Saddle pointHL only · also saddle, saddles, saddle points
An equilibrium point of a linear system whose eigenvalues are real with opposite signs; trajectories approach it along the eigenvector of the negative eigenvalue and leave along the eigenvector of the positive eigenvalue, so it is unstable.
Sample space diagramalso possibility diagram, possibility space, sample space grid, sample space diagrams
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 number by which a stretch or enlargement multiplies distances; for a stretch these are distances from its invariant line, which a scale factor greater than 1 increases and a scale factor between 0 and 1 reduces.
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.
Second derivativeHL only · also f''(x), f″(x), d²y/dx², second-order derivative, second derivatives
The derivative of the derivative of a function, written f′′(x) or dx2d2y
Second derivative testHL only · also second-derivative test, f″ test, second derivative test for maxima and minima
A test for the nature of a stationary point at x=a, where f′(a)=0: if f′′(a)<0
Second-order differential equationHL only · also second order DE, second-order DE, second order differential equation, second-order ODE, second-order differential equations
A differential equation in which the highest derivative of the unknown function is the second derivative, such as x¨=f(x,x˙,t).
Semi-log graphHL only · also semi-log plot, log-linear graph, semi-logarithmic graph, semi-log graphs
A graph in which one axis (usually the vertical) is logarithmic and the other is linear, or equivalently a graph of lny (or logy) against x; exponential relationships appear on it as straight lines.
SI unitalso SI units, International System of Units, SI
A unit of the International System of Units, the internationally agreed system of measurement whose base units include the metre (length), the kilogram (mass) and the second (time).
Sierpinski triangleHL only · also Sierpinski gasket, Sierpiński triangle, Sierpinski sieve
The fractal obtained from a solid triangle by repeatedly removing the middle triangle joining the midpoints of the sides of every remaining triangle; it is made of three copies of itself, each enlarged with scale factor 21.
Sigma notationalso summation notation, capital sigma
A way of writing a sum as ∑k=abuk, meaning the sum of the terms uk
Sign conventionalso cash-flow sign convention, TVM sign convention, signs in the TVM solver
The rule in a finance solver that money paid out by the person whose point of view is taken is entered as a negative number, and money received by them as a positive number.
A number line for x marked with the values where a function (usually f′) is zero, showing whether the function is positive or negative on each interval between them.
Signed areaHL only · also net area, negative integral, net signed area, signed areas
The value of the definite integral ∫abf(x)dx, in which regions above the x-axis count as positive and regions below the x
Significance levelalso α, alpha, level of significance
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.
Similar solidsalso similar shapes, volume scale factor
Solids with the same shape, in which every length of one is k times the corresponding length of the other, so that corresponding areas are in the ratio k2 and volumes in the ratio k3.
Simple harmonic motionHL only · also SHM, simple harmonic oscillator, harmonic motion
Motion in which the acceleration is proportional to the displacement from equilibrium and directed towards it, x¨=−ω2x with ω>0; it is a periodic oscillation with period ω2π
Simple interest
Interest calculated only on the original principal, so that the same amount is added in every period; on a principal C at r% a year, the total interest over n years is I=100Crn
Simple random samplingalso simple random sample, simple random samples, SRS
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.
A method of numerical integration that estimates ∫abydx by fitting parabolas through groups of three neighbouring points, using an even number n of strips of width h.
Simulationalso simulations, simulate, Monte Carlo method, Monte Carlo simulation
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.
Sinusoidal modelalso sinusoidal function, sine model, cosine model, sinusoid
A model of the form f(x)=asin(bx)+d or f(x)=acos(bx)+d
SIR modelHL only · also susceptible–infected–recovered model, compartmental model, SIR equations, susceptible-infected-recovered model
A coupled system of differential equations for an epidemic in which a population is divided into susceptible (S), infected (I) and recovered (R) groups, with new infections at a rate proportional to SI and recoveries at a rate proportional to I.
Introduced in
Sitealso sites, seed, seeds, generator, generators, Voronoi site
One of the given points around which a Voronoi diagram is built, such as a town, a shop, a phone mast or a weather station.
Sketchalso sketching, sketch a graph, sketches, sketch graph
To represent a function by a graph, labelled as appropriate, that gives a general idea of its shape and shows its relevant features, without needing to be drawn to scale.
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.
The distance from the apex of a cone or pyramid to the edge of its base, measured along the sloping surface; for a pyramid it is the height of a triangular face, measured from the midpoint of a base edge.
A control in graphing software that changes the value of a parameter continuously, redrawing the graph at once so that the effect of the parameter on its key features can be seen.
Slope fieldHL only · also direction field, slope fields, direction fields
A diagram of a first-order differential equation dxdy=f(x,y) made of short line segments drawn at the points of a grid, each with gradient equal to f(x,y)
Smooth curvealso smooth curves, smooth
A continuous curve drawn freehand through plotted points without corners, gaps or straight-line segments, following the gradual bending of the function between the points.
Solution curveHL only · also integral curve, solution curves, integral curves
The graph of a particular solution of a differential equation; on a slope field it is a smooth curve that is tangent to the segments at every point it passes through.
In the HL use of mathematics criterion, mathematics at the level of the HL course, or SL mathematics used in a more complex way than an SL student could reasonably be expected to manage.
Space diagonalalso diagonal of a cuboid
A line segment joining two vertices of a solid that do not lie on the same face, passing through the inside of the solid; in a cuboid it joins opposite corners.
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.
The solid made of all the points within a fixed distance r (the radius) of a fixed point (the centre); its surface is the set of points exactly r from the centre.
Stable equilibriumHL only · also stable, sink, attractor, stable equilibrium point, asymptotically stable
An equilibrium point that every trajectory starting close to it approaches as t→∞; for x˙=Mx this happens when both eigenvalues are negative or are complex with negative real part.
Standard normal distributionHL only · also Z ~ N(0, 1), z-value, z-distribution, standard normal, z-score distribution
The normal distribution with mean 0 and variance 1, written Z∼N(0,1); any normal variable X∼N(μ,σ2)
Standard positionHL only · also angle in standard position, measured anticlockwise from the positive x-axis
An angle drawn with its vertex at the origin, its first arm along the positive x-axis, and measured anticlockwise to its second arm; a negative angle is measured clockwise.
State vectorHL only · also state vectors, state matrix, column state matrix, state probability vector
A column vector sn 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.
Steady stateHL only · also steady-state vector, steady state vector, steady state probability vector, stationary distribution, long-term state
A state vector s of a Markov chain with Ts=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.
Strip widthalso interval width, step width, width of a strip
The equal width h of each of the n strips into which the interval a≤x≤b is divided for the trapezoidal rule, given by h=nb−a
Strongly connected graphHL only · also strongly connected
A directed graph in which, for every ordered pair of vertices, there is a sequence of edges, each followed in its own direction, leading from the first vertex to the second.
Successalso failure, probability of success, successes
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.
Sum of functionsalso (f + g)(x), sum of two functions, adding functions
The function f+g whose output at each input is the sum of the outputs of f and g, so (f+g)(x)=f(x)+g(x)
Sum of square residuals (SS_res)HL only · also SS_res, SSres, sum of squared residuals, residual sum of squares, SSR
The sum of the squares of the residuals of a model, SSres=∑(yi−y^i)2
Sum rulealso term-by-term differentiation, differentiating term by term, difference rule, sum and difference rule
The derivative of a sum or difference of functions is the sum or difference of their derivatives, so dxd(f(x)±g(x))=f′(x)±g′(x)
Sum to infinityHL only · also S∞, sum of an infinite geometric series, sum of an infinite geometric sequence, sums to infinity
The limit of the partial sums Sn of an infinite series as n increases without bound, when that limit exists; for a geometric series with ∣r∣<1 it is S∞=1−ru1
Supplementary anglesHL only · also supplement, supplementary, sin(180° − θ) = sin θ
Two angles whose sum is 180° (π radians); supplementary angles have equal sines and cosines of opposite sign.
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 square matrix that is unchanged when reflected in its leading diagonal, so the entry in row i, column j equals the entry in row j, column i for every i and j
System of linear equationsalso simultaneous equations, linear system, systems of equations, system of equations, linear systems
A set of two or more linear equations in the same unknowns, all of which must be satisfied by the same values at the same time.
Systematic samplingalso systematic sample, systematic random sampling
A method of choosing a sample by selecting every kth member from an ordered list of the population, starting from a randomly chosen member among the first k.