Summary of statistical notation

Mathematical notation in statistics can often be a source of confusion. Here is a brief summary of some commonly used conventions:

• Upper case Roman letters are used to denote random variables (e.g. $X$, $Y$, $Z$, etc.) whereas lower case Roman letters are used to denote their specific values (e.g. $x$, $y$, and $z$). For example, the probability of a (generic) random variable $X$ exceeding a (specific) value $x$ is given by $\Pr({X>x})$. Therefore, sample data such as measurements are denoted by lower case Roman letters (e.g. a data sample with values $\{x_1,x_2,\ldots,x_n\}$).

• Unobservable quantities such as model parameters and noise are denoted using lower case Greek letters. For example, the linear regression model that explains random variable $Y$ in terms of $X$ is given by $Y=X\beta+\alpha+\epsilon$.

• The hat symbol is used to denote estimated and predicted values. For example, $\hat{\beta}$ is an estimate of the model parameter $\beta$, and $\hat{Y}$ is a prediction of the random variable $Y$.

• Bold face upper case Roman letters are used to denote data matrices containing multiple variables. For example, the $(n\times p)$ data matrix ${\bf X}$ contains elements $x_{ij}$ where row index $i=1,2,\ldots,n$ refers to the data items/objects, and column index $j=1,2,\ldots,p$ refers to the variables.

• The symbol $n$ is often used to denote the sample size (the number of data objects in the sample).

• The population mean $\mu_X$ is written in terms of the expectation operator as $\mu_X=E(X)$ whereas the sample mean is denoted using an overline (e.g. $\overline{x}$). Sometimes climate studies using the quantum mechanics bra-ket notation $<.>$ to denote expectation but this non-standard practice should be avoided.