Maths AA terms: S

54 of the Maths AA glossary’s terms, each with the part of the syllabus where you first meet it.

IB Maths AA glossary: S | Augment Education

S

Samplealso random sample: 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.
Introduced in 4.1 Populations, samples and sampling
Sample spacealso U: The set of all possible outcomes of a trial, written $U$ . The number of outcomes in it is $n(U)$ .
Introduced in 4.5 Concepts of probability
Sample space diagramalso possibility diagram, table of outcomes: A grid that lists every outcome of two combined trials, such as rolling two dice, with one trial along each side.
Introduced in 4.6 Venn diagrams, tree diagrams and combined events
Sampling frame: A list of all the members of a population, from which a sample is selected.
Introduced in 4.1 Populations, samples and sampling
Sampling without replacementalso with replacement: 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.
Introduced in 4.6 Venn diagrams, tree diagrams and combined events
ScalarHL only: A quantity with size only and no direction, such as a mass, a time or a temperature; a real number. Multiplying a vector by a scalar changes its length and, if the scalar is negative, reverses its direction.
Introduced in 3.12 Vectors
Scalar productHL only · also dot product: The number $\boldsymbol{v} \cdot \boldsymbol{w} = v_1w_1 + v_2w_2 + v_3w_3 = |\boldsymbol{v}||\boldsymbol{w}|\cos\theta$
Scalar projectionHL only · also component of a vector along another, scalar resolute: The signed length of the shadow of $\boldsymbol{v}$ on the direction of $\boldsymbol{w}$ : $\frac{\boldsymbol{v} \cdot \boldsymbol{w}}{|\boldsymbol{w}|} = |\boldsymbol{v}|\cos\theta$
Scatter diagramalso scatter graph, scatter plot: 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.
Introduced in 4.4 Linear correlation and regression
Scientific notationalso standard form: A way of writing a number as $a \times 10^k$ , where $1 \le a < 10$ and $k$ is an integer. It makes very large and very small numbers easy to read, compare and calculate with.
Second derivativealso f″(x): The derivative of the derivative, written $f''(x)$ or $\frac{d^2y}{dx^2}$
Second derivative test: A way to classify a stationary point at $x = a$ : if $f''(a) < 0$ it is a local maximum, if $f''(a) > 0$
Sectoralso minor sector, major sector: The region of a circle bounded by two radii and the arc between them. Its area is $A = \frac{1}{2}r^2\theta$ when $\theta$ is in radians.
Introduced in
Segment (of a circle)also minor segment, major segment: The region of a circle bounded by a chord and the arc it cuts off. Its area is the sector area minus the triangle area: $\frac{1}{2}r^2(\theta - \sin\theta)$ .
Introduced in 3.4 Radians, arcs and sectors
Self-inverse functionalso self-inverse: A function that is its own inverse, so $f^{-1}(x) = f(x)$ and $f(f(x)) = x$ for every $x$
SensitivityHL only · also true positive rate: The probability that a test gives a positive result for someone who has the condition, $P(\text{positive} \mid \text{condition})$ .
Introduced in 4.13 Bayes’ theorem
Separation of variablesHL only · also variables separable, separable differential equation: A method for solving a differential equation that can be written as $g(y)\frac{dy}{dx} = f(x)$ : integrate both sides, giving $\int g(y)\,dy = \int f(x)\,dx$
Sequencealso sequences: An ordered list of numbers, such as $2, 5, 8, 11, \ldots$ , in which each number has a position: first, second, third and so on.
Introduced in 1.2 Arithmetic sequences and series
Seriesalso sum of a sequence: The sum of the terms of a sequence, such as $2 + 5 + 8 + 11$ . The sum of the first $n$ terms is written $S_n$
Sheaf of planesHL only · also sheaf: Three or more planes that all contain the same line, like the pages of a partly open book.
Introduced in 3.18 Intersections of lines and planes
Sigma notationalso summation notation, Σ: A short way of writing a sum with the Greek capital letter $\Sigma$ . For example, $\sum_{k=1}^{n} u_k$ means $u_1 + u_2 + \cdots + u_n$
Sign diagramalso sign chart: A number line showing where an expression such as $f'(x)$ is positive, negative or zero, with the values where it is zero (or undefined) marked as boundaries.
Introduced in 5.2 Increasing and decreasing functions
Signed areaalso net area: The value of a definite integral read as an area: regions above the $x$ -axis count as positive and regions below count as negative, so $\int_a^b f(x)\,dx$ is the area above minus the area below.
Introduced in 5.11 Definite integrals and areas
Significant figuresalso s.f., sig figs: The digits of a number that carry meaning about its size, counted from the first non-zero digit. For example, $0.004\,070$ has four significant figures: 4, 0, 7 and the final 0.
Introduced in 1.1 Scientific notation
Simple interest: Interest calculated only on the original amount, so the same interest is added every year. The balances form an arithmetic sequence.
Introduced in 1.4 Financial applications of geometric sequences
Simple random samplingalso simple random sample: 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.
Introduced in 4.1 Populations, samples and sampling
Simulationalso simulate: 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.
Introduced in 4.5 Concepts of probability
Sine rule: In any triangle $ABC$ , $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Sinusoidal functionalso sinusoid, sine wave, sinusoidal model: A function of the form $y = a\sin(b(x + c)) + d$ (or the same with cosine): a sine curve stretched and translated, with a constant amplitude and period.
Introduced in 3.7 Circular functions
Sketch (a graph): To draw a graph by hand that shows its general shape and key features (intercepts, maximum and minimum points, asymptotes, endpoints) labelled with their values, without plotting it to scale.
Introduced in 2.3 The graph of a function
Skew linesHL only · also skew: Two lines in three dimensions that are not parallel and do not intersect. Skew lines cannot lie in a common plane.
Introduced in 3.15 Coincident, parallel, intersecting and skew lines
Skewnessalso positive skew, negative skew, skewed: 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.
Introduced in 4.2 Presenting data
Slope fieldHL only · also direction field: A diagram of short line segments at the points of a grid, each with gradient equal to the value of $\frac{dy}{dx}$ given by a differential equation at that point. Solution curves follow the segments.
Introduced in 5.18 First-order differential equations
SpecificityHL only · also true negative rate: The probability that a test gives a negative result for someone who does not have the condition, $P(\text{negative} \mid \text{no condition})$ .
Introduced in 4.13 Bayes’ theorem
Speed: The magnitude of the velocity, $|v|$ : how fast a particle moves, whatever its direction. Speed is never negative.
Introduced in 5.9 Kinematics
Sphere: The set of all points in space at a fixed distance $r$ , the radius, from a fixed point, the centre. A solid sphere has volume $\frac{4}{3}\pi r^3$ and surface area $4\pi r^2$
Standard deviationalso σ: 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.
Introduced in 4.3 Measures of central tendency and dispersion
Standard deviation of a random variableHL only: The square root of the variance of a random variable, $\sigma = \sqrt{\operatorname{Var}(X)}$ , measured in the same units as $X$
Standard integralHL only · also standard integrals: An integral whose result is known directly from a standard derivative read backwards, such as $\int \sec^2 x\,dx = \tan x + C$ or $\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$
Standard limitHL only · also standard limits: A limit whose value is used so often that it is quoted as a known result, such as $\lim_{x \to 0}\frac{\sin x}{x} = 1$ or $\lim_{x \to 0}\frac{e^x - 1}{x} = 1$
Standard normal distributionalso Z ~ N(0, 1): 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.
Introduced in 4.12 Standardising normal variables
Standardisationalso standardising, standardization: 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.
Introduced in 4.12 Standardising normal variables
Stationary pointalso turning point, critical point: A point on the graph of $y = f(x)$ where the gradient is zero, $f'(x) = 0$ , so the tangent is horizontal. It is a local maximum, a local minimum or a stationary point of inflexion.
Introduced in 5.8 Stationary points, optimisation and inflexion
Stationary point of inflexionalso horizontal point of inflexion: A point of inflexion where the gradient is also zero. The sign of $f'(x)$ is the same on both sides, so the point is stationary but neither a maximum nor a minimum, as at the origin on $y = x^3$ .
Introduced in 5.8 Stationary points, optimisation and inflexion
Step lengthHL only · also step size: The constant amount $h$ by which $x$ increases at each step of a numerical method such as Euler's method.
Introduced in 5.18 First-order differential equations
Stratified samplingalso stratum, strata: 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.
Introduced in 4.1 Populations, samples and sampling
Stretchalso scale factor: A transformation that multiplies the distance of every point from an axis by a scale factor. $y = pf(x)$ is a vertical stretch with scale factor $p$ ; $y = f(qx)$ is a horizontal stretch with scale factor $\frac{1}{q}$
Strict inequalityHL only · also non-strict inequality: An inequality using $<$ or $>$ , so equality is not allowed. A non-strict inequality uses $\le$ or $\ge$ and includes the values where both sides are equal.
Introduced in 2.15 Solving inequalities
Sum rule (for derivatives)also difference rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives: if $f(x) = g(x) \pm h(x)$ , then $f'(x) = g'(x) \pm h'(x)$
Sum to infinityalso S∞, infinite sum: The limit $S_\infty$ that the partial sums $S_n$ of a series approach as $n$ increases without bound. For a geometric series with $|r| < 1$
Surface area: The total area of all the faces and curved surfaces that enclose a solid, measured in square units.
Introduced in 3.1 Distance, midpoint, volume and surface area
Synthetic divisionHL only: A short method for dividing a polynomial by a linear factor $(x - a)$ that works with the coefficients only: bring down, multiply by $a$ , add, and repeat.
Introduced in 2.12 Polynomial functions
System of linear equationsHL only · also simultaneous equations, linear system: A set of two or more linear equations in the same unknowns, to be satisfied at the same time. A solution is a set of values of the unknowns that satisfies every equation.
Introduced in 1.16 Systems of linear equations
Systematic sampling: 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 $k = \frac{N}{n}$ .
Introduced in 4.1 Populations, samples and sampling