Chapter 2 Notation Summary

This chapter provides a summary of notation used throughout STM1001, designed to be used as a reference. For convenience, the chapter is divided into sections based on the topic for which the notation is most relevant, starting with a section containing general notation used throughout STM1001. The final section contains a consolidated table of all notation used throughout STM1001.

2.1 General notation used throughought STM1001

The table below provides a summary of general notation used throughout STM1001. Most of this notation is introduced early on in the subject, during Topics 1-4, and is then used throughout the subject.

Notation	Meaning	Comments
$n$	Sample size
$x_i$	The $i$ th $x$ value, usually from a list of $n$ $x$ values, e.g. ( $x_1, x_2, \ldots , x_n$ ), where $i$ can take any value from 1 to $n$
$\displaystyle \sum$	Summation sign
$\displaystyle \sum_{i=1}^{n}$	The sum from $i=1$ to $n$	See the previous chapter for an example
$\mu$	Population mean	This is a Greek letter pronounced 'm-yoo'
$\overline{x}$	Sample mean	The line on top of the $x$ is referred to as a 'bar', so that $\overline{x}$ is pronounced ' $x$ bar'
$\sigma^2$	Population variance	$\sigma$ is a Greek letter pronounced 'sigma'
$s^2$	Sample variance
$\sigma$	Population standard deviation	$\sigma$ is a Greek letter pronounced 'sigma'
$s$	Sample standard deviation
Q1	Quartile 1: 25% quantile
Q2	Quartile 2: 50% quantile, and also the median
Q3	Quartile 3: 75% quantile
$\rho$	Population correlation	This is a Greek letter pronounced 'rho'
$r$	Sample correlation
$\|x\|$	The 'absolute value' of some number $x$	See the previous chapter for an example
$X$	A random variable (discrete or continuous)
$\text{E}(X)$	Expected value (or mean) of $X$
$\text{Var}(X)$	Variance of $X$
$\text{SD}(X)$	Standard deviation of $X$
$\overline{X}$	Sample mean (random)
$\pm$	Plus or minus. For example, $x \pm a = (x - a, x + a)$

2.2 Topics 3 and 4: Probability, Distributions and Sampling Distributions

The following table provides a summary of notation used in Topics 3 and 4.

The lecture slides for Topic 3 can be found here.

The readings for Topic 3 can be found here.

The lecture slides for Topic 4 can be found here.

The readings for Topic 4 can be found here.

Notation	Meaning	Comments
$\Omega$	Sample space	This is a Greek letter pronounced 'omega'
$\emptyset$	Null event (or null set)	Also sometimes referred to as 'empty set'
$P(A)$	The probability of event $A$
$A^C$	The complement of $A$ , or "not $A$ "	Sometimes denoted as $A^\prime$
$X$	A random variable (discrete or continuous)
$P(X = x)$	The probability that the random variable $X$ takes the value $x$ . For example, let $X$ denote the number of times you check this document. Then $P(X=2)$ denotes the probability that you check this document exactly 2 times.
$\text{E}(X)$	Expected value (or mean) of $X$
$\text{Var}(X)$	Variance of $X$
$\text{SD}(X)$	Standard deviation of $X$
$\overline{X}$	Sample mean (random)
$X \sim N(\mu, \sigma^2)$	$X$ follows a Normal distribution with mean $\mu$ and variance $\sigma^2$	To define a Normal distribution, we need to know the mean and variance
$Z \sim N(0, 1)$	$Z$ follows the "Standard Normal distribution", that is, a Normal distribution with $\mu = 0$ and $\sigma^2 = 1$ . The usual convention is to use $Z$ instead of $X$ when using the standard Normal distribution.	The variance of $Z$ is $\sigma^2 = 1^2 = 1$ . The standard deviation is also 1, since $\sqrt{\sigma^2} = \sqrt{1^2} = 1$
$z$	$z$ -score, defined as $z = \displaystyle \frac{x - \mu}{\sigma}$	For a given value of $x$ , the corresponding $z$ -score can be thought of as its "standardised" value
$\pm$	Plus or minus. For example, $x \pm a = (x - a, x + a)$

2.3 Topics 5 and 6: Hypothesis testing and $t$ -tests

The following table provides a summary of notation used in Topics 5 and 6.

The lecture slides for Topic 5 can be found here.

The readings for Topic 5 can be found here.

The lecture slides for Topic 6 can be found here.

The readings for Topic 6 can be found here.

Notation	Meaning	Comments
$\pm$	Plus or minus. For example, $x \pm a = (x - a, x + a)$
$t$ -distribution	The distribution used for $t$ -tests	To define a $t$ -distribution, we need to know the degrees of freedom
$\text{df}$	Degrees of freedom
$T \sim t_{\text{df}}$	$T$ follows a $t$ -distribution with degrees of freedom equal to $\text{df}$ . For example, if $\text{df} = 1$ , then $T \sim t_1$
$H_0$	Null hypothesis
$H_1$	Alternative hypothesis	Sometimes denoted $H_a$
$\mu_0$	The population mean under the null hypothesis
$T$	Random test statistic
$\overline{X}$	Sample mean (random)
$\text{S}$	Estimator of the standard deviation of $X$
$\text{SE}$	Estimator of the standard error, i.e. standard deviation of the sample mean	Equal to $\displaystyle\frac{S}{\sqrt{n}}$
$t$	Observed test statistic	Sometimes called the ' $t$ value'
$\bar{x}$	Observed sample mean
$\text{s}$	Observed standard deviation
$\text{se}$	Observed standard error, i.e. observed standard deviation of the sample mean	Equal to $\displaystyle\frac{s}{\sqrt{n}}$
$p$	$p$ -value
$\alpha$	Significance level, such that if $p < \alpha$ , we reject $H_0$	Usually $\alpha = 0.05$ , but different values for $\alpha$ can be chosen
Type I error	The error that occurs when we reject $H_0$ when $H_0$ is true
Type II error	The error that occurs when we fail to reject $H_0$ when $H_0$ is false
$t_{\text{df,}1 - \alpha/2}$	The value from the $t_{\text{df}}$ distribution such that $P(T \leq t_{\text{df,}1 - \alpha/2}) = 1 - \alpha/2$ , i.e. the $(1 - \alpha/2)$ th quantile. For example, if $\alpha = 0.05$ , $1 - \alpha/2 = 1 - 0.05/2 = 1 - 0.025 = 0.975.$ We then have that $P(T \leq t_{\text{df,}0.975}) = 0.975$
$d$	Effect size, Cohen's $d$	We use Cohen's $d$ for $t$ -tests

2.4 Topic 7: One-way ANOVA

The following table provides a summary of notation used in Topic 7.

The lecture slides for Topic 7 can be found here.

The readings for Topic 7 can be found here.

Notation	Meaning	Comments
$H_0$	Null hypothesis
$H_1$	Alternative hypothesis	Sometimes denoted $H_a$
$p$	$p$ -value
$\alpha$	Significance level, such that if $p < \alpha$ , we reject $H_0$	Usually $\alpha = 0.05$ , but different values for $\alpha$ can be chosen
Type I error	The error that occurs when we reject $H_0$ when $H_0$ is true
Type II error	The error that occurs when we fail to reject $H_0$ when $H_0$ is false
ANOVA	ANalysis Of VAriance
$F$ -distribution	The distribution used for ANOVA Hypothesis tests	To define the $F$ -distribution, we need to know $d_1$ and $d_2$
$d_1$	Degrees of freedom 1
$d_2$	Degrees of freedom 2
$N$	Total sample size (in the context of ANOVA)
$k$	Number of groups (in the context of ANOVA)
$F_{d_1, d_2}$	$F$ -distribution with degrees of freedom 1 equal to $d_1$ and degrees of freedom 2 equal to $d_2$ . For example, if $d_1 = 3$ and $d_2 = 45$ , then our distribution is $F_{3, 45}$
$\eta^2$	Effect size, 'eta squared'	$\eta$ is a Greek letter pronounced 'eta'. We use $\eta^2$ for One-way ANOVA tests

2.5 Topic 8: Correlation and Simple Linear Regression

The following table provides a summary of notation used in Topic 8.

The lecture slides for Topic 8 can be found here.

The readings for Topic 8 can be found here.

Notation	Meaning	Comments
$\rho$	Population correlation	This is a Greek letter pronounced 'rho'
$r$	Sample correlation
$H_0$	Null hypothesis
$H_1$	Alternative hypothesis	Sometimes denoted $H_a$
$p$	$p$ -value
$\alpha$	Significance level, such that if $p < \alpha$ , we reject $H_0$	Usually $\alpha = 0.05$ , but different values for $\alpha$ can be chosen
Type I error	The error that occurs when we reject $H_0$ when $H_0$ is true
Type II error	The error that occurs when we fail to reject $H_0$ when $H_0$ is false
$x$	The explanatory variable, also referred to as the independent variable or predictor variable
$y$	The response variable, also referred to as the dependent variable
$\beta_0$	Intercept coefficient in the simple linear regression model	$\beta$ is a Greek letter pronounced 'beta'. $\beta_0$ is pronounced 'beta nought'
$\beta_1$	Slope coefficient in the simple linear regression model	$\beta$ is a Greek letter pronounced 'beta'. $\beta_1$ is pronounced 'beta 1'
$\epsilon$	Random error term in the simple linear regression model
$\widehat{y}$	The estimated value for $y$ based on a simple linear regression model	The " $\widehat{}$ " symbol is referred to as a 'hat' and is normally used to denote an estimate, so that $\widehat{y}$ is pronounced ' $y$ -hat'
$\widehat{\beta}_0$	The estimated value for $\beta_0$	The " $\widehat{}$ " symbol is referred to as a 'hat' and is normally used to denote an estimate, so that $\widehat{\beta}_0$ is pronounced ' $\beta_0$ -hat'
$\widehat{\beta}_1$	The estimated value for $\beta_1$	The estimated value for $\beta_0$
$R^2$	Coefficient of Determination. This value can be used to evaluate the fit of a simple linear regression model and is also the correlation squared.

2.6 Topic 9: Hypothesis testing for one and two sample proportions

The following table provides a summary of notation used in Topic 9.

The lecture slides for Topic 9 can be found here.

The readings for Topic 9 can be found here.

Notation	Meaning	Comments
$H_0$	Null hypothesis
$H_1$	Alternative hypothesis	Sometimes denoted $H_a$
$p$	$p$ -value
$\alpha$	Significance level, such that if $p < \alpha$ , we reject $H_0$	Usually $\alpha = 0.05$ , but different values for $\alpha$ can be chosen
Type I error	The error that occurs when we reject $H_0$ when $H_0$ is true
Type II error	The error that occurs when we fail to reject $H_0$ when $H_0$ is false
$p$	In the context of a one-sample test of proportions, $p$ is the proportion of a population with a certain characteristic	Not to be confused with the $p$ -value in the context of hypothesis testing
$n$	In the context of a one-sample test of proportions, $n$ either is the number of observations in a random sample, or the number of independent trials
$x$	In the context of a one-sample test of proportions, $x$ is either the number of observations in the sample that have a certain characteristics, or the number of success in $n$ trials
$p_0$	The population proportion under the null hypothesis
$\widehat{p}$	The estimate of $p$
$p_1$	The proportion of Population 1 (or Group 1) with a certain characteristic
$p_2$	The proportion of Population 2 (or Group 2) with a certain characteristic
$n_1$	The sample size from Population 1 (or Group 1)
$n_2$	The sample size from Population 2 (or Group 2)
$x_1$	The number of individuals in the sample from Population 1 (or Group 1) exhibiting the trait of interest
$x_2$	The number of individuals in the sample from Population 2 (or Group 2) exhibiting the trait of interest
$\hat{p}_1$	The estimate of $p_1$
$\hat{p}_2$	The estimate of $p_2$

2.7 Topic 10: Chi-squared tests for categorical data

The following table provides a summary of notation used in Topic 10.

The lecture slides for Topic 10 can be found here.

The readings for Topic 10 can be found here.

Notation	Meaning	Comments
$\displaystyle \sum$	Summation sign
$\displaystyle \sum_{i=1}^{n}$	The sum from $i=1$ to $n$	See the previous chapter for an example
$H_0$	Null hypothesis
$H_1$	Alternative hypothesis	Sometimes denoted $H_a$
$p$	$p$ -value
$\alpha$	Significance level, such that if $p < \alpha$ , we reject $H_0$	Usually $\alpha = 0.05$ , but different values for $\alpha$ can be chosen
Type I error	The error that occurs when we reject $H_0$ when $H_0$ is true
Type II error	The error that occurs when we fail to reject $H_0$ when $H_0$ is false
$\chi^2$ -distribution	The distribution used for Chi-squared tests	$\chi$ is a Greek letter pronounced 'ky'
$\chi^2_{\text{df}}$	$\chi^2$ -distribution with degrees of freedom equal to $\text{df}$ . For example, if $\text{df}=5$ , then our distribution is $\chi^2_5$
$X^2$	The random test statistic for a Chi-squared test. $X^2 \sim \chi^2_{\text{df}}$ under $H_0$
$\chi^2$	The observed test statistic for a Chi-squared test
$O_i$	The observed frequency in the $i$ th category in a Chi-squared goodness of fit test
$E_i$	The expected frequency for the $i$ th category in a Chi-squared goodness of fit test
$k$	The number of categories in a Chi-squared goodness of fit test
$O_{ij}$	The observed frequency in the $i$ th row and the $j$ th column in a Chi-squared test of independence
$E_{ij}$	The expected frequency of the $i$ th row and the $j$ th column in a Chi-squared test of independence
$r$	The number of rows in a Chi-squared test of independence	Not to be confused with the sample correlation $r$ in the context of correlation
$c$	The number of columns in a Chi-squared test of independence

2.8 Topic 11: Statistical Power and Sample Size Calculation

The following table provides a summary of notation used in Topic 11.

The lecture slides for Topic 11 can be found here.

The readings for Topic 11 can be found here.

Notation	Meaning	Comments
$H_0$	Null hypothesis
$H_1$	Alternative hypothesis	Sometimes denoted $H_a$
$p$	$p$ -value
$\alpha$	Significance level, such that if $p < \alpha$ , we reject $H_0$	Usually $\alpha = 0.05$ , but different values for $\alpha$ can be chosen
Type I error	The error that occurs when we reject $H_0$ when $H_0$ is true
Type II error	The error that occurs when we fail to reject $H_0$ when $H_0$ is false
$\alpha$	The probability of a Type I Error	This is also the significance level
$\beta$	The probability of a Type II Error	Not to be confused with $\beta_0$ or $\beta_1$ in the context of simple linear regression

2.9 Complete table of all notation used throughout STM1001

The following table provides a summary of all notation used in STM1001. All notation summarised in the previous sections of this chapter are provided in the consolidated table below.

Notation	Meaning	Comments
$n$	Sample size
$x_i$	The $i$ th $x$ value, usually from a list of $n$ $x$ values, e.g. ( $x_1, x_2, \ldots , x_n$ ), where $i$ can take any value from 1 to $n$
$\displaystyle \sum$	Summation sign
$\displaystyle \sum_{i=1}^{n}$	The sum from $i=1$ to $n$	See the previous chapter for an example
$\mu$	Population mean	This is a Greek letter pronounced 'm-yoo'
$\overline{x}$	Sample mean	The line on top of the $x$ is referred to as a 'bar', so that $\overline{x}$ is pronounced ' $x$ bar'
$\sigma^2$	Population variance	$\sigma$ is a Greek letter pronounced 'sigma'
$s^2$	Sample variance
$\sigma$	Population standard deviation	$\sigma$ is a Greek letter pronounced 'sigma'
$s$	Sample standard deviation
Q1	Quartile 1: 25% quantile
Q2	Quartile 2: 50% quantile, and also the median
Q3	Quartile 3: 75% quantile
$\rho$	Population correlation	This is a Greek letter pronounced 'rho'
$r$	Sample correlation
$\|x\|$	The 'absolute value' of some number $x$	See the previous chapter for an example
$\Omega$	Sample space	This is a Greek letter pronounced 'omega'
$\emptyset$	Null event (or null set)	Also sometimes referred to as 'empty set'
$P(A)$	The probability of event $A$
$A^C$	The complement of $A$ , or "not $A$ "	Sometimes denoted as $A^\prime$
$X$	A random variable (discrete or continuous)
$P(X = x)$	The probability that the random variable $X$ takes the value $x$ . For example, let $X$ denote the number of times you check this document. Then $P(X=2)$ denotes the probability that you check this document exactly 2 times.
$\text{E}(X)$	Expected value (or mean) of $X$
$\text{Var}(X)$	Variance of $X$
$\text{SD}(X)$	Standard deviation of $X$
$\overline{X}$	Sample mean (random)
$X \sim N(\mu, \sigma^2)$	$X$ follows a Normal distribution with mean $\mu$ and variance $\sigma^2$	To define a Normal distribution, we need to know the mean and variance
$Z \sim N(0, 1)$	$Z$ follows the "Standard Normal distribution", that is, a Normal distribution with $\mu = 0$ and $\sigma^2 = 1$ . The usual convention is to use $Z$ instead of $X$ when using the standard Normal distribution.	The variance of $Z$ is $\sigma^2 = 1^2 = 1$ . The standard deviation is also 1, since $\sqrt{\sigma^2} = \sqrt{1^2} = 1$
$z$	$z$ -score, defined as $z = \displaystyle \frac{x - \mu}{\sigma}$	For a given value of $x$ , the corresponding $z$ -score can be thought of as its "standardised" value
$\pm$	Plus or minus. For example, $x \pm a = (x - a, x + a)$
$t$ -distribution	The distribution used for $t$ -tests	To define a $t$ -distribution, we need to know the degrees of freedom
$\text{df}$	Degrees of freedom
$T \sim t_{\text{df}}$	$T$ follows a $t$ -distribution with degrees of freedom equal to $\text{df}$ . For example, if $\text{df} = 1$ , then $T \sim t_1$
$H_0$	Null hypothesis
$H_1$	Alternative hypothesis	Sometimes denoted $H_a$
$\mu_0$	The population mean under the null hypothesis
$T$	Random test statistic
$\text{S}$	Estimator of the standard deviation of $X$
$\text{SE}$	Estimator of the standard error, i.e. standard deviation of the sample mean	Equal to $\displaystyle\frac{S}{\sqrt{n}}$
$t$	Observed test statistic	Sometimes called the ' $t$ value'
$\bar{x}$	Observed sample mean
$\text{s}$	Observed standard deviation
$\text{se}$	Observed standard error, i.e. observed standard deviation of the sample mean	Equal to $\displaystyle\frac{s}{\sqrt{n}}$
$p$	$p$ -value
$\alpha$	Significance level, such that if $p < \alpha$ , we reject $H_0$	Usually $\alpha = 0.05$ , but different values for $\alpha$ can be chosen
Type I error	The error that occurs when we reject $H_0$ when $H_0$ is true
Type II error	The error that occurs when we fail to reject $H_0$ when $H_0$ is false
$t_{\text{df,}1 - \alpha/2}$	The value from the $t_{\text{df}}$ distribution such that $P(T \leq t_{\text{df,}1 - \alpha/2}) = 1 - \alpha/2$ , i.e. the $(1 - \alpha/2)$ th quantile. For example, if $\alpha = 0.05$ , $1 - \alpha/2 = 1 - 0.05/2 = 1 - 0.025 = 0.975.$ We then have that $P(T \leq t_{\text{df,}0.975}) = 0.975$
$d$	Effect size, Cohen's $d$	We use Cohen's $d$ for $t$ -tests
ANOVA	ANalysis Of VAriance
$F$ -distribution	The distribution used for ANOVA Hypothesis tests	To define the $F$ -distribution, we need to know $d_1$ and $d_2$
$d_1$	Degrees of freedom 1
$d_2$	Degrees of freedom 2
$N$	Total sample size (in the context of ANOVA)
$k$	Number of groups (in the context of ANOVA)
$F_{d_1, d_2}$	$F$ -distribution with degrees of freedom 1 equal to $d_1$ and degrees of freedom 2 equal to $d_2$ . For example, if $d_1 = 3$ and $d_2 = 45$ , then our distribution is $F_{3, 45}$
$\eta^2$	Effect size, 'eta squared'	$\eta$ is a Greek letter pronounced 'eta'. We use $\eta^2$ for One-way ANOVA tests
$x$	In the context of simple linear regression, $x$ is the explanatory variable, also referred to as the independent variable or predictor variable
$y$	In the context of simple linear regression, $y$ is the response variable, also referred to as the dependent variable
$\beta_0$	Intercept coefficient in the simple linear regression model	$\beta$ is a Greek letter pronounced 'beta'. $\beta_0$ is pronounced 'beta nought'
$\beta_1$	Slope coefficient in the simple linear regression model	$\beta$ is a Greek letter pronounced 'beta'. $\beta_1$ is pronounced 'beta 1'
$\epsilon$	Random error term in the simple linear regression model
$\widehat{y}$	The estimated value for $y$ based on a simple linear regression model	The " $\widehat{}$ " symbol is referred to as a 'hat' and is normally used to denote an estimate, so that $\widehat{y}$ is pronounced ' $y$ -hat'
$\widehat{\beta}_0$	The estimated value for $\beta_0$	The " $\widehat{}$ " symbol is referred to as a 'hat' and is normally used to denote an estimate, so that $\widehat{\beta}_0$ is pronounced ' $\beta_0$ -hat'
$\widehat{\beta}_1$	The estimated value for $\beta_1$	The estimated value for $\beta_0$
$R^2$	Coefficient of Determination. This value can be used to evaluate the fit of a simple linear regression model and is also the correlation squared.
$p$	In the context of a one-sample test of proportions, $p$ is the proportion of a population with a certain characteristic	Not to be confused with the $p$ -value in the context of hypothesis testing
$n$	In the context of a one-sample test of proportions, $n$ either is the number of observations in a random sample, or the number of independent trials
$x$	In the context of a one-sample test of proportions, $x$ is either the number of observations in the sample that have a certain characteristics, or the number of success in $n$ trials
$p_0$	The population proportion under the null hypothesis
$\widehat{p}$	The estimate of $p$
$p_1$	In the context of a two-sample test of proportions, $p_1$ is the proportion of Population 1 (or Group 1) with a certain characteristic
$p_2$	In the context of a two-sample test of proportions, $p_2$ is the proportion of Population 2 (or Group 2) with a certain characteristic
$n_1$	In the context of a two-sample test of proportions, $n_1$ is the sample size from Population 1 (or Group 1)
$n_2$	In the context of a two-sample test of proportions, $n_2$ is the sample size from Population 2 (or Group 2)
$x_1$	In the context of a two-sample test of proportions, $x_1$ is the number of individuals in the sample from Population 1 (or Group 1) exhibiting the trait of interest
$x_2$	In the context of a two-sample test of proportions, $x_2$ is the number of individuals in the sample from Population 2 (or Group 2) exhibiting the trait of interest
$\hat{p}_1$	The estimate of $p_1$
$\hat{p}_2$	The estimate of $p_2$
$\chi^2$ -distribution	The distribution used for Chi-squared tests	$\chi$ is a Greek letter pronounced 'ky'
$\chi^2_{\text{df}}$	$\chi^2$ -distribution with degrees of freedom equal to $\text{df}$ . For example, if $\text{df}=5$ , then our distribution is $\chi^2_5$
$X^2$	The random test statistic for a Chi-squared test. $X^2 \sim \chi^2_{\text{df}}$ under $H_0$
$\chi^2$	The observed test statistic for a Chi-squared test
$O_i$	The observed frequency in the $i$ th category in a Chi-squared goodness of fit test
$E_i$	The expected frequency for the $i$ th category in a Chi-squared goodness of fit test
$k$	The number of categories in a Chi-squared goodness of fit test
$O_{ij}$	The observed frequency in the $i$ th row and the $j$ th column in a Chi-squared test of independence
$E_{ij}$	The expected frequency of the $i$ th row and the $j$ th column in a Chi-squared test of independence
$r$	The number of rows in a Chi-squared test of independence	Not to be confused with the sample correlation $r$ in the context of correlation
$c$	The number of columns in a Chi-squared test of independence
$\alpha$	The probability of a Type I Error	This is also the significance level
$\beta$	The probability of a Type II Error	Not to be confused with $\beta_0$ or $\beta_1$ in the context of simple linear regression