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 |