Notation and symbols

The symbols used throughout Statistics Quest: Greek letters for population parameters, Latin letters for sample statistics, hats for estimates, bars for averages and Σ for sums.

\sum_{i=1}^{n} x_i = x_1 + x_2 + \cdots + x_n

Summation notation.

$\sum$: Capital sigma: add up the terms that follow.
$i$: The index; it runs from the lower limit (1) to the upper limit ( $n$ ).
$x_i$: The $i$ -th observation.
$n$: The sample size (number of observations).

When to use

Use this page to decode a formula. The same quantity is written with a different symbol depending on whether it describes the whole population (a fixed, usually unknown parameter) or a sample (a statistic you can compute and which varies from sample to sample).

Quantity	Population parameter	Sample statistic
Size	$N$	$n$
Mean	$\mu$ (mu)	$\xbar$ (x-bar)
Standard deviation	$\sigma$ (sigma)	$s$
Variance	$\sigma^2$	$s^2$
Proportion	$p$ (some books use $\pi$ )	$\phat$ (p-hat)
Correlation	$\rho$ (rho)	$r$
Regression slope	$\beta_1$	$b_1$ or $\hat\beta_1$

Other conventions:

A hat marks an estimate of a parameter or a predicted value: $\hat\sigma$ , $\hat y$ .
A bar marks an average: $\xbar$ , $\ybar$ , $\bar d$ (mean difference), $\bar R$ (mean range).
$\E(X)$ , $\Var(X)$ , $\SD(X)$ : expected value, variance and standard deviation of a random variable; $\SE$ : standard error (the SD of a statistic across samples).
$P(A \given B)$ : probability of $A$ given $B$ . $\Cov$ , $\Corr$ : covariance and correlation.
$H_0$ , $H_1$ : null and alternative hypotheses; $\alpha$ : significance level; $\beta$ : Type II error rate (in testing) or a regression coefficient (in models) — context decides.
$z_{q}$ , $t_{q,\,df}$ , $\chi^2_{q,\,df}$ , $F_{q;\,d_1,d_2}$ : the $q$ -quantile of the named distribution; e.g. $z_{0.975} \approx 1.96$ .
Subscripts $(i)$ in parentheses mean sorted order: $x_{(1)}$ is the minimum.

How to interpret

Ask "parameter or statistic?" first. A sentence such as " $\mu = 30$ " is a claim about the population; " $\xbar = 31.2$ " is a fact about your sample.

Common mistakes

Writing hypotheses about statistics ( $H_0: \xbar = 30$ ) instead of parameters ( $H_0: \mu = 30$ ).
Mixing up $s$ (sample SD, divisor $n - 1$ ) and $\sigma$ (population SD).
Confusing the standard deviation of the data with the standard error of the mean.

Example

Data 3, 5, 10: $n = 3$ , $\sum x_i = 18$ , $\xbar = 18/3 = 6$ . The sample mean $\xbar = 6$ is a statistic estimating the unknown population mean $\mu$ .