BCS Written Mathematics key terms

The sum of a set of values divided by the number of values, giving a single central figure that balances the data. It is sensitive to very large or very small observations.

The middle value of an ordered data set, or the average of the two middle values when the count is even. Half the observations lie at or below it.

The value or values that occur most frequently in a data set. A distribution may have one mode, several modes, or none at all.

A measure of how far values typically sit from the mean, found by averaging the squared deviations, taking the square root. A larger value means greater spread.

The mean of the squared deviations of values from their arithmetic mean. It measures dispersion in squared units and is the square of the standard deviation.

One of three values that divide an ordered data set into four parts of roughly equal size. The second quartile coincides with the median.

The difference between the upper and lower quartiles, covering the middle half of the data. It is resistant to extreme values and useful for describing spread.

An arrangement of objects in which order matters. The number of permutations of r objects chosen from n distinct objects is n! divided by (n minus r) factorial.

A selection of objects in which order does not matter. The count of combinations of r objects from n is n! divided by r! times (n minus r) factorial.

The product of all positive integers from 1 up to a given whole number n, written n!. By convention the factorial of zero equals one.

Two events that cannot both happen in the same trial, so the probability of either occurring is the sum of their separate probabilities.

Events whose outcomes do not influence one another, so the probability that both occur equals the product of their individual probabilities.

The probability that an event occurs given that another event is known to have occurred, written as the probability of both divided by the probability of the given event.

The probability model for the number of successes in a fixed number of independent trials, each with the same two possible outcomes and the same success probability.

A complete listing or rule giving each possible value of a random variable together with its probability, where all probabilities are non-negative and sum to one.

The long-run average of a random variable, computed by multiplying each possible value by its probability and adding the results.

A single number calculated from a square matrix that indicates whether the matrix is invertible. For a two-by-two matrix it equals ad minus bc.

The matrix which, when multiplied by a given square matrix, yields the identity matrix. It exists only when the determinant is non-zero.

A square matrix whose determinant equals zero, so it has no inverse and its rows or columns are linearly dependent.

A square matrix with ones on the main diagonal and zeros elsewhere. Multiplying any conformable matrix by it leaves that matrix unchanged.

The matrix obtained by interchanging the rows and columns of a given matrix. The transpose of a product reverses the order of the factors.

A square matrix that equals its own transpose, so the entry in row i and column j matches the entry in row j and column i.

The signed minor obtained by deleting the row and column of an entry and attaching a plus or minus sign according to its position. Cofactors are used to expand determinants.

The determinant of the smaller matrix left after deleting one row and one column from a square matrix. Minors feed into cofactor expansion.

A method of solving a system of linear equations using determinants, replacing one column of the coefficient matrix at a time with the constants.

Two or more equations that must be satisfied by the same set of unknown values at once. Solutions may be found by elimination, substitution, or matrix methods.

An equation in which the highest power of the unknown is two, usually written in the form ax squared plus bx plus c equals zero with a not zero.

The quantity b squared minus four ac in a quadratic equation. Its sign reveals whether the roots are real and distinct, real and equal, or complex.

A sequence in which each term after the first is obtained by adding a fixed constant. Its nth term and sum are found from the first term and common difference.

A sequence in which each term after the first is obtained by multiplying by a fixed ratio. Its sum to n terms and infinite sum have standard formulas.

The sum of the terms of a sequence, written with sigma notation. Series may be finite or infinite, and infinite ones may converge or diverge.

The exponent to which a fixed base must be raised to produce a given number. Logarithms turn products into sums and solve equations with the unknown in an exponent.

A function in which the variable appears in the exponent, such as e to the power x. Its rate of growth is proportional to its current value.

The instantaneous rate at which a function changes with respect to its variable, defined as the limit of a difference quotient. Geometrically it gives the slope of the tangent.

The limit of a sum of many small products, interpreted as signed area under a curve. Indefinite integrals give antiderivatives; definite integrals give numerical values.

The value a function or sequence approaches as its input approaches a specified point or grows without bound. Limits underpin both derivatives and integrals.

A function whose graph has no breaks, jumps, or holes over an interval, so small changes in input cause small changes in output.

The result linking differentiation and integration as inverse processes, and showing that a definite integral can be evaluated from any antiderivative.

A straight line that touches a curve at a point and has the same slope as the curve there. Its slope equals the derivative at that point.

The line through a point on a curve that is perpendicular to the tangent there. Its slope is the negative reciprocal of the derivative.