Maths AI terms in Statistics and probability

A state of a Markov chain that cannot be left once it is entered, so that the probability of moving from it to itself is 1.

The set of values of the test statistic for which the null hypothesis is not rejected; it contains every possible value outside the critical region.

The probability, assuming the null hypothesis is true, that the test statistic falls in the critical region; for a discrete test it is usually less than the nominal significance level.

If $X\sim \mathrm{P}\mathrm{o}(m_1)$ and $Y\sim \mathrm{P}\mathrm{o}(m_2)$ are independent, then $X+Y\sim \mathrm{P}\mathrm{o}(m_1+m_2)$.

The extent to which two rankings of the same items place them in the same order, measured by $r_s$, where 1 means identical orders and −1 means reversed orders.

The statement, written ${\mathrm{H}}_1$, that is accepted if the null hypothesis is rejected; it describes a difference, an association or a departure from a model.

A tendency for the values of two variables to change together, so that knowing one gives information about the other; the pattern need not be linear.

A random trial with exactly two possible outcomes, called success and failure, where success has a fixed probability $p$.

A systematic tendency for a sample, or for the way data are collected, to give results that differ from the true population values in a particular direction.

An estimator whose expected value differs from the population parameter it estimates, so that it tends to overestimate or to underestimate.

A question whose wording, order or answer options encourage a particular response, so that the answers do not reflect respondents' true views; a question free of this is an unbiased question.

The number of ways to choose $r$ items from $n$ when order does not matter, written $\left(\genfrac{}{}{0px}{}{n}{r}\right)$ or ${}^n{C}_r$; in a binomial distribution it counts the sequences of $n$ trials with exactly $r$ successes.

The probability distribution of the number of successes $X$ in a fixed number $n$ of independent trials, each with the same probability of success $p$; it is written $X\sim \mathrm{B}(n,p)$.

A hypothesis test for a population proportion $p$, in which the number of successes in $n$ independent trials is modelled by $\mathrm{B}(n,p_0)$ when the null hypothesis $p=p_0$ is true.

Data in which two variables are measured or observed for each individual, so that each individual gives an ordered pair of values $(x,y)$.

A joint distribution of two random variables in which each variable is normally distributed and, for any value of one, the other is normally distributed with a mean that depends linearly on it.

A diagram drawn against a scale in which a box spans the lower to the upper quartile with a line at the median, and whiskers extend from the box to the smallest and largest values that are not outliers; any outliers are marked separately with a cross.

A relationship in which a change in one variable directly produces a change in the other variable.

A collection of data from every member of a population.

For a random sample of $n$ independent observations from any population with mean $\mu$ and finite variance ${\sigma }^2$, the distribution of the sample mean $\bar{X}$ approaches the normal distribution $\mathrm{N}(\mu ,\frac{{\sigma }^2}{n})$ as $n$ becomes large.

The continuous probability distribution, taking only non-negative values, that the chi-squared statistic follows when the null hypothesis is true; its shape is set by the number of degrees of freedom.

A test of whether observed frequencies in a set of categories are consistent with the frequencies predicted by a stated distribution.

A test of whether two categorical variables are independent in a population, using the observed frequencies in a contingency table.

A range of values grouped together as one class in a frequency distribution, written as an inequality such as $10\le t<15$.

A measure of how well a regression model fits bivariate data, equal to the proportion of the variation in the dependent variable that is explained by the model; it lies between 0 and 1 for a least squares model, with values close to 1 showing a close fit.

An event formed from two or more events using union ("or"), intersection ("and") or complement ("not").

Merging adjacent categories of a chi-squared table, adding their observed frequencies and their expected frequencies, so that every expected frequency is greater than 5.

The event that $A$ does not occur, written $A^{\mathrm{\prime }}$ and read "not $A$"; it contains every outcome in the sample space that is not in $A$.

The probability that an event $A$ occurs given that an event $B$ has occurred, written $\mathrm{P}(A\mid B)$; it is found by treating $B$ as the new sample space.

A range of values, calculated from a sample, constructed by a method that captures the true value of a population parameter in a stated proportion of all possible samples.

The percentage of all possible samples for which the method used to construct a confidence interval would produce an interval containing the true value of the parameter.

The lower and upper end points of a confidence interval.

A variable, not included in the analysis, that affects both of the variables being studied and so creates or strengthens a correlation between them.

The extent to which a test covers all the aspects of the quantity it is intended to measure, in the right proportions; it is judged by experts who compare the content of the test with a full description of what should be measured.

A table of observed frequencies that classifies each member of a sample by two categorical variables, one for the rows and one for the columns.

Quantitative data that can take any value within an interval, usually obtained by measuring.

A random variable that can take any value in an interval of real numbers, so that its probabilities are given by areas under a curve rather than by a list of separate values.

A method in which the sample consists of members of the population who are easy to reach or who volunteer, with no random selection and no fixed numbers from each category.

A relationship between two variables in which one tends to change as the other changes, described by its direction (positive or negative) and its strength.

The extent to which the results of a test agree with an established external measure (the criterion) of the same quantity, taken at the same time (concurrent validity) or later (predictive validity); it is judged by a correlation coefficient between test scores and criterion scores.

The set of values of the test statistic for which the null hypothesis is rejected at a given significance level.

The boundary value of a test statistic for a given significance level; a test statistic beyond the critical value leads to rejecting the null hypothesis.

Fitting a model of the form $y=a{x}^3+b{x}^2+cx+d$ to bivariate data by least squares.

The running total of the frequencies up to and including a given class, which gives the number of data values below that class's upper class boundary.

A graph of cumulative frequency (vertical axis) against the upper class boundaries (horizontal axis), with the points joined by a smooth increasing curve or by straight line segments.

The probability that a random variable takes a value less than or equal to a given value, $\mathrm{P}(X\le r)$, for a discrete random variable it is found by adding the probabilities of all values up to and including $r$.

A statement, fixed before the data are collected, of which observed values will lead to rejecting the null hypothesis and which will not.

The number of frequencies in a chi-squared calculation that can vary freely once the totals are fixed, written $\nu$ (or df).

The difference between a data value and the mean of the data, $x-\bar{x}$; it is positive for values greater than the mean and negative for values less than it.

Quantitative data that can take only particular separate values, usually whole numbers obtained by counting.

A random variable that can take only separate, countable values, such as whole numbers, with gaps between them.

A probability distribution in which a random variable takes each of a finite number of values with the same probability; with $n$ values, each has probability $\frac{1}{n}$.

For any normal distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two standard deviations and about 99.7% within three standard deviations.

Outcomes that all have the same probability of occurring, so that with $n(U)$ such outcomes each has probability $\frac{1}{n(U)}$.

The single numerical value that an estimator gives for one particular sample, used as a best guess of a population parameter.

An approximation to the mean of grouped data, calculated with the mean formula using the mid-interval value of each class in place of the unknown individual values.

A value of a distribution's parameter, such as a mean, a standard deviation or a probability of success, that is calculated from the sample data because it is not known or given in advance.

A rule or formula, applied to sample data, that is used to estimate a population parameter; because the sample is random, an estimator is itself a random variable.

A set of outcomes from the sample space, usually described in words and named with a capital letter; the event occurs when the outcome of a trial is one of its outcomes.

The number of observations predicted for a category if the null hypothesis were true, written $f_e$; it is the total frequency multiplied by the probability of the category under the null hypothesis.

The average number of times an event is expected to occur in $n$ trials, equal to $n\times \mathrm{P}(A)$ when the event has probability $\mathrm{P}(A)$ in each trial.

The long-run mean value of a random variable over a very large number of repetitions; for a discrete random variable $X$ it is $\mathrm{E}(X)=\sum x\mathrm{P}(X=x)$, the sum of each value multiplied by its probability.

Fitting a model of the form $y=a{b}^x$, or equivalently $y=a{\mathrm{e}}^{kx}$, to bivariate data with positive $y$-values by least squares.

For a positive integer $n$, the product of all the positive integers from 1 to $n$, written $n!=n\times (n-1)\times \cdots \times 2\times 1$, with $0!=1$.

A game in which the expected gain of a player is zero, $\mathrm{E}(X)=0$, where $X$ is the player's gain.

A test result that wrongly indicates that a condition is absent when it is present.

A test result that wrongly indicates that a condition is present when it is not.

The minimum, lower quartile, median, upper quartile and maximum of a data set.

The number of times a particular value, or a value within a particular group, occurs in a data set.

A table that lists each value, or each group of values, of a variable together with its frequency.

The amount a player receives from a game minus the amount paid to play, so that a loss is a negative gain.

A device in which balls fall through rows of pins, bouncing left or right at each pin, and collect in slots at the bottom, so that the number of balls in each slot follows a binomial distribution.

Data that have been sorted into classes, so that the table gives the number of values in each class rather than each individual value.

A diagram for grouped continuous data in which each class is shown by a bar standing on its class interval, with no gaps between bars; with equal class widths, the height of each bar is its frequency.

The organiser's expected profit as a percentage of the amount staked, equal to $-\mathrm{E}(X)$ per unit staked, where $X$ is the player's gain.

A procedure that uses sample data to decide whether there is enough evidence to reject a stated claim about a population.

Describes random variables that all have the same probability distribution and are mutually independent.

Events for which the occurrence of one does not change the probability of the other, so that $\mathrm{P}(A\mid B)=\mathrm{P}(A)$ and $\mathrm{P}(A\cap B)=\mathrm{P}(A)\mathrm{P}(B)$.

Random variables for which the value taken by any one of them does not change the probability distribution of any of the others.

Repeated trials in which the outcome of any trial does not change the probabilities of the outcomes of any other trial.

The state vector ${\mathit{s}}_0$ that describes a Markov chain before the first step, giving the probability of starting in each state or the number starting in each state.

The difference between the upper and lower quartiles, $Q_3-Q_1$, which measures the spread of the middle 50% of the data.

The event $A\cap B$ that both $A$ and $B$ occur; it contains only the outcomes that are in $A$ and in $B$.

Finding the value $k$ of a normally distributed variable for which the area on one side of $k$, or a central area around the mean, equals a given probability.

The curve of a chosen family (such as quadratic or exponential) whose parameter values make the sum of square residuals for the data as small as possible.

A straight line drawn by eye on a scatter diagram through the mean point, following the trend of the points so that they are balanced on either side of it.

A random variable of the form $a_1{X}_1\pm a_2{X}_2\pm \cdots \pm a_n{X}_n$, where $X_1,\dots ,X_n$ are random variables and $a_1,\dots ,a_n$ are constants.

A relationship between two variables in which the points of a scatter diagram lie close to a straight line, so that equal increases in $x$ go with roughly equal changes in $y$.

A new random variable $Y=aX+b$ formed from a random variable $X$ by multiplying every value by a constant $a$ and adding a constant $b$; it has $\mathrm{E}(Y)=a\mathrm{E}(X)+b$ and $\mathrm{V}\mathrm{a}\mathrm{r}(Y)=a^2\mathrm{V}\mathrm{a}\mathrm{r}(X)$.

The limit of the probability of being in a given state after $n$ steps of a Markov chain as $n$ becomes very large; for a regular chain it equals the corresponding entry of the steady state.

The value below which one quarter (25%) of the data lie when the data are arranged in order; it is written $Q_1$.

Half the width of a confidence interval, the distance from the point estimate to either confidence limit; for a z-interval it equals $z\frac{\sigma }{\sqrt{n}}$.

A sequence of random states, observed at equal steps, in which the probability of moving to each state at the next step depends only on the current state, and these probabilities stay the same at every step.

The property that the probability of the next state depends only on the current state, and not on the states that came before it.

Data in which each value in one sample is linked to exactly one value in the other, because both come from the same individual or from two individuals deliberately matched on relevant characteristics.

The sum of all the values in a data set divided by the number of values; written $\bar{x}$, or $\mu$ when the data are a whole population.

The point $(\bar{x},\bar{y})$ whose coordinates are the mean of the $x$-values and the mean of the $y$-values of a set of bivariate data.

The mean number of times an event occurs per unit of time or space, which gives the parameter $m$ of a Poisson distribution once it is multiplied by the size of the interval.

A single value that represents the centre or typical value of a data set; the mean, the median and the mode are the three standard measures.

A single value that describes how spread out the values of a data set are; the range, interquartile range, variance and standard deviation are measures of dispersion.

The middle value of a data set arranged in order of size, so that half of the values are less than or equal to it and half are greater than or equal to it.

The value halfway between the lower and upper boundaries of a class interval, used to represent every value in that class when estimating with grouped data.

Values that should be in a data set but were not recorded or cannot be used.

The class interval with the highest frequency in a grouped frequency table whose classes all have equal width.

The value that occurs most often in a data set; a data set can have more than one mode, or none if every value occurs equally often.

A relationship between two variables in which one variable always increases, or always decreases, as the other increases, at a rate that may change.

Carrying out many hypothesis tests on the same data, which greatly increases the probability that at least one test gives a significant result by chance even when every null hypothesis is true.

Events that cannot occur at the same time, so that $A\cap B$ contains no outcomes and $\mathrm{P}(A\cap B)=0$.

A relationship in which larger values of one variable tend to go with smaller values of the other, so the points of a scatter diagram fall from left to right.

The process of fitting a curve that is not a straight line, such as a quadratic, cubic, exponential, power or sine function, to bivariate data by choosing the values of its parameters that give the closest fit.

The symmetric bell-shaped graph of a normal distribution, in which the area under the curve between two values equals the probability that the variable lies between them.

A continuous probability distribution that is symmetric and bell-shaped about its mean $\mu$, with spread given by its standard deviation $\sigma$; it is written $X\sim \mathrm{N}(\mu ,{\sigma }^2)$.

The probability that a normally distributed variable lies in a given interval, equal to the area under the normal curve over that interval.

The statement of no difference, no association or no change from a model, written ${\mathrm{H}}_0$, that is assumed true while a hypothesis test is carried out.

The number of observations actually recorded in a category of a sample, written $f_o$.

A hypothesis test for the mean of a normal population with unknown standard deviation, using the statistic $t=\frac{\bar{x}-{\mu }_0}{s_{n-1}\mathrm{/}\sqrt{n}}$, which follows the t-distribution with $n-1$ degrees of freedom when the null hypothesis is true.

A test whose alternative hypothesis states the direction of the difference from the value in the null hypothesis.

Data whose values can be placed in a meaningful order, but where the differences between values are not equal or not measured.

One possible result of a single trial of a random experiment.

A data value that is more than $1.5\times \mathrm{I}\mathrm{Q}\mathrm{R}$ below the lower quartile or more than $1.5\times \mathrm{I}\mathrm{Q}\mathrm{R}$ above the upper quartile.

The situation in which count data have a variance noticeably larger than their mean, which suggests that a Poisson model is not suitable because the events are clustered or the rate varies.

The probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one calculated from the sample.

A one-sample t-test applied to the differences within matched pairs, testing whether the population mean difference ${\mu }_d$ equals zero.

A method of checking reliability in which two different but equivalent versions of a test, designed to measure the same thing at the same level of difficulty, are given to the same group of people, and the two sets of scores are compared using a correlation coefficient; strong positive correlation indicates a reliable test.

The population, or the probability distribution, from which the observations in a sample are drawn; its mean $\mu$ and variance ${\sigma }^2$ describe a single observation.

A triangular array of numbers in which each row begins and ends with 1 and every other number is the sum of the two numbers above it; row $n$ lists the binomial coefficients $\left(\genfrac{}{}{0px}{}{n}{0}\right),\left(\genfrac{}{}{0px}{}{n}{1}\right),\dots ,\left(\genfrac{}{}{0px}{}{n}{n}\right)$.

A number $r$ with $-1\le r\le 1$ that measures the strength and direction of the linear relationship between two variables; $r=1$ and $r=-1$ mean that all the points lie exactly on a straight line with positive or negative gradient, and $r=0$ means no linear correlation.

The $k$th percentile is the value below which $k\mathrm{\%}$ of the data lie when the data are arranged in order.

A question that asks for private or sensitive information, so that respondents may refuse to answer or may answer untruthfully.

The probability distribution of the number of occurrences $X$ of an event in a fixed interval of time or space, when the events occur independently and at a uniform average rate; it has one parameter $m$, the mean number of occurrences in the interval, and is written $X\sim \mathrm{P}\mathrm{o}(m)$.

A hypothesis test for the mean of a Poisson distribution, in which the observed count in a stated interval is modelled by $\mathrm{P}\mathrm{o}(m_0)$ when the null hypothesis $m=m_0$ is true.

The complete set of individuals, items or measurements that a statistical investigation wants to describe or draw conclusions about.

The product-moment correlation coefficient between two variables in the whole population, written $\rho$; the sample correlation coefficient $r$ is its estimate.

A fixed numerical value that describes a whole population or probability model and that is usually unknown.

A relationship in which larger values of one variable tend to go with larger values of the other, so the points of a scatter diagram rise from left to right.

The probability of rejecting the null hypothesis when it is false, for a specified value of the parameter under the alternative hypothesis; it equals $1-\mathrm{P}(\text{Type II error})$.

Fitting a model of the form $y=a{x}^b$ to bivariate data with positive $x$- and $y$-values by least squares.

A question that is clear and unambiguous, with a stated time frame and units where needed, so that every respondent interprets it in the same way.

A number from 0 to 1 that measures how likely an event is to occur, where 0 means the event is impossible and 1 means it is certain; the probability of an event $A$ is written $\mathrm{P}(A)$.

A list, table or function that gives every possible value of a random variable together with the probability that the variable takes that value.

Fitting a model of the form $y=a{x}^2+bx+c$ to bivariate data by least squares.

Data that describe a quality or category, such as colour, nationality or a yes/no answer, rather than a numerical amount.

Data that are numerical values of a quantity that can be counted or measured, so that calculations such as a mean are meaningful.

A set of written questions, given in the same form to every respondent, used to collect data from the members of a sample.

A method in which a fixed number of members is required from each category of the population, and members are chosen by the data collector, not at random, until each category's number is reached.

A sample chosen by a process in which every member of the population has a known, non-zero chance of being selected, decided by chance rather than by the person collecting the data.

A variable whose value is a number determined by the outcome of a random process, so that each possible value has a probability.

The difference between the largest and smallest values in a data set, maximum − minimum.

The position of a value when all the values of a variable are arranged in order, with 1 for the first value in the chosen order.

The straight line $y=ax+b$ that makes the sum of the squares of the vertical distances from the data points to the line as small as possible; it is used to predict values of $y$ from values of $x$.

A Markov chain whose transition matrix $T$ has some power $T^k$ in which every entry is strictly positive.

The number of times an event occurs in a set of trials divided by the total number of trials; it is used as an estimate of the probability of the event.

A variable that has a direct bearing on the research question being investigated, chosen from all the variables available because it measures the quantity of interest or a factor that could affect it.

The extent to which data can be trusted to be accurate and consistent, so that collecting them again in the same way would give similar results.

The difference between an observed value of the dependent variable and the value predicted by a model, $y_i-{\widehat{y}}_i$; it is positive when the data point lies above the model's graph and negative when it lies below.

A scatter diagram of the residuals of a model, $y_i-{\widehat{y}}_i$, plotted against the independent variable $x_i$; residuals scattered randomly about zero suggest a suitable model, while a curved or other systematic pattern shows that the model's form is wrong.

A summary statistic that changes little when a few extreme values or outliers are added, removed or changed; the median and the interquartile range are robust, while the mean, range and standard deviation are not.

A subset of a population from which data are actually collected, used to draw conclusions about the population.

The mean of the values in a sample, $\bar{x}=\frac{1}{n}\sum x_i$; as an estimator, $\bar{X}=\frac{X_1+X_2+\cdots +X_n}{n}$, it is an unbiased estimator of the population mean $\mu$.

The set of all possible outcomes of a random experiment, written $U$; the number of outcomes in it is $n(U)$.

A table or grid that shows every outcome of a two-stage experiment, with the outcomes of one stage along the top, those of the other stage down the side, and one cell for each combined outcome.

The probability distribution of a statistic over all possible random samples of a given size $n$ from the same population.

The list, register or map of the members of a population from which a sample is actually selected.

A graph of bivariate data in which each pair of values $(x,y)$ is plotted as a single point, with the independent variable on the horizontal axis and the dependent variable on the vertical axis.

The threshold probability, chosen before a test is carried out, below which the p-value leads to rejecting the null hypothesis; it equals the probability of rejecting the null hypothesis when it is in fact true.

A method of choosing a sample of size $n$ so that every member of the population has the same chance of being chosen and every possible sample of size $n$ is equally likely.

The use of random numbers, generated by hand, by a calculator or by a computer, to imitate a random process many times so that probabilities can be estimated from relative frequencies.

Fitting a model of the form $y=a\sin (bx+c)+d$ to bivariate data by least squares, where $\mathrm{\mid }a\mathrm{\mid }$ is the amplitude, $\frac{2\pi }{b}$ the period (with $x$ in radians) and $d$ the principal axis.

A distribution that is not symmetric because its values are spread out further on one side of the centre, giving a longer tail on that side; the skew is positive when the long tail is towards larger values and negative when it is towards smaller values.

A number from −1 to 1 that measures the strength and direction of the association between two variables, found by calculating Pearson's product-moment correlation coefficient for the ranks of the data instead of the data values.

A correlation between two variables that have no direct causal connection, arising from coincidence or from a confounding variable that affects both.

The square root of the variance, $\sigma =\sqrt{\frac{\sum f_i(x_i-\mu {)}^2}{n}}$; it measures the typical distance of the data values from the mean, in the same units as the data.

The square root of the variance of a random variable, $\sigma =\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X)}$, measured in the same units as the variable itself.

The standard deviation of the sampling distribution of the sample mean, $\frac{\sigma }{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ the sample size.

The normal distribution with mean 0 and variance 1, written $Z\sim \mathrm{N}(0,1)$; any normal variable $X\sim \mathrm{N}(\mu ,{\sigma }^2)$ is converted to it by $Z=\frac{X-\mu }{\sigma }$.

One of the possible conditions a system can be in at a given step of a Markov chain; at each step the system is in exactly one state.

A column vector ${\mathit{s}}_n$ whose entries give, for each state of a Markov chain, the probability of being in that state (or the number of members in that state) after $n$ steps.

A value calculated only from the data in a sample; it varies from one sample to another.

Describes a result whose p-value is less than the chosen significance level, so that the null hypothesis is rejected at that level.

A state vector $\mathit{s}$ of a Markov chain with $T\mathit{s}=\mathit{s}$ whose entries are non-negative and sum to 1, so that a chain with this distribution keeps the same distribution at every later step.

A method in which the population is divided into groups called strata, and a simple random sample is taken from each stratum, with the number taken from each stratum proportional to the stratum's size.

A question with a fixed set of answer options, the same for every respondent, so that every answer falls into one of the given categories.

The outcome of a Bernoulli trial that is being counted; the other outcome is called failure, and the probability of success $p$ and of failure $1-p$ add to 1.

The sum of the squares of the residuals of a model, $S{S}_{res}=\sum (y_i-{\widehat{y}}_i{)}^2$; it measures the total squared error of the model, and a smaller value means a closer fit to the same data.

A distribution whose values are spread in the same way on both sides of its centre, so that one half of its histogram is roughly a mirror image of the other.

A method of choosing a sample by selecting every $k$th member from an ordered list of the population, starting from a randomly chosen member among the first $k$.

A continuous probability distribution that is symmetric and bell-shaped about 0 like the standard normal distribution but has heavier tails, with a shape fixed by its number of degrees of freedom $\nu$; for a sample of size $n$ from a normal population, $\frac{\bar{X}-\mu }{s_{n-1}\mathrm{/}\sqrt{n}}$ has a t-distribution with $\nu =n-1$.

A confidence interval for the mean of a normal population with unknown standard deviation, given by $\bar{x}\pm t\frac{s_{n-1}}{\sqrt{n}}$, where $t$ is the critical value of the t-distribution with $n-1$ degrees of freedom for the confidence level.

A test of whether the means of two populations are equal, using independent random samples from each; the pooled two-sample t-test assumes both populations are normally distributed with equal variances.

The part of a distribution far from the mean on one side; the lower (left) tail holds the smallest values and the upper (right) tail the largest values.

A single value calculated from the sample data that measures how far the data are from what the null hypothesis predicts.

A method of checking reliability in which the same test is given to the same group of people on two occasions under the same conditions, and the two sets of scores are compared using a correlation coefficient; strong positive correlation indicates a reliable test.

A probability found by reasoning from a model of the experiment rather than from observed data; when the outcomes are equally likely it equals $\frac{n(A)}{n(U)}$, the number of outcomes in the event divided by the number in the sample space.

The equal ranks given to equal data values, each equal to the mean of the positions the tied values occupy together.

A directed graph whose vertices are the states of a Markov chain and whose edges are arrows labelled with the transition probabilities, including loops for staying in the same state.

The square matrix $T$ of a Markov chain whose entry $T_{ij}$ in row $i$ and column $j$ is the probability of moving from state $j$ to state $i$ in one step, so that every entry lies between 0 and 1 and each column sums to 1.

The probability that a Markov chain moves from one given state to another given state in one step; $T_{ij}$ denotes the probability of moving from state $j$ to state $i$.

A branching diagram for a sequence of events, in which each branch represents an outcome at one stage and is labelled with its probability given the outcomes on the branches before it.

A single performance of a random experiment.

A test whose alternative hypothesis states only that a parameter differs from the value in the null hypothesis, in either direction.

A table that classifies a set of people or items by two features, with one feature in the rows, the other in the columns, a count in each cell and totals in the margins.

Rejecting the null hypothesis when it is true.

Not rejecting (accepting) the null hypothesis when it is false.

The value $s_{n-1}^2=\frac{\sum f_i(x_i-\bar{x}{)}^2}{n-1}=\frac{n}{n-1}{s}_n^2$, calculated from a sample of size $n$, whose estimator has expected value equal to the population variance ${\sigma }^2$.

An estimator whose expected value equals the population parameter it estimates, so that over all possible samples it neither overestimates nor underestimates on average.

The property that the mean number of events per unit of time or space stays the same throughout the period or region being modelled, so that the mean number in any interval is proportional to the interval's size.

The event $A\cup B$ that $A$ occurs or $B$ occurs or both occur; it contains every outcome that is in $A$, in $B$ or in both.

The set of everything being considered in a problem, written $U$; in probability it is the sample space, the set of all possible outcomes.

A question that the respondent answers freely in their own words, with no answer options given.

The value at the top end of a class interval, which is the boundary it shares with the next class.

The value below which three quarters (75%) of the data lie when the data are arranged in order; it is written $Q_3$.

The extent to which a test or other measuring instrument measures what it is intended to measure, so that conclusions drawn from its results are justified.

The mean of the squared deviations of the data values from their mean, ${\sigma }^2=\frac{\sum f_i(x_i-\mu {)}^2}{n}$; it measures spread in squared units of the data.

The expected value of the squared deviation of a random variable from its mean, $\mathrm{V}\mathrm{a}\mathrm{r}(X)=\mathrm{E}((X-\mu {)}^2)$, where $\mu =\mathrm{E}(X)$; it measures how widely the values of $X$ spread about its mean in the long run.

A diagram in which a rectangle represents the universal set and closed curves (usually circles) inside it represent events or sets, with overlapping regions showing outcomes that belong to more than one event.

A method of repeated selection in which each item is returned before the next one is chosen, so the probabilities are the same at every draw.

A method of repeated selection in which items are not returned once chosen, so the number of items, and usually the probabilities, change after each draw.

A confidence interval for the mean of a normal population with known standard deviation $\sigma$, given by $\bar{x}\pm z\frac{\sigma }{\sqrt{n}}$, where $z$ is the critical value of the standard normal distribution for the confidence level.

A hypothesis test for the mean of a normal population with known standard deviation $\sigma$, using the statistic $z=\frac{\bar{x}-{\mu }_0}{\sigma \mathrm{/}\sqrt{n}}$, which follows the standard normal distribution when the null hypothesis is true.