu\) is a fixed (if unknown) number and the interval’s endpoints are fixed numbers — there is no more randomness left to attach a 95% probability to this particular interval. The 95% describes the method’s long-run track record, not this one outcome.
MAC Study tie-back (Week 7): \(49.8 \pm 1.96(2.5) = 49.8 \pm 4.9 \rightarrow (44.9, 54.7)\). The correct reading is “this interval was produced by a procedure that captures the true mean visit duration in 95% of hypothetical repeated samples” — not “there is a 95% chance the true mean visit duration is between 44.9 and 54.7.” The same distinction holds for the companion proportion CI, \(0.38 \pm 1.96(0.0485) \approx (0.285, 0.475)\).
\(L(\theta)\) vs. a probability distribution over \(\theta\)
\(L(\theta)\) is a function of \(\theta\) built from data that are already fixed — it says how consistent each candidate value of \(\theta\) is with the data you actually observed, but it is not required to (and typically does not) integrate to 1 over \(\theta\), so it is not itself a probability distribution over \(\theta\). That changes only when a prior \(\pi(\theta)\) is introduced and combined with the likelihood to produce a genuine posterior distribution \(\pi(\theta \mid \text{data})\) — a Week 12 idea, not a Week 5–6 one.
MAC Study tie-back (Week 5): for the small pilot survey (\(n = 5\), \(k = 2\)), the likelihood kernel \(L(\pi) \propto \pi^2(1-\pi)^3\) evaluates to \(0.02048\) at \(\pi = 0.2\), \(0.03456\) at \(\pi = 0.4\), and \(0.02304\) at \(\pi = 0.6\). Those three numbers rank candidate values of \(\pi\) by how well each explains the fixed data in hand (\(\pi = 0.4\) wins, foreshadowing the Week 6 MLE \(\hat\pi = k/n = 0.4\)) — they are not, and were never claimed to be, probabilities of \(\pi\) itself, and they do not sum or integrate to 1.
Bias, Variance, and MSE vs. each other
\(\operatorname{Bias}(\hat\theta) = E[\hat\theta] - \theta\) measures systematic offset — does the estimator aim at the right target on average. \(\operatorname{Var}(\hat\theta)\) measures spread — how much the estimator would bounce around across repeated samples, regardless of where it’s centered. \(\operatorname{MSE}(\hat\theta) = \operatorname{Var}(\hat\theta) + \operatorname{Bias}(\hat\theta)^2\) combines both into a single number, which is why an estimator can have zero bias and still lose to a biased competitor on MSE if the biased one has much lower variance.
MAC Study tie-back (Week 4): the \((n-1)\)-divisor derivation shows the naive \(n\)-divisor variance estimator has bias \(-6.25\) (from \(E\!\left[\tfrac{1}{n}\sum(x_i-\bar x)^2\right] = \tfrac{35}{36}(225) = 218.75\) against the hypothetical true \(\sigma^2 = 225\)). Separately, the shrinkage estimator \(\hat\theta = 0.9\bar x\), compared against the hypothetical true \(\mu = 48\), has \(\operatorname{Bias} = -4.8\) and \(\operatorname{Var}(\hat\theta) = 0.81(6.25) = 5.0625\), giving \(\operatorname{MSE} = 5.0625 + 23.04 = 28.1025\) — much worse than the unbiased \(\bar x\)’s \(\operatorname{MSE} = \operatorname{Var} = 6.25\) in this particular setup. Zero bias does not automatically mean “better”: here it means much better, but the comparison has to be made on MSE, not on bias alone.
Type I vs. Type II error, and power
A Type I error is rejecting \(H_0\) when \(H_0\) is actually true; its rate is controlled directly by \(\alpha\), chosen before the data are seen. A Type II error is failing to reject \(H_0\) when a specific \(H_a\) is actually true; its rate is \(\beta\), and \(\beta\) can only be computed against one specific alternative value at a time — there is no single “the” Type II error rate the way there is a single \(\alpha\). Power \(= 1 - \beta\) is the complementary “correctly reject” rate against that same specific alternative.
MAC Study tie-back (Week 9): against the specific hypothetical alternative \(\mu = 50\), with a one-sided \(\alpha = 0.05\) test, the critical value is \(\bar x = 45 + 1.645(2.5) = 49.11\), giving \(\text{Power} = P(\bar X > 49.11 \mid \mu = 50) = P(Z > -0.355) = \Phi(0.355) \approx 0.639\), so \(\beta \approx 0.361\). Change the stipulated alternative to \(\mu = 55\) instead of \(\mu = 50\) and both numbers would change — power is never a property of the test alone, only of the test against a stated alternative.
A frequentist CI vs. a Bayesian credible interval
Both are intervals meant to summarize uncertainty about a parameter, and both can look numerically similar, but they answer different questions. The frequentist CI (above) is a statement about the procedure’s long-run coverage rate, with \(\theta\) treated as a fixed unknown constant. A Bayesian credible interval is a statement about the posterior distribution \(\pi(\theta \mid \text{data})\) itself — an interval that actually contains 95% of the posterior probability, because in the Bayesian framework \(\theta\) is treated as having a genuine probability distribution (updated from the prior by the data), so “95% probability \(\theta\) is in this interval” is the correct reading of a credible interval, in a way it never is for a frequentist CI.
MAC Study tie-back (Week 12): starting from prior \(\text{Beta}(3,7)\) (mean \(0.30\)) and updating on the full survey (\(n=100\), \(k=38\)) gives posterior \(\text{Beta}(41,69)\), posterior mean \(41/110 \approx 0.373\), posterior SD \(\approx 0.0459\), and a normal-approximation 95% credible interval \(0.373 \pm 1.96(0.0459) \approx (0.283, 0.463)\). Compare this directly to the Week 7 frequentist CI for the same usage rate, \((0.285, 0.475)\): the two intervals are close in this case (a similar sample size and a not-too-informative prior), but only the credible interval supports the sentence “there is a 95% probability \(\pi\) falls in \((0.283, 0.463)\), given the prior and the data.” The frequentist interval never supports that sentence, no matter how tempting it is to read it that way.
How this connects to the rest of the course
- Weeks 1–3 introduce the parameter/statistic distinction (\(\theta\) vs. \(\hat\theta\); \(\mu,\pi\) vs. \(\bar x, \hat p\)) and the standard-error notation that every later week reuses.
- Week 4 derives \(\operatorname{Bias}\), \(\operatorname{Var}\), and \(\operatorname{MSE}\) from scratch, including the \((n-1)\)-divisor result referenced above.
- Weeks 5–6 develop \(L(\theta)\), \(\ell(\theta)\), and \(\hat\theta_{\text{MLE}}\), and the likelihood-vs-probability distinction that matters until Week 12 reframes it.
- Week 7 develops the CI notation and the procedure-vs-probability distinction in full.
- Week 8 develops the p-value notation and the p-value-vs-\(P(H_0\text{ true})\) distinction in full.
- Week 9 develops \(\alpha\), \(\beta\), and Power against a stated alternative.
- Weeks 10–11 reuse the CI and testing notation under resampling (bootstrap SE, permutation p-values), flagging where an estimated SE is close-but-not-identical to a known-\(\sigma\) SE.
- Week 12 introduces \(\pi(\theta)\), \(L(\theta\mid\text{data})\), and \(\pi(\theta\mid\text{data})\), and is where the CI-vs-credible-interval distinction above is developed in full.
- Week 13 puts the frequentist, likelihood, bootstrap/randomization, and Bayesian notations side by side on the same MAC Study usage-rate question.
Public vs. graded
These notes, the examples, and the practice here are public and ungraded — study material only. No graded prompts, answer keys, rubrics, point values, or due dates appear on this site. Graded inference checkpoints, quizzes, homework, labs, the midterm, the project, and the final live in Blackboard (the LMS), which is authoritative for due dates, submissions, and grades. If this page and Blackboard ever disagree, follow Blackboard.