Notation
Notation isn’t just about using arbitrary symbols to represent quantities. Consistent use of notation can help reveal the structure and relationships present in a collection of ideas, such as statistical inference, and can help clarify the roles of the various quantities used in data analysis. To emphasise the structure in the notation we include a brief overview here of the different types of notation and where they have been used.
Greek Letters
Greek letters are frequently used in mathematics for a range of purposes. You will have seen the Greek letter ‘p’, [latex]\pi[/latex], used for the area of the unit circle.
The most important role of Greek letters for us has been to signify a population parameter. The table below shows the ones we have used in this role and others, listed alphabetically.
Greek symbols
| Letter | Role | Chapter |
|---|---|---|
| [latex]\alpha[/latex] | probability of a Type I error | Chapter 15 |
| [latex]\beta[/latex] | probability of a Type II error | Chapter 15 |
| population intercept ([latex]\beta_0[/latex]) & slope ([latex]\beta_1[/latex]) | Chapter 18 | |
| [latex]\eta[/latex] | population median | Chapter 24 |
| [latex]\theta[/latex] | arbitrary population parameter | Chapter 18 |
| [latex]\lambda[/latex] | Poisson mean | Chapter 11 |
| [latex]\mu[/latex] | population mean | Chapter 10 |
| [latex]\pi[/latex] | used in Normal distribution | Chapter 12 |
| [latex]\rho[/latex] | population correlation | Chapter 18 |
| [latex]\sigma[/latex] | population standard deviation | Chapter 10 |
| [latex]\phi[/latex] | signal-to-noise ratio | Chapter 15 |
| [latex]\chi[/latex] | [latex]\chi^2[/latex] distribution | Chapter 22 |
The whole Greek alphabet, showing the names and English equivalents of these letters, is given for reference in the below.
The Greek alphabet
| A | [latex]\alpha[/latex] | alpha | a |
| B | [latex]\beta[/latex] | beta | b |
| [latex]\Gamma[/latex] | [latex]\gamma[/latex] | gamma | g |
| [latex]\Delta[/latex] | [latex]\delta[/latex] | delta | d |
| E | [latex]\epsilon[/latex] | epsilon | e |
| Z | [latex]\zeta[/latex] | zeta | z |
| H | [latex]\eta[/latex] | eta | e |
| [latex]\Theta[/latex] | [latex]\theta[/latex] | theta | th |
| I | [latex]\iota[/latex] | iota | i |
| K | [latex]\kappa[/latex] | kappa | k |
| [latex]\Lambda[/latex] | [latex]\lambda[/latex] | lambda | l |
| M | [latex]\mu[/latex] | mu | m |
| N | [latex]\nu[/latex] | nu | n |
| [latex]\Xi[/latex] | [latex]\xi[/latex] | xi | ks |
| O | o | omicron | o |
| [latex]\Pi[/latex] | [latex]\pi[/latex] | pi | p |
| P | [latex]\rho[/latex] | rho | r |
| [latex]\Sigma[/latex] | [latex]\sigma[/latex] | sigma | s |
| T | [latex]\tau[/latex] | tau | t |
| Y | [latex]\upsilon[/latex] | upsilon | u |
| [latex]\Phi[/latex] | [latex]\phi[/latex] | phi | f |
| X | [latex]\chi[/latex] | chi | ch |
| [latex]\Psi[/latex] | [latex]\psi[/latex] | psi | ps |
| [latex]\Omega[/latex] | [latex]\omega[/latex] | omega | o |
Capital Letters
We have generally used capital letters to denote random variables, with lowercase letters used for particular outcomes of these. For example, [latex]\overline{x}[/latex] denotes a particular number, the mean from a sample, whereas [latex]\overline{X}[/latex] denotes the random process of taking a random sample and returning the mean.
We have had two main uses for random variables. Firstly we have thought of them as models for sampling from populations. The random variable [latex]X[/latex] might be the height of a randomly chosen female, for instance.
The second use has been to discuss the sampling distribution of statistics. In Chapter 5 we used the sample mean [latex]\overline{x}[/latex] to summarise the location of an observed distribution, while in Chapter 13 we used the random variable [latex]\overline{X}[/latex] to think about how [latex]\overline{x}[/latex] would change from sample to sample.
Other Symbols
The table below shows a list of some of the other symbols used in this book, together with the first section that discusses their use and meaning.
Other symbols
| Symbol | Role | Section |
| [latex]M[/latex] | sample median | Chapter 4 |
| [latex]Q_j[/latex] | [latex]j[/latex]th quartile | Chapter 4 |
| [latex]n[/latex] | sample size | Chapter 5 |
| [latex]\overline{x}[/latex] | sample mean | |
| [latex]s[/latex] | sample standard deviation | Chapter 5 |
| [latex]r[/latex] | Pearson correlation coefficient | Chapter 7 |
| [latex]p[/latex] | population proportion | Chapter 8 |
| [latex]\hat{p}[/latex] | sample proportion | |
| [latex]P(\cdot)[/latex] | probability | Chapter 8 |
| [latex]N[/latex] | population size | Chapter 10 |
| [latex]E(\cdot)[/latex] | expected value | |
| [latex]\mathrm{var}(\cdot)[/latex] | variance | Chapter 10 |
| [latex]\mathrm{sd}(\cdot)[/latex] | standard deviation | |
| [latex]e[/latex] | base of natural logarithms | Chapter 12 |
| [latex]z[/latex] | [latex]z[/latex] score | Chapter 12 |
| [latex]\mathrm{se}(\cdot)[/latex] | standard error | Chapter 14 |
| [latex]t[/latex] | [latex]t[/latex] statistic | Chapter 14 |
| df | degrees of freedom | |
| [latex]t^{*}[/latex] | critical [latex]t[/latex] statistic | Chapter 14 |
| [latex]z^{*}[/latex] | critical [latex]z[/latex] score | Chapter 17 |
| OR | odds ratio | Chapter 17 |
| [latex]F[/latex] | [latex]F[/latex] statistic | Chapter 19 |
| [latex]S[/latex] | signed-rank statistic | Chapter 24 |
| [latex]W[/latex] | Wilcoxon statistic | Chapter 24 |
| [latex]H[/latex] | Kruskal-Wallis statistic | Chapter 24 |
| [latex]r_S[/latex] | Spearman correlation coefficient | Chapter 24 |
