Appendix D — Notation
This appendix collects every symbol this book uses, so you have one place to look when a formula in a later chapter reuses a letter from an earlier one. It describes this book’s choices and not a universal standard, because there is no universal standard: statistics notation varies by field, by textbook, and sometimes by chapter within one textbook. Expect other sources to write the sample mean as \(\bar{x}\) where this book writes \(\hat{M}\), and to use \(P(\cdot)\) where this book uses \(Prob(\cdot)\). The organizing idea here is a single distinction, carried consistently: a hat marks a quantity computed from data, and a bare letter marks the theoretical quantity it corresponds to. Once you have that, most formulas can be read before you know the details, because the decoration tells you whether you are looking at something you can compute or something you can only estimate. The tables below are grouped by topic in roughly the order the book introduces them.
D.1 Hats and No Hats
The book distinguishes the sample (the data you observed) from the population (the theoretical quantity you want to know) using hats.
| Concept | Sample (data) | Population (theory) |
|---|---|---|
| Observation | \(\hat{X}_i\) | \(X_i\) |
| Mean | \(\hat{M}\) | \(\mu\) or \(\mathbb{E}[X_i]\) |
| Variance | \(\hat{V}\) | \(\sigma^2\) or \(\mathbb{V}[X_i]\) |
| Std deviation | \(\hat{S}\) | \(\sigma\) |
| Probability | \(\hat{p}_x\) | \(p_x\) |
| Regression coeff | \(\hat{b}_k\) | \(\beta_k\) |
| Predicted value | \(\hat{Y}_i\) or \(\hat{y}_i\) | – |
| Residual | \(\hat{e}_i\) | \(\epsilon_i\) |
Use \(\hat{\cdot}\) for anything computed from data, and a bare Greek or Roman letter for the population quantity. The pairing is what makes the central question of inference expressible in one line: you have \(\hat{M}\) and you want \(\mu\).
D.2 Data and Indices
| Symbol | Meaning |
|---|---|
| \(\hat{X}_i\) | Value of the \(i\)th observation |
| \(n\) | Sample size |
| \(i\) | Observation index (\(i = 1, \ldots, n\)) |
| \(k\) | Variable index |
| \(j\) | Secondary observation index (pairwise sums) |
| \(g\) | Group index |
| \(K\) | Number of explanatory variables |
| \(G\) | Number of groups |
The index \(k\) always counts variables and \(i\) always counts observations, so a doubly-subscripted \(\hat{X}_{ik}\) can be read without checking the surrounding text.
D.3 Descriptive Statistics
| Symbol | Meaning |
|---|---|
| \(\hat{M}\) | Sample mean |
| \(\hat{V}\) | Sample variance |
| \(\hat{S}\) | Sample standard deviation |
| \(\tilde{M}\) | Sample median |
| \(\hat{\text{MAD}}\) | Median absolute deviation |
| \(\hat{IQR}\) | Interquartile range |
| \(W_x\) | Unnormalized weight |
| \(w_x\) | Normalized weight (\(\sum w_x = 1\)) |
D.4 Population Parameters
| Symbol | Meaning |
|---|---|
| \(\mu\) | Population mean |
| \(\sigma^2\) | Population variance |
| \(\sigma\) | Population standard deviation |
| \(p\) | Bernoulli or Binomial success probability |
| \(\lambda\) | Poisson or Exponential rate parameter |
| \(\theta\) | Generic parameter |
| \(\alpha\) | Significance level (Type I error) |
| \(\rho\) | Population correlation |
D.5 Operators and Probability
Expectation operators are written in blackboard bold with square brackets.
| Operator | Meaning |
|---|---|
| \(\mathbb{E}[X_i]\) | Expected value |
| \(\mathbb{V}[X_i]\) | Variance |
| \(\mathbb{C}[X_i, Y_i]\) | Covariance |
| \(\mathbb{R}[X_i, Y_i]\) | Correlation |
| \(\mathbb{E}[X_i \mid Y_i]\) | Conditional expectation |
The square brackets are the book’s standard, in place of parentheses such as \(\mathbb{V}(\cdot)\). Reserving brackets for operators means a parenthesis is always function application, which keeps expressions like \(\mathbb{V}[m(x)]\) unambiguous.
| Symbol | Meaning |
|---|---|
| \(Prob(X_i = x)\) | Probability mass or density |
| \(Prob(X_i \leq x)\) | Cumulative probability |
| \(Prob(X_i = x \mid Y_i = y)\) | Conditional probability |
| \(F(x)\) | CDF (population) |
| \(\hat{F}(x)\) | Empirical CDF (sample) |
| \(f(x)\) | PDF (continuous) |
| \(\mathbf{1}(\cdot)\) | Indicator function |
The probability function is written \(Prob(\cdot)\) rather than \(Pr(\cdot)\) or \(P(\cdot)\), which leaves the single letters \(p\) and \(P\) free to denote probabilities and proportions.
D.6 Distributions
| Distribution | Notation | Parameters |
|---|---|---|
| Normal | \(N(\mu, \sigma^2)\) | mean, variance |
| Bernoulli | \(X_i \in \{0, 1\}\) | \(p\) |
| Binomial | \(\binom{n}{k} p^k (1-p)^{n-k}\) | \(n\), \(p\) |
| Poisson | \(\text{Poisson}(\lambda)\) | \(\lambda\) |
| Uniform | \(\text{Uniform}(a, b)\) | \(a\), \(b\) |
| Student t | \(t_{df}\) | degrees of freedom |
| F | \(F_{df_1, df_2}\) | two df parameters |
| Exponential | \(\text{Exp}(\lambda)\) | rate \(\lambda\) |
The Normal is parameterized by its variance, not its standard deviation, so \(N(0, 4)\) has standard deviation \(2\). R’s rnorm() takes the standard deviation instead, which is a common source of off-by-a-square-root errors.
D.7 Regression
Simple regression:
\[\hat{Y}_i = b_0 + b_1 \hat{X}_i + e_i\]
Multiple regression:
\[\hat{Y}_i = \sum_{k=0}^{K} b_k \hat{X}_{ik} + e_i\]
| Symbol | Meaning |
|---|---|
| \(b_0\) | Intercept (sample) |
| \(b_k\) | Slope coefficient (sample) |
| \(\hat{b}_k\) | OLS estimate of \(b_k\) |
| \(b_k^*\) | Optimal or best-fit coefficient |
| \(\beta_k\) | Population coefficient |
| \(\epsilon_i\) | Population error term |
| \(e_i\) | Sample residual |
| \(\hat{C}_{XY}\) | Sample covariance |
| \(\hat{V}_X\) | Sample variance of \(X\) |
| \(\hat{R}^2\) | Coefficient of determination |
| \(\hat{R}^2_{adj}\) | Adjusted R-squared |
Sums of Squares.
| Symbol | Meaning |
|---|---|
| \(\hat{TSS}\) | Total sum of squares |
| \(\hat{ESS}\) | Explained sum of squares |
| \(\hat{RSS}\) | Residual sum of squares |
| \(\hat{BSS}\) | Between-group sum of squares |
| \(\hat{WSS}\) | Within-group sum of squares |
Matrix Notation.
| Symbol | Meaning |
|---|---|
| \(\mathbf{X}\) | Design matrix |
| \(\mathbf{Y}\) | Outcome vector |
| \(\mathbf{X}'\) | Transpose |
| \((\mathbf{X}'\mathbf{X})^{-1}\) | Inverse cross-product |
Transposes are written with a prime, \(\mathbf{X}'\), rather than a superscript \(T\), which would collide with the letter used for totals and for time. Bold always means a vector or a matrix, so \(\mathbf{X}\) and \(X\) are different objects.
D.8 Local Regression and Nonparametrics
| Symbol | Meaning |
|---|---|
| \(m(x)\) | True conditional mean function |
| \(\hat{m}(x)\) | Estimated conditional mean |
| \(h\) or \(h_k\) | Bandwidth (half-width of a window around \(x\)) |
| \(k(\hat{X}_i, x, h)\) | Kernel weight function |
| \(L\) | Number of bins |
span |
LOESS bandwidth, as a share of the sample |
| \(\hat{b}_0(x, h)\) | Local intercept |
| \(\hat{\beta}_k(\mathbf{x})\) | Local marginal effect |
| \(\mathbf{H}\) | Bandwidth matrix |
| \(\mathbf{K}(\mathbf{x})\) | Kernel weight matrix |
The letter \(h\) is a half-width everywhere it appears, measured from a point \(x\) out to the edge of the window. So a histogram bin is \(2h\) wide, a kernel gives zero weight once \(|\hat{X}_{i}-x| > h\), and a local regression at \(x\) fits on the observations within \(h\) of it. These are all the same idea, which is how much data around a point gets used. LOESS is the one exception: its span fixes the share of observations in each window instead of the width, so the window widens where data are sparse rather than staying at a fixed \(h\).
The letter \(k\) does double duty here, distinguished by whether it takes arguments. A bare subscript \(k\) is the variable index, so \(h_{k}\) is the bandwidth for variable \(X_{k}\) and \(\hat{\beta}_{k}(\mathbf{x})\) is the marginal effect of that variable at the point \(\mathbf{x}\). Written with arguments, \(k(\cdot)\) is the kernel weight function. Capital \(K\) is the number of explanatory variables, and bold \(\mathbf{K}(\mathbf{x})\) is the matrix of kernel weights.
D.9 Hypothesis Testing
| Symbol | Meaning |
|---|---|
| \(H_0\) | Null hypothesis |
| \(H_A\) | Alternative hypothesis |
| \(\hat{t}\) | t-statistic |
| \(\hat{F}\) | F-statistic |
| \(\hat{SE}\) | Standard error |
| \(s_{\hat{b}_k}\) | SE of a regression coefficient |
| \(q(\alpha/2)\) | Critical quantile |
| \(z\) | Standard normal score |
D.10 Sampling and Resampling
| Symbol | Meaning |
|---|---|
| \(M\) | Estimator (random variable, before seeing data) |
| \(\hat{M}\) | Estimate (realized, after seeing data) |
| \(SE(M)\) | Standard error of an estimator |
| \(\sigma / \sqrt{n}\) | SE formula for the mean |
| \(\hat{M}_b^{\text{boot}}\) | Bootstrap replicate mean |
| \(\hat{SE}^{\text{boot}}\) | Bootstrap standard error |
| \(\hat{SE}^{\text{jack}}\) | Jackknife standard error |
| \(\hat{M}_i^{\text{jack}}\) | Leave-one-out mean |
The distinction between \(M\) and \(\hat{M}\) is the one students most often lose. \(M\) is a recipe whose value varies across samples, so it has a distribution and a standard error, while \(\hat{M}\) is the one number your data produced.
D.11 ANOVA
| Symbol | Meaning |
|---|---|
| \(\hat{Y}_{ig}\) | Observation \(i\) in group \(g\) |
| \(\hat{M}_g\) | Group \(g\) mean |
| \(\hat{M}_Y\) | Grand mean |
| \(n_g\) | Group size |
| \(\hat{D}\) | Difference in means |
| \(\hat{F}\) | F-statistic |
| \(F_{G-1, n-G}\) | F-distribution reference |
D.12 Sub- and Superscripts
| Pattern | Meaning | Example |
|---|---|---|
| \(_i\) | Observation index | \(\hat{X}_i\) |
| \(_k\) | Variable index | \(b_k\), \(\hat{X}_{ik}\) |
| \(_j\) | Secondary observation index (pairwise sums only) | \(\hat{X}_j\) in \(\sum_{j \neq i}\) |
| \(_g\) | Group index | \(\hat{M}_g\) |
| \(_{x}\) or \(_{xy}\) | Outcome value | \(\hat{p}_x\), \(\hat{p}_{xy}\) |
| \(^*\) | Optimal value | \(b_k^*\) |
| \(^{(b)}\) | Bootstrap replicate | \(\hat{X}_i^{(b)}\) |
| \(^{\text{boot}}\) | Bootstrap variant | \(\hat{SE}^{\text{boot}}\) |
| \(^{\text{jack}}\) | Jackknife variant | \(\hat{SE}^{\text{jack}}\) |
| \(_{[i]}\) | Leave-one-out | \(\hat{y}_{[i]}\) |
| \(_{\text{null}}\) | Under the null hypothesis | \(M_{\text{null}}\) |
| \(_{\text{obs}}\) | Observed value | \(t_{\text{obs}}\) |
D.13 Reference Card
- Hats on data. \(\hat{X}_i\), \(\hat{M}\), \(\hat{V}\), \(\hat{b}_k\) for anything computed from a sample.
- Greek for the population. \(\mu\), \(\sigma\), \(\beta_k\), \(\epsilon_i\).
- Blackboard bold operators. \(\mathbb{E}\), \(\mathbb{V}\), \(\mathbb{C}\), \(\mathbb{R}\), always with square brackets.
- Probability. \(Prob(\cdot)\), not \(P(\cdot)\) or \(Pr(\cdot)\).
- Transpose. A prime, \(\mathbf{X}'\), not \(\mathbf{X}^T\).
- Bold for vectors and matrices. \(\mathbf{X}\) is not \(X\).
- \(_k\) for variables and \(_i\) for observations. \(_j\) appears only as a second observation index in pairwise sums.
Further Reading.
- Code Style – how these symbols map onto object names in R.
- Sampling – where the \(M\) against \(\hat{M}\) distinction is introduced and used.