NCEA Level 2 Mathematics and Statistics key terms

The number that multiplies a variable in an algebraic term. In 5x² the coefficient is 5; in −y it is −1. Coefficients stay fixed while the variable changes value.

The small raised number showing how many times the base is multiplied by itself. In 2⁵ the index is 5 and the base is 2, so 2⁵ = 2×2×2×2×2 = 32.

Terms that have exactly the same variable part, so they can be added or subtracted to give a single term. 3x and −7x are like terms; 3x and 3x² are not.

Rewriting an expression as a product of its factors, which is the reverse of expanding. For example x² + 5x + 6 factorises to (x + 2)(x + 3).

An equation in which the highest power of the variable is two, usually written ax² + bx + c = 0 with a ≠ 0. It can have two, one or no real solutions.

The U-shaped curve that is the graph of a quadratic function. It opens upwards when the x² coefficient is positive and downwards when it is negative, with a vertex at its turning point.

The point where a parabola reaches its maximum or minimum value and changes direction. For y = a(x − h)² + k the vertex is (h, k).

The value b² − 4ac from the quadratic formula. It tells how many real roots a quadratic has: positive gives two, zero gives one, negative gives none.

Two or more equations containing the same variables that must be solved together. The solution is the set of values satisfying every equation at once, found by substitution or elimination.

A statement comparing two expressions using <, >, ≤ or ≥ rather than equals. Solving it gives a range of values, often shown on a number line or as a shaded region.

A sequence in which each term after the first is found by adding a fixed number, called the common difference, to the previous term, for example 4, 7, 10, 13.

A list of numbers in which each term is obtained by multiplying the previous term by a fixed value called the common ratio, for example 3, 6, 12, 24 where the ratio is 2.

In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides, written a² + b² = c², where c is the side opposite the right angle.

A ratio linking an angle of a right-angled triangle to two of its sides: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.

A direction measured clockwise from north and written as a three-digit angle, for example 045° or 210°. Bearings describe the direction of one place from another.

The angle measured upwards from the horizontal to a line of sight, for example when looking at the top of a tall building from the ground.

A relationship that holds in any triangle: a/sin A = b/sin B = c/sin C, where each side is opposite its matching angle. It is used when a side and its opposite angle are known.

A formula that works in any triangle: c² = a² + b² − 2ab cos C. It finds a side when two sides and the included angle are known, or an angle when all three sides are known.

The area of any triangle equals half the product of two sides multiplied by the sine of the angle between them, written Area = ½ab sin C.

The distance along the curved part of a circle between two points, calculated as a fraction of the circumference: arc length = (θ/360) × 2πr for a central angle θ in degrees.

A pie-slice shaped part of a circle enclosed by two radii and an arc. Its area is (θ/360) × πr², where θ is the central angle in degrees.

The amount of three-dimensional space an object occupies, measured in cubic units. Common formulas are V = lwh for a cuboid and V = πr²h for a cylinder.

The total area of all the outer faces and curved surfaces of a three-dimensional object, measured in square units such as cm² or m².

The multiplier that enlarges or reduces a shape. Lengths are multiplied by the scale factor k, areas by k² and volumes by k³.

The entire group of individuals or items that a statistical investigation is about, such as all Year 12 students in a school. Conclusions from a sample are generalised to this group.

A smaller group selected from a population and used to gather data. Because studying everyone is often impractical, a well-chosen sample allows estimates to be made about the whole population.

A selection method in which every member of the population has an equal chance of being chosen, which helps reduce bias. Drawing names from a hat is a simple example.

A systematic tendency for a sample, question or measurement to favour one outcome over another, so results do not accurately represent the population being studied.

The middle value when data are ordered from smallest to largest. If there is an even number of values, the median is the mean of the two middle values.

The difference between the upper quartile and the lower quartile, written IQR = UQ − LQ. It describes the spread of the middle 50% of the data and resists extreme values.

A measure of spread describing the typical distance of data values from the mean. A larger standard deviation means the values are more spread out.

A data value lying well outside the general pattern. A common rule labels values as outliers when they are more than 1.5 × IQR below the lower quartile or above the upper quartile.

A graph showing the smallest value, lower quartile, median, upper quartile and largest value of a data set, with a box for the middle 50% and whiskers for the rest.

A graph plotting one numerical variable against another as points, used to look for a relationship such as a trend or association between the two variables.

A measure of how strongly two numerical variables are linearly related, taking values from −1 for a perfect negative relationship through 0 for no linear relationship to +1 for a perfect positive relationship.

A number between 0 and 1 describing how likely an event is, where 0 means impossible and 1 means certain. It can be estimated from long-run relative frequency.

The set of all possible outcomes of a probability experiment. For one roll of a fair dice the sample space is {1, 2, 3, 4, 5, 6}.

Two events that cannot both happen in the same trial, such as rolling a 2 and a 5 on a single dice. Their combined probability is the sum of their separate probabilities.

Events where the occurrence of one does not change the probability of the other occurring. For independent events P(A and B) = P(A) × P(B).

A range of plausible values for a population parameter, built from sample data. A 95% interval is produced by a method that captures the true value in about 95% of repeated samples.