Maths AA terms in Statistics and probability

A formula that reverses a conditional probability: $P(B\mid A)=\frac{P(B)P(A\mid B)}{P(B)P(A\mid B)+P(B^{\mathrm{\prime }})P(A\mid B^{\mathrm{\prime }})}$.

A single random trial with exactly two possible outcomes, called success (with probability $p$) and failure (with probability $1-p$).

A systematic tendency for a sample or a method of collecting data to give results that differ from the true values for the population in the same direction.

The distribution of the number of successes $X$ in $n$ independent trials, each with the same probability of success $p$. Written $X\sim B(n,p)$, with $P(X=r)=\left(\genfrac{}{}{0px}{}{n}{r}\right)p^r(1-p{)}^{n-r}$.

Data in which two variables are measured or recorded for each individual or item, giving pairs of values $(x,y)$ that can be plotted and compared.

A diagram of a data set drawn on a scale: a box from $Q_1$ to $Q_3$ with a line at the median, and whiskers to the smallest and largest values that are not outliers.

A relationship in which a change in one variable directly produces a change in another. Correlation alone never shows causation: both variables may depend on a third, or the link may be coincidence.

A survey that collects data from every member of the population rather than from a sample.

A range of values used to group data, such as $50\le t<55$. Its boundaries are the values that separate it from the neighbouring classes; its width is the upper boundary minus the lower boundary.

The square $r^2$ of Pearson's correlation coefficient, with $0\le r^2\le 1$: the proportion of the variation in $y$ that the linear regression on $x$ accounts for.

An event $A$ and the event $A^{\mathrm{\prime }}$ ("not $A$"), which contains every outcome in $U$ that is not in $A$. Exactly one of them occurs, so $P(A)+P(A^{\mathrm{\prime }})=1$.

The probability that $A$ occurs given that $B$ has occurred, $P(A\mid B)=\frac{P(A\cap B)}{P(B)}$, where $P(B)>0$.

Numerical data that can take any value in an interval and are obtained by measuring, such as height, mass or time. Recorded values are rounded to some degree of accuracy.

A random variable that can take any value in an interval of real numbers, such as a time or a length. Its probabilities are areas under a probability density function, and $P(X=a)=0$ for any single value $a$.

The distribution of a continuous random variable whose probability density function is constant, $f(x)=\frac{1}{b-a}$, on an interval $a\le x\le b$ and zero elsewhere.

A non-random sampling method that uses members of the population who are easy to reach, such as the first people met.

A relationship between two variables in which one tends to increase (positive correlation) or decrease (negative correlation) as the other increases. Linear correlation is when the points lie close to a straight line.

The running total of the frequencies up to a given value or upper class boundary. Plotted against the upper boundaries, it gives a cumulative frequency graph.

The probability that a random variable takes a value less than or equal to a given value, $P(X\le x)$. A GDC's binomial cdf function gives it directly.

Events that are not independent: knowing that one has occurred changes the probability of the other, so $P(A\cap B)\mathrm{\ne }P(A)P(B)$.

Numerical data that can take only separate, distinct values (usually counts), such as the number of goals scored in a match.

A variable whose value is a numerical outcome of a random process and which can take only separate (countable) values, such as $0,1,2,\dots$. It is written with a capital letter, such as $X$.

A probability distribution in which each of the $n$ possible values of a discrete random variable has the same probability $\frac{1}{n}$, such as the score on a fair die.

Outcomes that all have the same chance of occurring, such as the six faces of a fair die. Only then does $P(A)=\frac{n(A)}{n(U)}$ apply.

A set of outcomes from the sample space, such as "an even number" when a die is rolled. An event occurs when the outcome of the trial is one of its outcomes.

Events that together cover the whole sample space, so at least one of them must occur. Mutually exclusive, exhaustive events $B_1,B_2,B_3$ form a partition of $U$.

The number of times an event is predicted to occur in $n$ trials when its probability is $p$, equal to $np$.

The long-run mean value of a random variable over many repetitions, written $E(X)$ or $\mu$. For a discrete random variable, $E(X)=\sum xP(X=x)$.

Using a regression line to predict a value outside the range of the data used to find it. Predicting inside that range is interpolation, which is much more reliable.

A game in which a player's expected gain is zero, so that $E(\text{gain})=0$: on average the player neither wins nor loses money over many plays.

A positive test result for someone who does not have the condition. A false negative is a negative result for someone who does.

A table listing each value or class of values in a data set together with its frequency, the number of times it occurs.

A diagram for grouped continuous data in which each class is drawn as a bar with no gaps between bars. With equal class widths, the height of each bar is the frequency.

Events for which the occurrence of one does not change the probability of the other: $P(A\cap B)=P(A)P(B)$, or equivalently $P(A\mid B)=P(A)$.

In bivariate data, the variable $x$ that is controlled or used to make predictions, plotted on the horizontal axis. The dependent variable $y$ is the one predicted, plotted on the vertical axis.

The difference between the upper and lower quartiles, $\mathrm{I}\mathrm{Q}\mathrm{R}=Q_3-Q_1$: the spread of the middle half of the data.

The event $A\cap B$ ("$A$ and $B$"), containing the outcomes that are in both $A$ and $B$.

Finding the value $k$ of a normally distributed variable $X$ that gives a stated probability, such as $P(X<k)=0.9$. It reverses a normal probability calculation: the area is known and the boundary is found.

The result that, as the number of trials increases, the relative frequency of an event tends to get closer to its probability.

For mutually exclusive events $B_1,B_2,\dots$ that cover the sample space, $P(A)=P(B_1)P(A\mid B_1)+P(B_2)P(A\mid B_2)+\dots$, the sum over every path of a tree diagram that ends in $A$.

The sum of the data values divided by the number of values; for a frequency table, $\bar{x}=\frac{\sum fx}{n}$.

The point $(\bar{x},\bar{y})$ whose coordinates are the means of the two variables in a set of bivariate data. Every regression line and every good line of best fit passes through it.

A single value that represents the centre or typical value of a data set: the mean, the median or the mode.

A single value describing how spread out a data set is: the range, the interquartile range, the variance or the standard deviation.

The middle value of a data set arranged in order; for an even number of values, the mean of the two middle values.

The value $m$ that splits the area under the probability density function into two halves, so that ${\int }_{-\mathrm{\infty }}^{m}f(x)dx=0.5$.

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

The class interval with the highest frequency in grouped data (with equal class widths).

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

The value of $x$ at which the probability density function $f(x)$ takes its greatest value on the domain.

Events that cannot occur together, so $P(A\cap B)=0$ and $P(A\cup B)=P(A)+P(B)$.

A continuous probability distribution with a symmetric, bell-shaped curve, fixed by its mean $\mu$ and variance ${\sigma }^2$ and written $X\sim N(\mu ,{\sigma }^2)$. Its mean, median and mode are equal, and the total area under the curve is $1$.

One possible result of a trial. Exactly one outcome occurs each time the trial is carried out.

A data value that lies unusually far from the rest of the data. A value is treated as an outlier when it is more than $1.5\times \mathrm{I}\mathrm{Q}\mathrm{R}$ below $Q_1$ or above $Q_3$.

A number $r$, with $-1\le r\le 1$, measuring the strength and direction of the linear correlation between two variables. Values near $\pm 1$ show strong linear correlation; values near $0$ show little or none.

The value below which a given percentage of the ordered data lie; the $p$th percentile has $p\mathrm{\%}$ of the data below it.

The complete set of individuals or items about which information is wanted in a statistical investigation.

The updated probability of an event after new evidence is taken into account, found with Bayes' theorem, such as $P(\text{disease}\mid \text{positive test})$.

The probability of an event before new evidence is taken into account, such as the proportion of a population that has a disease.

A number from $0$ to $1$ that measures how likely an event is: $0$ for an impossible event, $1$ for a certain one.

A function $f$ describing a continuous random variable $X$, with $f(x)\ge 0$ for all $x$ and ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx=1$, so that $P(a<X<b)={\int }_a^{b}f(x)dx$.

A list, table, graph or formula giving every possible value $x$ of a discrete random variable $X$ with its probability $P(X=x)$. The probabilities are all between $0$ and $1$ and add up to $1$.

One of the values that divide ordered data into four equal parts: the lower quartile $Q_1$ has a quarter of the data below it, the upper quartile $Q_3$ three quarters.

A non-random sampling method in which the interviewer fills a fixed number (quota) of members from each group of the population, choosing the individuals themselves.

The difference between the largest and smallest values in a data set.

The least squares 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 passes through the mean point $(\bar{x},\bar{y})$.

The least squares line $x=cy+d$ that minimises the sum of the squares of the horizontal distances from the data points to the line. It is used to predict $x$ from a given value of $y$.

The number of times an event occurs divided by the number of trials. It is used as an estimate of the probability of the event.

The difference between an observed value and the value predicted by a regression line, $y-\widehat{y}$ for the line of $y$ on $x$; on a scatter diagram it is the vertical distance from a point to the line.

A subset of a population from which data are collected. In a random sample every member of the population has a known, non-zero chance of being chosen.

The set of all possible outcomes of a trial, written $U$. The number of outcomes in it is $n(U)$.

A grid that lists every outcome of two combined trials, such as rolling two dice, with one trial along each side.

A list of all the members of a population, from which a sample is selected.

Choosing items one after another without returning each to the population, so the probabilities for later choices change. With replacement, each item is returned and the probabilities stay the same.

A graph of bivariate data in which each pair of values $(x,y)$ is plotted as a single point, used to see whether and how the two variables are related.

The probability that a test gives a positive result for someone who has the condition, $P(\text{positive}\mid \text{condition})$.

A sampling method in which every possible sample of the chosen size is equally likely to be selected, so every member of the population has the same chance of being chosen.

Imitating a random process with random numbers (from a calculator, a computer, dice or a table) so that a probability can be estimated by relative frequency.

The lack of symmetry in a distribution. A positively skewed distribution has a long tail to the right (high values); a negatively skewed one has a long tail to the left.

The probability that a test gives a negative result for someone who does not have the condition, $P(\text{negative}\mid \text{no condition})$.

The square root of the variance, $\sigma$: a measure of 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, $\sigma =\sqrt{\mathrm{Var}(X)}$, measured in the same units as $X$.

The normal distribution with mean $0$ and standard deviation $1$, written $Z\sim N(0,1)$. Standardising any normal variable gives a variable with this distribution.

Converting a value or a random variable into a $z$-score by subtracting the mean and dividing by the standard deviation, $Z=\frac{X-\mu }{\sigma }$, so that different normal distributions can be compared on one scale.

A sampling method in which the population is divided into groups (strata) and a random sample is taken from each group, in proportion to the size of the group.

A sampling method in which members are chosen at regular intervals from an ordered list, starting from a randomly chosen position among the first $k$, where $k=\frac{N}{n}$.

A probability found by reasoning about the outcomes rather than by experiment, for example $P(A)=\frac{n(A)}{n(U)}$ when all outcomes are equally likely.

A diagram of branches showing the outcomes of successive trials and their probabilities. Probabilities are multiplied along a path and added between paths.

A single performance of an experiment whose result is uncertain, such as one roll of a die or one toss of a coin.

A table of frequencies classified by two categories at once, with row and column totals, used to find probabilities of combined and conditional events.

The event $A\cup B$ ("$A$ or $B$"), containing the outcomes that are in $A$, in $B$, or in both.

The mean of the squared deviations from the mean, ${\sigma }^2=\frac{\sum f(x-\mu {)}^2}{n}$. Its units are the square of the data's units.

The expected squared distance of a random variable from its mean, $\mathrm{Var}(X)=E((X-\mu {)}^2)=E(X^2)-[E(X){]}^2$. It measures the spread of the distribution.

A diagram in which a rectangle represents the sample space $U$ and circles inside it represent events; overlapping regions show outcomes that belong to more than one event.

The number of standard deviations that a value $x$ lies from the mean, $z=\frac{x-\mu }{\sigma }$. A positive $z$-score means the value is above the mean; a negative one means it is below.

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.