Tamil Nadu HSE Second Year Statistics key terms

A square matrix whose determinant equals zero, so it has no inverse. In solving linear equations, a singular coefficient matrix signals either no solution or infinitely many solutions.

A method of solving a system of linear equations using determinants, where each unknown equals a ratio of two determinants with the coefficient determinant in the denominator.

The minor of an element is the determinant left after deleting its row and column; the cofactor attaches the sign (−1)^(i+j) to that minor. Cofactors build adjoints and inverse matrices.

The transpose of the cofactor matrix. For an invertible square matrix A, the inverse equals the adjoint divided by the determinant of A.

The order of the largest non-zero minor of the matrix. It tells how many independent equations a system really contains, which decides consistency.

A linear system is consistent when at least one solution exists. Comparing the ranks of the coefficient and augmented matrices decides consistency and whether solutions are unique or infinite.

The coefficient matrix of a linear system with the constant column attached as an extra column. Row operations on it solve the system and reveal inconsistency.

The complex number obtained by changing the sign of the imaginary part; the conjugate of a + bi is a − bi. Multiplying a number by its conjugate gives a real value a² + b².

The modulus is the distance of a complex number from the origin, and the argument is the angle its position vector makes with the positive real axis. Together they give polar form.

Writing a complex number as r(cos θ + i sin θ), where r is the modulus and θ the argument. It makes multiplication, division and powers easy to compute.

For a complex number in polar form, raising it to a power n multiplies the argument by n while the modulus is raised to n. It also generates roots of complex numbers.

The three solutions of z³ = 1, namely 1, ω and ω², where ω is a non-real cube root. They satisfy 1 + ω + ω² = 0 and simplify many algebraic problems.

Expressions such as sum of roots, sum of pairwise products, and product of roots, written in terms of coefficients. They let you compute related expressions without finding individual roots.

For a polynomial, the sum of roots, sum of products taken two at a time, and so on, are fixed by the ratios of its coefficients. These relations link roots to coefficients.

The discriminant of a quadratic decides whether roots are real and distinct, real and equal, or complex conjugates. For cubics, sign changes in the polynomial help locate real roots.

A polynomial equation whose coefficients read the same forwards and backwards, so if r is a root then 1/r is also a root. Such equations are reduced in degree by substitution.

The restricted range chosen for an inverse trigonometric function so that it becomes one-to-one and its inverse is a genuine function, for example −π/2 to π/2 for sine inverse.

The function that returns the angle whose sine is a given value between −1 and 1. Its value always lies in the principal branch from −π/2 to π/2.

The function giving the angle whose tangent is a given real number. Its principal range is the open interval from −π/2 to π/2, and it accepts every real input.

The set of inputs for which an inverse trigonometric function is defined, such as −1 to 1 for sine and cosine inverses, and all real numbers for tangent inverse.

If one task can be done in m ways and a second in n ways, the two together can be done in m × n ways. It underlies all permutation and combination formulas.

An arrangement of objects in which order matters. The number of arrangements of r objects chosen from n distinct objects is written nPr.

A selection of objects in which order does not matter. The number of ways to choose r objects from n distinct objects is written nCr and equals nPr divided by r factorial.

An arrangement of objects around a circle, where rotations of the same arrangement are treated as identical. The number of arrangements of n distinct objects in a circle is (n − 1) factorial.

The formula for any single term in the expansion of (a + b)^n, written using the binomial coefficient and powers of a and b. It locates a particular term without expanding fully.

The number nCr appearing in a binomial expansion, counting how many ways r factors are chosen from n. These coefficients form symmetric patterns in Pascal's triangle.

The term or pair of terms at the centre of a binomial expansion. When n is even there is one middle term; when n is odd there are two, found from the general term.

A sequence in which each term is obtained by adding a fixed constant, the common difference, to the previous term. Its nth term and sum have standard closed formulas.

A sequence in which each term is obtained by multiplying the previous term by a fixed constant, the common ratio. Its sum formula depends on whether the ratio has magnitude below one.

The average of two numbers, equal to half their sum. In an arithmetic progression, the middle term between two terms is their arithmetic mean.

For two positive numbers, the positive square root of their product. In a geometric progression, the middle term between two terms is their geometric mean.

A real-valued function that assigns a number to each outcome of a random experiment. Discrete random variables take countable values, each with a probability.

A rule giving the probability of each possible value of a discrete random variable. The probabilities must be non-negative and add up to one.

The distribution of the number of successes in n independent trials, each with the same success probability p. Its mean is np and variance is npq.

A distribution for counting rare events in a fixed interval, defined by a single parameter, the mean number of occurrences. Its mean and variance are both equal to that parameter.

A sampling method in which every sample of a given size has an equal chance of selection. It reduces bias and supports standard error formulas.

The probability distribution of a statistic, such as the sample mean, computed from all possible samples of a fixed size drawn from a population.

The standard deviation of a statistic's sampling distribution. For the sample mean it equals the population standard deviation divided by the square root of the sample size.

The result that the sampling distribution of the sample mean approaches a normal distribution as sample size grows, whatever the shape of the population distribution.

A single value computed from sample data and used as a best guess for an unknown population parameter, such as the sample mean estimating the population mean.

A range of values, built from sample data with a stated confidence level, believed to contain the unknown population parameter. Its width depends on variability and sample size.

A statistic whose expected value equals the parameter it estimates. The sample mean is unbiased for the population mean; the sample variance uses n − 1 to stay unbiased.

The null hypothesis states no effect or no difference and is tested; the alternative states the claim we suspect. The decision is framed as rejecting or not rejecting the null.

A Type I error rejects a true null hypothesis; a Type II error fails to reject a false null. Their probabilities trade off against each other for a fixed sample size.

The maximum probability of making a Type I error that a test allows, commonly 0.05 or 0.01. It fixes the critical values used in the decision rule.

The boundary value from a reference distribution that separates the rejection region from the acceptance region in a hypothesis test. It depends on the significance level and the test type.

A number between −1 and 1 measuring the strength and direction of a straight-line relationship between two variables. Zero indicates no linear relationship.

The straight line that best predicts one variable from another by minimising squared deviations. Two regression lines exist, one for predicting x and one for predicting y.

The slopes of the two regression lines, written bxy and byx. Their product equals the square of the correlation coefficient and their geometric mean gives its magnitude.

The square of the correlation coefficient, showing the proportion of variation in one variable explained by the other through the fitted regression line.