Notation glossary

Symbols and conventions used across the course

This glossary collects the symbols and modeling conventions used everywhere in the course. The same letters mean the same things every week, so once you learn a symbol here you can read any model on the site without relearning it. Keep this page open beside the weekly notes. When a week introduces a number — a slope, an $R^2$, a standard error — come back here to remind yourself what the symbol stands for and how to say it in a plain sentence.

A guiding distinction runs through the whole list: we separate the parameters (the unknown true values in the population, written with Greek letters) from the estimates (the values we compute from a sample, written with Latin letters or a hat). A model is a structured claim about the parameters; the data give us estimates of them, together with a statement of how uncertain those estimates are.

Core symbols

Symbol	Read it as	Plain-language meaning
$y$	“why”	The response (outcome) — the variable the model tries to explain or predict. In the recurring dataset this is `final`, the final exam score.
$x$, $x_1, x_2, \dots$	“ex,” “ex-one,” “ex-two”	The predictors (explanatory variables, inputs). A single $x$ for one predictor; $x_1, x_2, \dots$ once there are several.
$n$	“en”	The number of observations (rows, units). The recurring dataset has $n = 200$ students.
$\beta_0$	“beta-naught”	The intercept parameter — the model’s true (unknown) value of $y$ when every predictor is $0$. A population quantity.
$\beta_1$	“beta-one”	A slope parameter — the true (unknown) change in $y$ per one-unit change in a predictor, holding the others fixed.
$b_0, b_1$ (also $\hat\beta_0, \hat\beta_1$)	“bee-naught, bee-one”	The fitted estimates of $\beta_0, \beta_1$ computed from the sample. The hat-beta and the $b$ notation mean the same thing.
$\hat{y}$	“why-hat”	The fitted / predicted value the model gives for an observation: $\hat{y} = b_0 + b_1 x$ (with more terms when there are more predictors).
$e = y - \hat{y}$	“ee,” the residual	The residual — what the model missed for one observation: actual minus fitted. Positive means the model under-predicted; negative means it over-predicted.
$r$	“ar”	The correlation between two numeric variables, between $-1$ and $1$. Measures strength and direction of a linear association only.
$R^2$	“ar-squared”	The coefficient of determination — the proportion of variation in $y$ the model explains, between $0$ and $1$. For simple regression $R^2 = r^2$.
$s$	“ess”	The residual standard error — the typical size of a residual, in the units of $y$. A smaller $s$ means tighter predictions.
$\mathrm{SE}(b_1)$	“ess-ee of bee-one”	The standard error of a coefficient — how much the estimate $b_1$ would bounce around from sample to sample. The yardstick for its uncertainty.
$t$	“tee”	The test statistic $t = b_1 / \mathrm{SE}(b_1)$ — how many standard errors the estimate sits from $0$. Large $\|t\|$ is evidence the predictor matters.
CI	“see-eye”	A confidence interval — a range of plausible values for a parameter, e.g. a 95% CI for a slope. Width reflects uncertainty.
$\operatorname{logit}(p) = \log\!\big(\tfrac{p}{1-p}\big)$	“logit of pee”	The log-odds of a probability $p$. The scale on which logistic regression is linear; it maps a probability in $(0,1)$ to the whole number line.
OR $= e^{\beta_1}$	“odds ratio”	The odds ratio — exponentiating a logistic slope turns a log-odds change into a multiplicative factor on the odds, per one-unit change in the predictor.

We write math in $...$ / $$...$$ so it renders as MathML in the browser — never as an image. The fitted line for one predictor, for example, is

\[\hat{y} = b_0 + b_1 x,\]

and a residual for a single observation is $e = y - \hat{y}$.

Modeling conventions

The symbols above are only half the language. The other half is a set of recurring moves — habits of interpretation that apply no matter which model is on the page.

A slope is a holding-others-fixed comparison. Read $b_1$ as: “comparing observations that differ by one unit in this predictor but are equal on every other predictor in the model, the predicted $y$ differs by $b_1$, on average.” The phrase “holding constant” or “adjusting for” is doing real work — the slope is a comparison within the model you fit, not a universal truth about the variable.

Crude vs. adjusted. A crude slope comes from a model with that predictor alone; an adjusted slope comes from a model that also includes other predictors. When the adjusted slope differs from the crude one, the other predictors were confounding the relationship. Always say which kind you are reporting — a slope means different things depending on what is held fixed alongside it.

Indicator coding and a baseline. A categorical predictor enters a model through indicator (dummy) variables: one category is chosen as the baseline, and each remaining category gets a coefficient read as its difference from that baseline. So with three formats and in_person as baseline, the hybrid coefficient is “hybrid mean minus in_person mean,” and the baseline mean lives in the intercept. The baseline is a reference point, not a judgment about which category is “normal.”

An interaction means within-group slopes. When two predictors interact, the slope of one depends on the value of the other — the relationship is modified. We read an interaction by computing the slope separately within each group: a main slope for the baseline group, and that slope plus the interaction coefficient for the other group. Effect modification is a feature to report, not a nuisance to remove.

Logistic regression reports an odds ratio plus a predicted probability. Because logistic coefficients live on the log-odds scale, we translate them for human readers in two complementary ways. Exponentiate a coefficient to get the odds ratio, $\mathrm{OR} = e^{\beta_1}$, a multiplicative effect on the odds. And convert the fitted log-odds back to a predicted probability via

\[\hat{p} = \frac{1}{1 + e^{-(b_0 + b_1 x)}},\]

so you can say “at this predictor value, the model predicts about this chance.” Always give at least one predicted probability at a meaningful $x$, because an odds ratio alone hides the actual size of the chance.

How to use this page

When a weekly note shows model output, name each piece against this glossary before interpreting it: identify the response $y$ and the predictors $x$; separate the estimates $b_0, b_1$ from the parameters they estimate; read $R^2$ and $s$ as fit, and $\mathrm{SE}(b_1)$, $t$, and the CI as uncertainty. Then restate the slope as a holding-others-fixed comparison and decide whether it is crude or adjusted. For a logistic model, report both the odds ratio and a predicted probability. Doing this every time turns a table of numbers into a careful, defensible claim — which is the whole point of the course.

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 modeling checkpoints, labs, quizzes, homework/modeling memos, 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.

Symbol	Read it as	Plain-language meaning
\(y\)	“why”	The response (outcome) — the variable the model tries to explain or predict. In the recurring dataset this is `final`, the final exam score.
\(x\), \(x_1, x_2, \dots\)	“ex,” “ex-one,” “ex-two”	The predictors (explanatory variables, inputs). A single \(x\) for one predictor; \(x_1, x_2, \dots\) once there are several.
\(n\)	“en”	The number of observations (rows, units). The recurring dataset has \(n = 200\) students.
\(\beta_0\)	“beta-naught”	The intercept parameter — the model’s true (unknown) value of \(y\) when every predictor is \(0\). A population quantity.
\(\beta_1\)	“beta-one”	A slope parameter — the true (unknown) change in \(y\) per one-unit change in a predictor, holding the others fixed.
\(b_0, b_1\) (also \(\hat\beta_0, \hat\beta_1\))	“bee-naught, bee-one”	The fitted estimates of \(\beta_0, \beta_1\) computed from the sample. The hat-beta and the \(b\) notation mean the same thing.
\(\hat{y}\)	“why-hat”	The fitted / predicted value the model gives for an observation: \(\hat{y} = b_0 + b_1 x\) (with more terms when there are more predictors).
\(e = y - \hat{y}\)	“ee,” the residual	The residual — what the model missed for one observation: actual minus fitted. Positive means the model under-predicted; negative means it over-predicted.
\(r\)	“ar”	The correlation between two numeric variables, between \(-1\) and \(1\). Measures strength and direction of a linear association only.
\(R^2\)	“ar-squared”	The coefficient of determination — the proportion of variation in \(y\) the model explains, between \(0\) and \(1\). For simple regression \(R^2 = r^2\).
\(s\)	“ess”	The residual standard error — the typical size of a residual, in the units of \(y\). A smaller \(s\) means tighter predictions.
\(\mathrm{SE}(b_1)\)	“ess-ee of bee-one”	The standard error of a coefficient — how much the estimate \(b_1\) would bounce around from sample to sample. The yardstick for its uncertainty.
\(t\)	“tee”	The test statistic \(t = b_1 / \mathrm{SE}(b_1)\) — how many standard errors the estimate sits from \(0\). Large \(\|t\|\) is evidence the predictor matters.
CI	“see-eye”	A confidence interval — a range of plausible values for a parameter, e.g. a 95% CI for a slope. Width reflects uncertainty.
\(\operatorname{logit}(p) = \log\!\big(\tfrac{p}{1-p}\big)\)	“logit of pee”	The log-odds of a probability \(p\). The scale on which logistic regression is linear; it maps a probability in \((0,1)\) to the whole number line.
OR \(= e^{\beta_1}\)	“odds ratio”	The odds ratio — exponentiating a logistic slope turns a log-odds change into a multiplicative factor on the odds, per one-unit change in the predictor.