Glossary

adjacency matrix

An adjacency matrix for a non-directed graph with `n` vertices is a `n×n` matrix in which the entry in row `i` and column `j` is the number of edges joining the vertices `i`and `j`. In an adjacency matrix a loop is counted as one edge.

For a directed graph, the entry in row `i` and column `j` is the number of directed edges, arcs, joining the vertex `i` and `j` in the direction `i` to `j`.

algorithm

An algorithm is a precisely defined routine procedure that can be applied and systematically followed through to a conclusion.

angle of depression

When an observer looks at an object that is lower than the eye of the observer, the angle between the line of sight and the horizontal is called the angle of depression.

angle of elevation

When an observer looks at an object that is higher than the eye of the observer, the angle between the line of sight and the horizontal is called the angle of elevation.

bearing

The direction of a fixed point or the path of an object from the point of observation.

break-even point

The break-even point is the point at which revenue begins to exceed the cost of production.

categorical data

Data associated with a categorical variable is called categorical data.

categorical variable

A categorical variable is a variable whose values are categories.

Examples include blood group, A, B, AB or O, or house construction type, brick, concrete, timber, steel, other.

Categories may have numerical labels, e.g. the numbers worn by player in a sporting team, but these labels have no numerical significance, they merely serve as labels.

compass bearings

Compass bearings are specified as angles either side of North or South, that describe the direction of a fixed point or the path of an object. For example: a compass bearing of N50°E is found by facing North and moving through an angle of 50° towards East.

compound interest

The interest earned when each successive interest payment is added to the principal for the purpose of calculating the next interest payment.

e.g. if the principal (`p`) earns compound interest (`A`) at the interest rate (`i`) expressed as a percentage per period, then after (`n`) compounding periods the total amount accrued is:

`A = P(1 + i)^n`

When plotted on a graph, the total amount accrued is shown to grow exponentially.

continuous data

Data associated with a continuous variable is called continuous data.

continuous variable

A continuous variable is a numerical variable that can take any value that lies within an interval. In practice, the values taken are subject to accuracy of the measurement instrument used to obtain these values.

Examples include height, reaction time, temperature and systolic blood pressure.

cosine ratio

In any right-angled triangle,

`cosθ = (adjacent) /(hypoten use)`

discrete data

Discrete data is data associated with a discrete variable. Discrete data is sometimes called count data.

discrete variable

A discrete variable is a numerical variable that can take only integer values.

Examples include the number of people in a car, the number of decayed teeth in an 18-year‐old male etc.

effective annual rate of interest

The effective annual rate of interest `i_(effective)` is used to compare the interest paid on loans, or investments, with the same nominal annual interest rate `i` but with different compounding periods, daily, fortnightly, monthly, quarterly, annually, other. If the number of compounding periods per annum is `n`, then:

`i_(effective)=(1+i/n)^n-1`

extrapolation

In the context of fitting a linear relationship between two variables, extrapolation occurs when the fitted model is used to make predictions using values of the explanatory variable that are outside the range of the original data. Extrapolation is a dangerous process as it can sometimes lead to quite erroneous predictions.

five-number summary

A five-number summary is a method of summarising a set of data using the minimum value, the lower or first-quartile (Q1), the median, the upper or third-‐quartile (Q3) and the maximum value. Forms the basis for a boxplot.

Goods and Services Tax (GST)

The Goods and Services Tax (GST) is a broad sales tax of 10% on most goods and services and other items sold or consumed in Australia.

gradient

The gradient of a line describes its steepness, incline or grade.

Gradient is normally described by the ratio of the "rise" divided by the "run" between two points on a line.

identity matrix

A square matrix in which all of the elements in the leading diagonal are 1s and the remaining elements are 0s; identity matrices are designated by the letter 𝐼.

There is an identity matrix for each size, or order, of a square matrix.

interpolation

In the context of fitting a linear relationship between two variables, interpolation occurs when the fitted model is used to make predictions using values of the explanatory variable that lie within the range of the original data.

interquartile range

The interquartile range (IQR) is a measure of the spread within a numerical data set. It is equal to the upper quartile (Q3) minus the lower quartile (Q1); that is IQR = Q3 – Q1

The IQR is the width of an interval that contains the middle 50%, approximately, of the data values. To be exactly 50% the sample size must be a multiple of four.

inverse of a square matrix

The inverse of a square matrix A is written as A–1 and has the property that

`A A–1= A–1A= I`

Not all square matrices have an inverse. A matrix that has an inverse is said to be invertible.

leading diagonal

The leading diagonal of a square matrix is the diagonal that runs from the top left corner to the bottom right corner of the matrix.

linear equation

A linear equation in one variable x is an equation of the form:

`ax + b = 0`, e.g. `3x + 1 = 0`

A linear equation in two variables x and y is an equation of the form:

`ax + by + c = 0` e.g. `x – y + 5 = 0`

linear graph

A linear graph is a graph of a linear equation with two variables. If the linear equation is written in the form y = a + bx, then a represents the y-‐intercept and b represents the gradient of the linear graph.

matrix (matrices)

A matrix is a rectangular array of elements or entities displayed in rows and columns. Matrices are described as m×n, where m is the number of rows and n is the number of columns.

A square matrix has the same number of rows and columns.

A column matrix, or vector, has only one column.

A row matrix, or vector, has only one row.

matrix multiplication

Matrix multiplication is the process of multiplying a matrix by another matrix.

The product `AB` of two matrices `A` and `B` of size `m×n` and `p × q` respectively, is defined if `n = p`. If `n = p` the resulting matrix has size `m×q`.

mean

The arithmetic mean, `x ̅,` of a list of numbers is the sum of the data values divided by the number of values in the list.

In everyday language, the arithmetic mean is commonly called the average.

median

The median is the value in a set of ordered set of data values that divides the data into two parts of equal size. When there are an odd number of data values, the median is the middle value. When there is an even number of data values, the median is the arithmetic mean of the two central values.

mode

The mode is the most frequently occurring value in a data set.

outlier

An outlier in a set of data is an observation that appears to be inconsistent with the remainder of that set of data. An outlier is a surprising observation.

piecewise-linear graph

A graph consisting of one or more none overlapping line segments. Sometimes called a line segment graph.

Pythagoras’ theorem

The square of the hypotenuse of a right-angled triangle equals the sum of the squares of the lengths of the other two sides. As a rule: `c^2=a^2+b^2`, where `c` is the length of the hypotenuse.

scalar multiplication, of a matrix

Scalar multiplication is the process of multiplying a matrix by a scalar number.

In general, for the matrix A with elements `aij` the elements of `kA` are `kaij`.

scale factor

A number that scales or multiplies some quantity. In the equation y = kx, 𝑘 is the scale factor for 𝑥; if two or more figures are similar, their sizes can be compared. The scale factor is the ratio of the length of one side on one figure to the length of the corresponding side on the other figure. It is a measure of magnification; the change of size.

sequence

A sequence is an ordered list of numbers or objects.

For example: 1, 3, 5, 7 is a sequence of numbers that differs from the sequence 3, 1, 7, 5 as order matters.

A sequence maybe finite, for example, 1, 3, 5, 7, the sequence of the first four odd numbers, or infinite, for example, 1, 3, 5, … the sequence of all odd numbers.

similar figures

Two geometric figures are similar if they are of the same shape but not necessarily of the same size

sine ratio

In any right-angled triangle,

`sinθ= (opposite )/(hypoten use)`

singular matrix

A matrix is singular if `det A = 0`. A singular matrix does not have a multiplicative inverse.

simple interest

Simple interest is the interest (`I`) accumulated when the interest payment in each period is a fixed fraction of the principal, e.g. if the principle 𝑃 earns simple interest at the rate (`R`) expressed as a percentage per period, then after (`T`) periods the accumulated simple interest is:

`I=PRT`

When plotted on a graph, the total amount accrued is shown to grow linearly.

size (of a matrix)

Two matrices are said to have the same size, or order, if they have the same number of rows and columns. A matrix with `m` rows and n columns is said to be a `m×n` matrix.

standard deviation

The standard deviation is a measure of the variability or spread of a data set. It gives an indication of the degree to which the individual data values are spread around their mean.

The standard deviation of 𝑛 observations `x_1,x_2,…,x_n` is:

`s=sqrt((Σ(x_i-x ̅ )^2)/(n-1))`

step graph

A graph consisting of one or more non-‐overlapping horizontal line segments that follow a step-like pattern.

tangent ratio

In any right-angled triangle,

`tanθ = (opposite)/(adjacent)`

true bearings

True bearings are measured in degrees in a clockwise direction from the North line. Three figures are used to specify the direction. Thus, North is specified as 000°T, East is specified as 090°T, South-East is specified as 135°T.

zero matrix

A zero matrix is a matrix where all of its entries are zero.