- absolute value
The absolute value \(|x|\) of a real number \(x\) is the magnitude or size of \(x\) without regard to its sign, ie the distance of the number from the origin. Also known as the modulus.

Formally, \(|x|= x\) if \(x \ge 0\), \(|x|\) = \(-x\) if \(x < 0\).

- acceleration
Acceleration is the rate of change of velocity with respect to time. Examples of units used to measure acceleration include \(km/h^2\) or \(m/s^2\).

- ambiguous case
The ambiguous case refers to using the sine rule to calculate the size of an angle in a triangle where there are two possibilities for the angle, one obtuse and one acute, leading to two possible non-congruent triangles.

- amplitude
The amplitude of a trigonometric function, such as \(y = \sin x\) or \(y = \cos x\), is the distance from the centre (of motion) of the function to its peak or trough (ie its maximum or minimum values). For example, the functions \(y = A\sin x\) and \(y = A\cos x\) have amplitude \(|A|\).

- angle of inclination
The angle of inclination of a straight line is the angle the line makes with the \(x\)-axis, with the angle being measured anti-clockwise from the positive direction of the \(x\)-axis.

- annuity
An annuity is a compound interest investment from which payments are made or received on a regular basis for a fixed period of time.

- anti-differentiation
An anti-derivative, primitive or indefinite integral of a function \(f(x)\) is a function \(F(x)\) whose derivative is \(f(x)\), i.e. \(F'(x) = f(x)\).

The process of finding anti-derivatives is called anti-differentiation.

Anti-derivatives are not unique. If \(F(x)\) is an anti-derivative of \(f(x)\), then so too is the function \(F(x) + c\) where \(c\) is any number. We write \(\int {f(x) \, dx} = F(x) + c\) to denote the set of all anti-derivatives of \(f(x)\). The number \(c\) is called the constant of integration. For example, since \(\frac{d}{dx} \left( x^3 \right) = 3x^2\), we can write \(\int {3x^2 \, dx} = x^3 + c\).

- area of a triangle
The area of any triangle \(ABC\) is given by:

\[\text{\(Area\)} = \frac{1}{2}ab\sin C \]

\[(\text{or alternatively, \(Area\) =} \frac{1}{2}(\text{\(base\)} \times \text{\(perpendicular\) \(height\)}))\]

- arithmetic sequence
An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 2, 5, 8, 11, 14, 17, \( \ldots \) is an arithmetic sequence with common difference 3.

- arithmetic series
An arithmetic series is any sum whose terms form an arithmetic sequence.

- array
An array is an ordered collection of objects or numbers.

- asymptote
An asymptote to a curve is a line that the curve begins to imitate at infinity.

For example, the line with equation \(x=\frac{\pi}{2}\) is a vertical asymptote to the graph of \(y=\tan x\), and the line with equation \(y=0\) is a horizontal asymptote to the graph of \(y=\frac{1}{x}\).

- bearing
A bearing is a measure of direction from one point to another. Two types of bearings may be used: compass bearings and true bearings.

- bias
Bias depends upon the context but may generally refer to a systematic favouring of certain outcomes more than others, due to unfair influence (knowingly or otherwise).

- bivariate data
Bivariate data is data relating to two variables that have both been measured on the same set of items or individuals. For example, the arm spans and heights of 16-year-olds, the sex of primary school students and their attitude to playing sport.

- box- plot
A box-plot is a graphical display of a five-number summary.

In a box-plot, the ‘box’ (a rectangle) represents the interquartile range (IQR), with ‘whiskers’ reaching out from each end of the box towards maximum and minimum values in the data set (sometimes these values are represented beyond the extending whiskers individually to assist with assessing potential outliers). A line in the box is used to indicate the location of the median. Also known as a box-and-whisker plot.

- break-even point
The break-even point is the point at which income begins to exceed the cost of production for business.

- categorical data
Data associated with a categorical variable are called categorical data. Categorical data are sometimes called qualitative data.

- categorical variable
A categorical variable is a variable whose values are categories.

Examples include major blood type (A, B, AB or O) or principal construction type (brick, concrete, timber, steel, other).

Categories may have numerical labels, eg the numbers worn by players in a sporting team, but these labels have no numerical significance, they merely serve as labels.

- centre of motion
The centre of motion of a trigonometric function is the mean value that represents the centre of the graph. For example, the centre of motion of \(y = \cos x + k\) is \(k\).

- chain rule
The chain rule is a method for differentiating composite functions. It states that if \(h(x)=f\left(g(x)\right)\) then \(h'(x)=f'\left(g(x)\right)g'(x)\)

The chain rule is also expressed using notation \(\frac{du}{dx}=\frac{du}{dy} \times \frac{dy}{dx}\)

- compass bearing
Compass bearings are specified as angles either side of north or south. For example, a compass bearing of \(\text{N}50^\circ \text{E}\) is found by facing north and moving through an angle of \(50^\circ\) to the east.

- complement
The complement of an event refers to when the event does

**not**occur. For example, if \(A\) is the event of throwing a 5 on a die, then the complement of \(A\), denoted by \(\bar A\) or \(A^c\), is**not**throwing a 5 on a die.- composite functions
In a composite function, the output of one function becomes the input of a second function.

More formally, the composite of \(f\) and \(g\), acting on \(x\), can be written as \(\left(f\circ g\right)\)\((x)\) and means \(f\left(g(x)\right)\), with \(g(x)\) being performed first.

For example, if \(f(y)=\sqrt y\) and \(g(x)=x^2+3\), then \( (f \circ g )(x) = \sqrt{x^2+3} \).

- compound interest (and formula)
The interest earned by investing a sum of money (the principal) is called compound interest when each successive interest payment is added to the principal before calculating the next interest payment.

If the principal \(\$P\) earns compound interest at the rate of \(r\) per period as a decimal, then after \(n\) equal periods the principal plus interest (\(\$A\)) is given by the compound interest formula \(A=P(1+r)^n\).

- concavity
The concavity of a function is the rate of turning of the tangent. If the gradients of the tangents are decreasing, the function is concave down and if the gradients of the tangents are increasing, the function is concave up.

- conditional probability
The probability that an event \(A\) occurs can change if it becomes known that another event \(B\) occurs. The new probability is known as conditional probability and is written as \(P(A|B)\). If \(B\) has occurred, the sample space is reduced by discarding all outcomes that are not in the event \(B\).

The conditional probability of event \(A\) is given by \(P\left(A|B\right)=\frac{P(A \cap B)}{P(B)}\).

- continuity
*See*continuous function and discontinuous function.- continuous function
A function is continuous when sufficiently small changes in the input result in arbitrarily small changes in the output, ie its graph is a single unbroken curve.

- continuous random variable
A continuous random variable is a numerical variable that can take any value that lies along a continuum.

- cosine rule
The cosine rule for any triangle \(ABC\) is given by:

\[c^2 = a^2 + b^2 -2ab\cos C\].

- cumulative distribution function
Given a continuous random variable \(X\), the cumulative distribution function \(f(x\)) is the probability that \(X\) ≤ \(x\)

- cumulative frequency
The cumulative frequency is the accumulating total of frequencies within an ordered dataset.

- deciles
Deciles divide an ordered dataset into ten equal parts. See also quantiles.

- degree of a polynomial
The degree of a polynomial in one variable is the highest power of \(x\) occurring in the polynomial.

- dependent variable
A dependent variable in a function is the variable used to represent the output values of a function. The dependent variable is generally represented on the vertical axis of a graph.

- derivative
The derivative \(\frac{dy}{dx} = f'(x)\) of a function \(f(x)\) can be defined as \(\displaystyle{\lim_{h\to0} \frac{f\left(x+h\right) - f\left(x\right)}{h}}\). Also known as the gradient function, the first derivative or the rate of change of \(y\) with respect to \(x\).

- direct variation
Two variables are in direct variation if one is a constant multiple of the other. This can be represented by the equation \(y=kx\), where \(k\) is the constant of variation (or proportion). Also known as direct proportion, it produces a linear graph through the origin.

- discontinuous function
A discontinuous function has at least one break or gap in the graph of the function.

- discrete random variable
A discrete random variable is a numerical variable whose values can be listed.

Examples include the number of children in a family, shoe size or the number of days in a month.

- discriminant
The discriminant of a quadratic equation \(ax^2+bx+c=0\) is given by \(b^2-4ac\) and is used to identify the nature of the roots of the quadratic equation.

- domain
The domain of a function is the set of values of the independent variable for which the function is defined. Also known as the ‘input’ of a function.

*See also*function.- Euler’s number
Euler’s number \(e\) is an important irrational number used as the base of the natural logarithm. It is approximately equal to 2.71828 and can be defined by the limit: \( \displaystyle{ e \equiv \lim_{n \rightarrow \infty} \left( 1 + \frac1n \right)^{n} } \)

- even function
An even function has line symmetry about the vertical axis. Algebraically, a function is even if \(f\left(-x\right) = f\left(x\right)\), for all values of \(x\) in the domain.

- event
An event is a set of outcomes for a random experiment.

- expected value
In statistics, the expected value \(E(X)\) of a random variable \(X\) is a measure of the central tendency of its distribution. Also known as the expectation or mean. \(E(X)\) is calculated differently depending on whether the random variable is discrete or continuous.

- exponential function
An exponential function is a function in which the independent variable occurs as an exponent (or power/index) with a positive base. For example, \(y=2^x\) is an exponential function with \(x\) as the independent variable. The exponential function is defined to be \(y=e^x\) where \(e\) is Euler’s number.

- exponential graph
- An exponential graph is a graph drawn to represent an exponential function.
- exponential growth and decay
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value, resulting in its growth with time being an exponential function. Exponential decay occurs in the same way when the growth rate is negative.

- extrapolation
Extrapolation occurs when the fitted model is used to make predictions using values that are outside the range of the original data upon which the fitted model was based. Extrapolation far beyond the range of the original data is a dangerous process as it can sometimes lead to quite erroneous predictions.

- five-number summary
A five-number summary is a method of summarising a set of data using five summary statistics: the minimum value, the first quartile, the median, the third quartile and the maximum value.

- function
A function \(f\) is a rule that associates each element \(x\) in a set \(S\) with a unique element \(f(x)\) which we place in a set \(T\).

The functions most commonly encountered in elementary mathematics are real functions of real variables. For such functions, the set \(S\) is called the domain of \(f\) and the set \(T\) is called the range of \(f\). If we write \(y=f(x)\), then we say that \(x\) is the independent variable and \(y\) is the dependent variable.

For example, the formula \(f(x)=x^2\) defines the ‘squaring function’ that maps each real number \(x\) to its square \(x^2\).

- future value
The future value of an investment or annuity is the total value of the investment at the end of the term of the investment, including all contributions and interest earned.

- future value interest factors
Future value interest factors are the values of an investment at a specific date. A table of these factors can be used to calculate the future value of different amounts of money that are invested at a certain interest rate for a specified period of time.

- geometric sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio.

- geometric series
A geometric series is a sum whose terms form a geometric sequence.

- gradient
The gradient of a line is the steepness or slope of the line and can be measured using any two points on the line/interval.

Formally, if \(A\,(x_1,y_1)\) and \(B\,(x_2,y_2)\) are points in the Cartesian plane, where \(x_2-x_1 \ne 0\), the gradient of the line segment (or interval) \(AB\) is given by \(m=\frac{y_2-y_1}{x_2-x_1}\). If \(x_1 = x_2\), the line is said to have infinite gradient.

- Heron’s formula
Heron’s formula determines the area of a triangle given the lengths of its sides as \(a,b,c\). The formula is given by: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s=\frac{a+b+c}{2}\), called the semi-perimeter.

- horizontal line test
The horizontal line test is a method that can be used to determine whether a function is a one-to-one function. If any horizontal line intersects the graph of a function more than once then the function is not a one-to-one function.

- horizontal point of inflexion
A horizontal point of inflexion is a point of inflexion where the gradient of the curve is zero.

- identity
An identity is an equation involving a variable(s) that is true no matter what values are substituted for the variable(s). Identities can be used to manipulate the format of an expression.

- independent events (independence)
In probability, two events are independent of each other if the occurrence of one does not affect the probability of the occurence of the other. That is, \( P( A \vert B ) = P(A) \).

- independent variable
An independent variable in a function is the variable used to represent the input values of the function. The independent variable is generally represented on the horizontal axis of a graph.

- instantaneous rate of change
The instantaneous rate of change is the rate of change at a particular moment. For a differentiable function, the instantaneous rate of change at a point is the same as the slope/gradient of the tangent to the curve at that point. This is defined to be the value of the derivative at that particular point.

- interpolation
Interpolation occurs when a fitted model is used to make predictions using values that lie within the range of the original data.

- interquartile range (IQR)
The interquartile range \((IQR)\) is a measure of the spread within a numerical dataset. It is equal to the upper quartile \((Q_3)\) minus the lower quartile \((Q_1)\); that is, \(IQR=Q_3-Q_1\).

- interval notation
Interval notation is a notation for representing the endpoints of an interval as a pair of numbers or indicating a range by using the symbol for infinity. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. For example, \([3,8)\) is the interval of real numbers between 3 and 8, including 3 and excluding 8 or \(x\ge3\) can be represented as \([3,\infty)\).

- inverse variation
Inverse variation (or inverse proportion) exists when, as one variable increases, the other variable decreases. For example, if \(y\) is said to be 'directly inversely proportional' to \(x\), the equation is of the form \(y=\frac{k}{x}\), where \(k\) is a constant of variation (or proportion). If \(y\) is said to be inversely proportional to the square of \(x\), then \(y=\frac{k}{x^2}\)

- irrational number
An irrational number is a real number that cannot be expressed in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\ne0\).

- least-squares regression line
Least-squares regression is a method for finding a straight line that summarises the relationship between two variables, within the range of the dataset.

The least-squares regression line is the line that minimises the sum of the squares of the residuals. Also known as the least-squares line of best fit.

- limit
The limit of a function at a point \(a\), if it exists, is the value the function approaches as the independent variable approaches \(a\) .

The notation used is \[ \lim_{x\to a} \ {f\left(x\right)} = L \] ie the limit as \(x\) approaches \(a\) is \(L\).

- limiting sum of a geometric series
The limiting sum of a geometric series is the value (if any) that the series approaches as the number of terms increases without bound.

This value is found using the formula \(S = \frac{a}{1-r}\), provided \(|r|<1\).

- line of best fit
A line of best fit is a line drawn through a scatterplot of data points that represents the nature of the relationship between two variables.

- linear relationship/linear function
Two variables \(x\) and \(y\) are in a linear relationship (or form a linear function) if they are connected by an equation of the form \(y=mx+c\). Graphically, \(m\) is the gradient and \(c\) is the intercept with the vertical axis of the corresponding linear graph.

- local and global maximum and minimum
We say that \(f(x_0)\) is a local maximum of the function \(f(x)\) if \(f(x) \le f(x_0)\) for all values of \(x\) near \(x_0\). We say that \(f(x_0)\) is a global maximum of the function \(f(x)\) if \(f(x)\le f(x_0)\) for all values of \(x\) in the domain of \(f\).

We say that \(f(x_0)\) is a local minimum of the function \(f(x)\) if \(f(x)\ge f(x_0)\) for all values of \(x\) near \(x_0\). We say that \(f(x_0)\) is a global minimum of the function \(f(x)\) if \(f(x)\ge f(x_0)\) for all values of \(x\) in the domain of \(f\).

- logarithm
The logarithm of a positive number \(x\) to the base \(a\) is the power to which the given positive number \(a\) must be raised in order to produce the number \(x\). The logarithm of \(x\) to the base \(a\) is denoted by \(\log_ax\).

Algebraically: \(\log_ax = y \Leftrightarrow a^y = x\).

A logarithmic function has the form \(y=\log_a x\) where \(a\) is the base of the logarithm \((a>0)\).

- mean (average)
There are a number of different types of means used in mathematics and statistics. When dealing with a group of numbers, their mean (or arithmetic mean) is defined as the sum of these values divided by the number of values. Also known as their average.

- mean (discrete random variable)
*See*expected value.- measures of central tendency
Given a dataset, the measures of central tendency give values about which the data are scattered, or a measure of the centre of the data. Also known as measures of location. The two most common measures of central tendency are the mean and the median.

- measures of spread
Given a numerical dataset, its measures of spread describe how spread out the data is. Common measures of spread include the range, combinations of quantiles (deciles, quartiles, percentiles), the interquartile range, variance and standard deviation.

- median
The median of an ordered numerical dataset is the value that divides it into two equal parts. When the number of data values is odd, the median is the middle data value. When the number of data values is even, the median is the average of the two middle data values. The median as a measure of central tendency is suitable for both symmetric and skewed distributions as it is relatively unaffected by outliers.

- modality
Modality describes the number of modes in a set of data.

For example, data can be unimodal (having one mode), bimodal (having two modes) or multimodal (having many modes).

- mode
The mode is the most frequently occurring value in a set of data. There can be more than one mode. When a data set has one mode, it is called unimodal, and when it has more than one, it is called multi-modal.

- Newton’s laws of motion
Newton’s laws of motion consist of three fundamental laws of classical physics. The first states that a body continues in a state of rest or uniform motion in a straight line unless it is acted on by an external force. The second states that the rate of change of momentum of a moving body is proportional to the force acting to produce the change. The third states that if one body exerts a force on another, there is an equal and opposite force (or reaction) exerted by the second body on the first.

- normal distribution
The normal distribution is a type of continuous distribution where the mean, median and mode are equal and the scores are symmetrically arranged either side of the mean. The graph of a normal distribution is often called a ‘bell curve’ due to its shape, as shown below.

Formally, the normal distribution is defined by the probability density function:

\(\displaystyle{f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}}\), where \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation.

- normal random variable
A normal random variable is a variable which varies according to the normal distribution.

- normal (to a curve)
In calculus, the normal to a curve at a given point \(P\) is the straight line that is perpendicular to the tangent to the curve at that point.

- numerical data
Numerical data are data associated with a numerical variable. Also known as quantitative data.

- numerical variable
Numerical variables are variables whose values are numbers. Numerical variables can be either discrete or continuous.

- odd function
An odd function has point symmetry about the origin. Algebraically, a function is odd if \(f\left(-x\right)=-f\left(x\right)\), for all values of \(x\) in the domain.

- one-to-one function
In a one-to-one function, every element in the range of a function corresponds to exactly one element of the domain.

- outcome
An outcome is a single possible result from an experiment.

- outlier
An outlier in a dataset is a data value that appears to be inconsistent with the remainder of that dataset.

- Pareto chart
A Pareto chart is a type of chart that contains both bars and a line graph, where individual values are represented in descending order by the bars and the cumulative total is represented by the line graph.

- Pearson’s correlation coefficient
Pearson’s correlation coefficient is a statistic that measures the strength of the linear relationship between a pair of variables or datasets. Its value lies between -1 and 1 (inclusive). Also known as simply the correlation coefficient. For a sample, it is denoted by \(r\).

- percentiles
Percentiles divide an ordered dataset into 100 equal parts. See also quantiles.

More formally, it is a statistical measure indicating the value below which a given percentage of observations in a group of observations lie. For example, the 20th percentile is the value below which 20% of the observations may be found.

- period
The period of a periodic function is the distance required for the function to complete one full cycle.

- phase
When a trigonometric function is translated horizontally, the phase (or phase shift) is the magnitude of this translation.

For example, the phase of \(y=\sin\,(x-c)\) is \(c\).

- point of inflexion
A point of inflexion is a point on a curve at which the curve changes concavity, ie from being concave down to concave up, or vice versa.

- polynomial
A polynomial is an expression of the form \(a_nx^n + \cdots + a_2 x^2 + a_1 x + a_0 \), where \(n\) is a non-negative integer and \(a_n\ne0\).

Examples of polynomials include \(2x-5\), \(x^3-4x+7\), \(x^5+2x^3-x+8\), which are polynomials of degree 1, 3 and 5 respectively.

- population
The population in statistics is the entire dataset from which a statistical sample may be drawn.

- power function
A power function is a function of the form \(f(x) = kx^n\), where \(k\) and \(n\) are real numbers.

- present value
The present value of an investment or annuity is the single sum of money (or principal) that could be initially invested to produce a future value over a given period of time.

- probability density function
A probability density function of a continuous random variable is a function that describes the probabilities that the random variable lies within some range.

Formally, if \(p(x)\) is the probability density of the continuous random variable \(X\), then the probability that \(X\) takes a value in some interval \([a,b]\) is given by \(\int_a^bp(x)\,dx\).

- probability distributions
The probability distribution function identifies the possible values of a discrete random variable and its associated probabilities, along with each of its possible values.

- product rule
The product rule is a method for finding the derivative of the product of two functions.

If \(h(x) = f(x)g(x)\) then \(h'(x) = f(x)g'(x)+f'(x)g(x)\).

Or, using Leibniz notation, \(\frac{d}{dx}(uv) = u\frac{dv}{dx}+v\frac{du}{dx}\)

- quadratic equation
A quadratic equation is a polynomial equation of degree 2. The general quadratic equation in one variable is \(ax^2+bx+c=0\), where \(a\ne0\).

- quadratic function
A quadratic function is a function of the form \(y=ax^2+bx+c\), where \(a\ne0\). For example, \(y=3x^2+7\).

- quantiles
Quantiles are a set of values that divide an ordered data set into equal groups. Examples include quartiles, deciles and percentiles.

Formally in statistics, quantiles are cutpoints dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way.

- quartiles
Quartiles divide an ordered dataset into four equal parts.

There are three quartiles. The first or lower quartile \((Q_1)\), divides off (approximately) the lowest 25% of data values. The second quartile \((Q_2)\) is the median. The third or upper quartile \((Q_3)\), divides off (approximately) the highest 25% of data values.

*See also*quantiles.- quotient rule
The quotient rule is a method to find the derivative of the quotient of two functions.

If \( \displaystyle{ h(x) = \frac{f(x)}{g(x)} } \) then \( \displaystyle{ h'(x) = \frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2} } \).

Or, using Leibniz notation, \( \displaystyle{ \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2} } \)

- radian
A radian is a unit of angular measure. One radian is defined to be the angle subtended at the centre of a unit circle by an arc of length 1 unit.

- random variable
A random variable is a variable whose possible values are outcomes of a statistical experiment or a random phenomenon.

- range (of data)
The range is the difference between the largest and smallest observations in a dataset. It is sensitive to outliers.

- range (of function)
The range of a function is the set of \(y\) values in the graph of the function \(y=f(x)\). Also known as the ‘output’ of a function. See also function.

- rate of change
A rate of change of a function, \(y = f(x)\) is \( \frac{\Delta y}{\Delta x} \) where \( \Delta x \) is the change in \(x\) and \( \Delta y \) is the corresponding change in \(y\).

- rational number
A rational number is a real number that can be expressed in the form \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\ne0\).

- real number
The set of real numbers consists of the set of all rational and irrational numbers.

- reciprocal trigonometric functions
The reciprocal trigonometric functions are defined as follows:

\(\sec A = \frac{1}{\cos A},\cos A \ne0\)

\(\text{cosec}\,A = \frac{1}{\sin A}, \sin A\ne0\)

\(\cot A = \frac{1}{\tan A} =\frac{\cos A}{\sin A},\sin A \ne0\)- recurrence relation
A sequence of numbers satisfies a recurrence relation when each term can be found using some or all of the previous terms. Examples include compound interest and annuities.

- reducing balance loan
A reducing balance loan is a compound interest loan where the loan is repaid by making regular payments and the interest paid is calculated on the amount still owing (the reducing balance of the loan) after each payment is made.

- relation
Graphically, a relation between two real variables \(x\) and \(y\) is the set of ordered pairs \((x,y)\) that can be plotted on the Cartesian number plane.

Functions are a subset of the set of relations.

*See*the vertical line test.- relative frequency
Relative frequency is the proportion of times that an event has occurred in a repeated experiment. If an event \(E\) occurs \(r\) times when a chance experiment has been repeated \(n\) times, then the relative frequency of \(E\) is \(\frac{r}{n}\).

- residuals
The residual of an observed value is the difference between the observed value and the estimated value of the quantity of interest.

- sample
A sample is a subset of data from a statistical population.

- sample space
The sample space of an experiment is the set of all possible outcomes for that experiment.

- scatterplot
A scatterplot is a two-dimensional data plot using Cartesian coordinates to display the values of two variables in a bivariate dataset. Also known as a scatter graph.

- secant (line)
A secant is the straight line passing through two points on the graph of a function.

- second derivative
The second derivative is the derivative of the first derivative. It is denoted by \(f^{\prime\prime}(x)\) or \( \displaystyle{ \frac{d^2y}{dx^2} } \).

- sequence
In mathematics, a sequence is a set of numbers whose terms follow a prescribed pattern. Examples of mathematical sequences include arithmetic sequences and geometric sequences.

- series
A series is the sum of the terms of a particular sequence. The \(n\)th term of a series is the sum of the first \(n\) terms of the related sequence.

- set language and notation
A set is a collection of distinct objects called elements.

The language and notation used in the study of sets includes:

A set is a collection of objects, for example, '\(A\) is the set of the numbers 1, 3 and 5' is written as \(A=\lbrace 1,3,5\rbrace\).

Each object is an element or member of a set, for example, '1 is an element of set \(A\)' is written as \( 1 \in A \).

The number of elements in set \(A=\lbrace 1,3,5\rbrace\) is written as \( n\left(A\right) = 3 \), or \(|A|\) = 3.

The empty set is the set with no members and is written as \( \lbrace \rbrace \) or \(\varnothing\).

The universal set contains all elements involved in a particular problem.

\(B\) is a subset of \(A\) if every member of \(B\) is a member of \(A\) and is written as \( B \subset A \), ie '\(B\) is a subset of \(A\)'. \(B\) may also be equal to \(A\) in this scenario, and we can therefore write \(B \subseteq A\).

The complement of a set \(A\) is the set of all elements in the universal set that are not in \(A\) and is written as \( \bar A\) or \(A^c \).

The intersection of sets \(A\) and \(B\) is the set of elements which are in both \(A\) and \(B\) and is written as \( A \cap B \), ie '\(A\) intersection \(B\)'.

The union of sets \(A\) and \(B\) is the set of elements which are in \(A\) or \(B\) or both sets and is written as '\(A \cup B\), ie '\(A\) union \(B\)'.- sine rule
The sine rule for any triangle \(ABC\) is given by: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.\]

- sketch
A sketch is an approximate representation of a graph, including labelled axes, intercepts and any other important relevant features. Compared to the corresponding graph, a sketch should be recognisably similar but does not need to be exact.

- smooth and non-smooth functions
A smooth function is a continuous function for which the derivative is defined at all points in its domain. A non-smooth function has at least one point where the derivative is not defined.

- standard deviation
Generally, standard deviation is a measure of the spread of a dataset, by giving an indication of how far, on average, individual data values are spread around their mean. It is equal to the square root of the variance.

- stationary points
A stationary point on the graph \(y=f(x)\) of a differentiable function is a point where \(f'(x)=0\).

A stationary point could be a local or global maximum or minimum or a horizontal point of inflection.

- subtend
In geometry, an angle subtended by an arc or interval is the angle whose two rays pass through the endpoints of the arc/interval. A possible synonym for ‘subtends’ is ‘makes’.

- summary statistics
Summary statistics are the calculated values representing or summarising a dataset. For example, a five-number summary.

- tangent
The tangent (or tangent line) to a curve at a given point \(P\) can be described intuitively as the straight line that ‘just touches’ (but does not intersect) the curve at that point. At \(P\) the curve has ‘the same direction’ as the tangent. In this sense it is the best straight-line approximation to the curve at point \(P\).

- trapezoidal rule
The trapezoidal rule uses trapezia to approximate the area under a curve.

- tree diagram
A tree diagram is a diagram that can be used to determine the outcomes of a multi-step random experiment. A probability tree diagram has the probability for each stage written on the branches.

- true bearing
True bearings are measured in degrees clockwise from true north and are written with three digits being used to specify the direction. For example, the direction of north is specified \(000^\circ\), east is specified as \(090^\circ\), south is specified as \(180^\circ\) and north-west is specified as \(315^\circ\).

- turning point
A turning point occurs at a local maximum or local minimum point.

- uniform acceleration
Uniform (or constant) acceleration occurs in a type of motion where the rate of change of the velocity is constant.

- unit circle
A unit circle is a circle with radius 1.

- variance
In statistics, the variance \(Var(X)\) of a random variable \(X\) is a measure of the spread of its distribution. \(Var(X)\) is calculated differently depending on whether the random variable is discrete or continuous. It is equal to the square of the standard deviation.

- velocity
Velocity is the speed of an object in a given direction.

- vertical line test
The vertical line test determines whether a relation or graph is also a function. If a vertical line intersects or touches a graph at more than one point, then the graph is not a function.

- \(\boldsymbol{z}\)-score
A \(z\)-score is a statistical measurement of how many standard deviations a raw score is above or below the mean. A \(z\)-score can be positive or negative, indicating whether it is above or below the mean, or zero. Also known as a standardised score.