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NSW Syllabuses

Glossary

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Aboriginal and Torres Strait Islander Peoples

Aboriginal Peoples are the first peoples of Australia and are represented by over 250 language groups each associated with a particular Country or territory. Torres Strait Islander Peoples whose island territories to the north east of Australia were annexed by Queensland in 1879 are also Indigenous Australians and are represented by five cultural groups.

An Aboriginal and/or Torres Strait Islander person is someone who:

  • is of Aboriginal and/or Torres Strait Islander descent
  • identifies as an Aboriginal person and/or Torres Strait Islander person, and
  • is accepted as such by the Aboriginal and/or Torres Strait Islander community in which they live.
absolute error

The absolute error of a measurement is half of the smallest unit on the measuring device. The smallest unit is called the precision of the device.

allowable tax deductions

Allowable tax deductions are expenses incurred that are related to your job and profession and can be deducted from your salary to obtain your taxable income. These form part of an individual’s or company’s tax return.

ambiguous case in trigonometry

In trigonometry, the ambiguous case refers to using the sine rule to calculate the size of an angle in a triangle where there are two possibilities for the angle, one obtuse and one acute, leading to two possible triangles.

annuity

An annuity is a compound interest investment from which payments are made or received on a regular basis for a fixed period of time.

appreciated value

Appreciation is an increase in the value of an asset over time. An appreciated value is the value an asset has increased to over that time.

array ⚐

An array is an ordered collection of objects or numbers arranged in rows and columns.

asymptote ⚐

An asymptote is a line.

  • A horizontal asymptote is a horizontal line whose distance from the curve becomes as small as we please for all large values on the horizontal axis.
  • The line \( x=a \) is a vertical asymptote if the curve is not defined at \( x=a \) and the vertical values of the curve become as large as we please (positive or negative) as \( x \) approaches \( a \).
bearing

A bearing is a direction from one point on the Earth’s surface to another. Two types of bearings may be used: compass bearing and true bearings.

bias

Bias generally refers to a systematic favouring of certain outcomes more than others, due to unfair influence (knowingly or otherwise).

bivariate data

Bivariate data is data relating to two variables that have both been measured on the same set of items or individuals. For example the arm spans and heights of 16-year-olds, the sex of primary school students and their attitude to playing sport.

blood alcohol content (BAC)

Blood alcohol content measures the amount of alcohol present in the bloodstream, and may be used for legal purposes.

blood pressure

Blood pressure is the pressure exerted by circulating blood upon the walls of blood vessels. It is usually measured at a person's upper arm. Blood pressure is expressed in terms of the systolic (maximum) pressure over diastolic (minimum) pressure and is measured in millimetres of mercury (mm Hg).

box-plot

A box-plot is a graphical display of a five-number summary.

In a box-plot, the ‘box’ (a rectangle) represents the interquartile range (IQR) with ‘whiskers’ reaching out from each end of the box towards maximum and minimum values in the dataset. A line in the box is used to indicate the location of the median. Also known as a box-and-whisker plot.

break-even point ⚐

The break-even point is the point at which income and cost of production are equal.

categorical data

Data associated with a categorical variable is called categorical data. Also known as qualitative data.

categorical variable

A categorical variable is a variable whose values are categories.

Examples include major blood type (A, B, AB or O) or principal construction type (brick, concrete, timber, steel, other).

Categories may have numerical labels, for example postcodes, but these labels have no numerical significance, they merely serve as labels.

compass bearing

Compass bearings are specified as angles either side of north or south. For example, a compass bearing of \(\text N50^\circ\text E\) is found by facing north and moving through an angle of \(50^\circ\) to the East.

complement

The complement of an event refers to when the event does NOT occur. For example, if \(A\) is the event of throwing a 5 on a die, then the complement of \(A\), denoted by \(\bar A\) or \(A^c\), is throwing a number that is NOT 5 on a die.

constant of variation

Also known as the constant of proportionality. See direct or inverse variation.

continuous data

Continuous data is data associated with continuous variables and is a type of numerical data.

continuous variable

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

Examples include height, reaction time to a stimulus and systolic blood pressure.

critical path ⚐

The critical path is the sequence of network activities which combine to have the longest overall duration so as to determine the shortest possible time needed to complete a project.

cumulative frequency

The cumulative frequency is the accumulating total of frequencies within an ordered dataset.

cumulative frequency graph

A cumulative frequency graph or 'ogive' is a curve or series of straight lines representing the cumulative frequency for a given dataset.

deciles

Deciles divide an ordered dataset into ten equal parts. See also quantiles.

dependent variable

A dependent variable within a statistical model is one whose value depends upon that of another. It is represented on the vertical axis of a scatterplot. The dependent variable is also known as the outcome variable or the output of a function.

depreciation

Depreciation is a decrease in the value of an asset over time.

diastolic pressure

Diastolic pressure is the blood pressure in the arteries when the heart muscle is relaxed between beats.

direct variation

Two variables are in direct variation if one is a constant multiple of the other. This can be represented by the equation \(y=kx\), where \(k\) is the constant of variation (or proportion). Also known as direct proportion, it produces a linear graph through the origin.

directed networks ⚐

A directed network is a network whose edges have arrows and travel is only possible in the direction of the arrows.

discrete data

Discrete data is data associated with discrete variables and is a type of numerical data.

discrete variable

A discrete variable is a numerical variable whose values can be listed.

Examples include the number of children in a family, shoe size or the number of days in a month.

dividend

A dividend of a share is a sum of money paid by a company to its shareholders out of its profits.

dividend yield

A dividend yield is the dividend expressed as a percentage of the current share price.

earliest starting time (EST)

The earliest starting time is the earliest time that any activity can be started after all prior activities have been completed.

edge (in networks)

In a network diagram, an edge refers to a line which joins vertices to each other. Also called an arc.

elevation views

Elevation views are scale drawings showing what a building looks like from the front, back and sides.

event

An event is a set of outcomes for a random experiment.

exponential function

An exponential function is a function in which the independent variable occurs as an exponent (or power/index) with a positive base. For example, \(y=2^x\) is an exponential function where \(x\) is the independent variable.

exponential model

Creating an exponential model involves fitting an exponential graph and/or function to a practical situation or set of data.

extrapolation

Extrapolation occurs when the fitted model is used to make predictions using values that are outside the range of the original data upon which the fitted model was based. Extrapolation far beyond the range of the original data is a dangerous process as it can sometimes lead to quite erroneous predictions.

five-number summary

A five-number summary is a method for summarising a dataset using five statistics: the minimum value, the first quartile, the median, the third quartile and the maximum value.

float time

Float time is the amount of time that a task in a project network can be delayed without causing a delay to subsequent tasks.

flow capacity ⚐

The flow capacity of a network can be found using the maximum-flow minimum-cut theorem and depends upon the capacity of each edge in the network.

fuel consumption rate

The fuel consumption rate of a vehicle measures how much fuel it uses and is usually measured in litres per 100 kilometres (L/100 km).

future value

The future value of an investment or annuity is the total value of the investment at the end of the term of the investment, including all contributions and interest earned.

future value interest factors ⚐

Future value interest factors are the values of an investment at a specific date. A table of these factors can be used to calculate the future value of different amounts of money that are invested at a certain interest rate for a specified period of time.

gross pay

Gross pay is the total income per pay period (weekly, fortnightly, monthly as appropriate).

GST

GST is an abbreviation for the Goods and Services Tax which, in Australia, is a flat percentage of tax levied on most goods and services.

heart rate

Heart rate is the speed of a heartbeat in beats per minute (bpm) and measures the number of contractions of the heart per minute.

income tax

Income tax is a government tax levied on taxable income.

independent variable

An independent variable within a statistical model is one whose outcomes are not due to those of another variable and is represented on the horizontal axis of a scatterplot. The independent variable is also referred to as the input of a function.

interpolation

Interpolation occurs when a fitted model is used to make predictions using values that lie within the range of the original data.

interquartile range (IQR)

The interquartile range is a measure of the spread within a numerical dataset. It is equal to the difference between the upper quartile \(\left(Q_3\right)\) and the lower quartile \(\left(Q_1\right)\); that is, \(IQR=Q_3-Q_1\).

inverse variation

Two variables are in inverse variation (or inverse proportion) if one is a constant multiple of the reciprocal of the other. Hence, as one variable increases, the other variable decreases.

For example, if \(y\) is inversely proportional to \(x\), they are connected by the equation \(y=\frac{k}{x}\), where \(k\) is a constant of variation (or proportion).

Königsberg Bridge problem ⚐

The Königsberg Bridge problem asked whether the seven bridges of the old city of Königsberg could all be crossed only once during a single trip that starts and finishes at the same place.

Kruskal's algorithm ⚐

Kruskal's algorithm finds a minimum-spanning tree for a connected weighted network graph.

latest starting time (LST)

The latest starting time is the latest time an activity may be started after all prior activities have been completed and without delaying the project.

least-squares regression line ⚐

Least-squares regression is a method for finding a straight line that summarises the relationship between two variables, within the range of the dataset.

The least-squares regression line is the line that minimises the sum of the squares of the residuals. Also known as the least-squares line of best fit.

limits of accuracy

The limits of accuracy for a recorded measurement are the possible upper and lower bounds for the actual measurement.

line of best fit ⚐

A line of best fit is a line drawn through a scatterplot of data points that most closely represents the relationship between two variables.

maximum-flow minimum-cut theorem

The maximum-flow minimum-cut theorem states that the flow through a network cannot exceed the value of any cut in the network and that the maximum flow equals the value of the minimum cut, ie it identifies the ‘bottle-neck’ in the system.

measures of central tendency

Measures of central tendency are the values about which the set of data values for a particular variable are scattered. They are a measure of the centre or location of the data.

The two most common measures of central tendency are the mean and the median.

measures of spread

Measures of spread describe how similar or varied the set of data values are for a particular variable.

Common measures of spread include the range, combinations of quantiles (deciles, quartiles, percentiles), the interquartile range, variance and standard deviation.

minimum spanning tree ⚐

A minimum spanning tree is a spanning tree of minimum length in a connected, undirected network. It connects all the vertices together with the minimum total weighting for the edges.

modality

Modality describes the number of peaks in a set of data.

For example data can be unimodal (having one peak), bimodal (having two peaks) or multimodal (having many peaks).

net pay

Net pay is the remaining amount of gross pay after tax and other deductions have been made.

network

A network is a group or system of interconnecting objects which can be represented as a diagram of connected lines (called edges) and points (called vertices). For example a rail network.

network diagram

A network diagram is a representation of a group of objects called vertices that are connected together by lines called edges. Also known as a network graph.

nominal data

Nominal data is a type of categorical data that has no natural order in which the categories may be placed, for example eye colour.

normal distribution

The normal distribution is a type of continuous distribution where the mean, median and mode are equal and the scores are symmetrically arranged either side of the mean. The graph of a normal distribution is often called a ‘bell curve’ due to its shape, as shown below.

graph demonstrating bell curve

 

numerical data

Numerical data is data associated with a numerical variable.

Also known as quantitative data.

numerical variable

Numerical variables are variables whose values are numbers. Numerical variables can be either discrete or continuous.

ordinal data

Ordinal data is a type of categorical data where the possible categorical responses have a natural order. For example level of happiness: very unhappy, unhappy, neutral, happy, very happy.

outcome

An outcome is a single possible result from an experiment.

outlier

An outlier in a dataset is a data value that appears to be inconsistent with the remainder of that dataset.

parabola

A parabola is the graph of a quadratic function. The vertex of a parabola is its highest or lowest point (or turning point). The parabola has an axis of symmetry through its vertex.

Pareto chart ⚐

A Pareto chart is a type of chart that contains both a bar and a line graph, where individual values are represented in descending order by the bars and the cumulative total is represented by the line graph.

path

A path in a network diagram is a walk in which all of the edges and all the vertices are different. A path that starts and finishes at different vertices is said to be open, while a path that starts and finishes at the same vertex is said to be closed. There may be multiple paths between the same two vertices.

Pay As You Go (PAYG) tax

Pay As You Go tax is a system for making regular tax instalments which are removed from gross pay towards the expected income tax liability for that financial year.

Pearson’s correlation coefficient ⚐

Pearson’s correlation coefficient is a statistic that measures the strength of the linear relationship between a pair of variables or datasets. Its value lies between -1 and 1 (inclusive). Also known as simply the correlation coefficient. For a sample, it is denoted by \(r\).

percentage error

The percentage error of a measurement is the absolute error expressed as a percentage of the recorded measurement.

percentiles

Percentiles divide an ordered dataset into 100 equal parts. See also quantiles.

More formally, it is a statistical measure indicating the value below which a given percentage of observations in a group of observations lie. For example the 20th percentile is the value below which 20% of the observations may be found.

piecework

Piecework is employment where a worker is paid a fixed rate for each item produced or action performed regardless of the time taken.

population

The population in statistics is the entire dataset from which a statistical sample may be drawn.

present value

The present value of an investment is the single sum of money (or principal) that could be initially invested to produce a future value over a given period of time.

Prim’s algorithm ⚐

Prim's algorithm determines a minimum spanning tree for a connected weighted network.

quadratic function

A quadratic function is a function of the form \(y=ax^2+bx+c\) where \(a\ne0\). For example, \(y=3x^2+7\)

quadratic model

Creating a quadratic model involves fitting a quadratic graph and/or function to a set of data or creating a model to describe a practical situation.

quantiles

Quantiles are a set of values that divide an ordered dataset into equal groups. Examples include quartiles, deciles and percentiles.

Formally in statistics, quantiles are cutpoints dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way.

quartiles

Quartiles divide an ordered dataset into four equal parts.

There are three quartiles. The first or lower quartile (\(Q_1\)), divides off (approximately) the lowest 25% of data values. The second quartile (\(Q_2\)) is the median. The third or upper quartile (\(Q_3\)), divides off (approximately) the highest 25% of data values. See also quantiles.

radial survey

A radial survey can be used to measure the area of an irregular block of land. In a radial survey, a central point is chosen within the block of land and measurements are taken along intervals from this point to each vertex. The angles between these intervals at the central point are also measured and recorded.

rate

A rate is a particular kind of ratio in which the two quantities are measured in different units. For example the ratio of distance to time, known as speed, is a rate because distance and time are measured in different units (such as kilometres and hours). The value of the rate depends on the units in which the quantities are expressed.

ratio

A ratio is a quotient or proportion of two numbers, magnitudes or algebraic expressions. It is often used as a measure of the relative size of two objects. For example the ratio of the length of a side of a square to the length of a diagonal is \(1:\sqrt2 \,\); that is, \(\frac{1}{\sqrt2}\).

reciprocal function

A function where the independent variable, \(x\), is the denominator in a fraction. Examples of reciprocal functions include those of the form \(y=\frac{k}{x}\). See also inverse variation.

reciprocal model

Creating a reciprocal model involves fitting a reciprocal graph and/or a function to a practical situation or set of data.

rectangular hyperbola

The graph of a reciprocal function is a type of rectangular hyperbola. 

A rectangular hyperbola is a hyperbola for which the asymptotes are perpendicular.

recurrence relation

A recurrence relation occurs when each successive application uses the resultant value of the previous application to generate the next value. Examples include compound interest and annuities.

reducing balance loan

A reducing balance loan is a compound interest loan where the loan is repaid by making regular payments and the interest paid is calculated on the amount still owing (the reducing balance of the loan) after each payment is made.

relative frequency

Relative frequency is a measure of the number of times that an event has occurred in a repeated experiment. If an event \(E\) occurs \(r\) times when a chance experiment has been repeated \(n\) times, then the relative frequency of \(E\) is \(\frac{r}{n}\).

sample space

The sample space of a chance experiment is the set of all possible outcomes for that experiment.

sampling

Sampling is the selection of a subset of data from a statistical population. Methods of sampling include:

  1. systematic sampling – sample data is selected from a random starting point and using a fixed periodic interval
  2. self-selecting sampling – non-probability sampling where individuals volunteer themselves to be part of a sample
  3. simple random sampling – sample data is chosen at random where each member has an equal probability of being chosen
  4. stratified sampling – after dividing the population into separate groups or strata, a random sample is then taken from each group/strata in an equivalent proportion to the size of that group/strata in the population.

A sample can be used to estimate the characteristics of the statistical population.

scale factor

A scale factor is a number that scales, or multiplies, or reduces some quantity.

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 or reduction, the change of size.

scatterplot

A scatterplot is a two-dimensional data plot using Cartesian coordinates to display the values of two variables in a bivariate dataset. Also known as a scatter graph.

share

A share is one of the equal parts into which a company's capital is divided, entitling the shareholder to a portion of the company’s profits.

shortest path ⚐

A shortest path in a network diagram is a path between two vertices in a network where the sum of the weights of its edges are minimised.

sketch

A sketch is an approximate representation of a graph, including labelled axes, intercepts and any other important relevant features. Compared to the corresponding graph, a sketch should be recognisably similar but does not need to be exact.

spanning tree ⚐

A spanning tree of an undirected network diagram is a diagram which includes all the vertices of the original network connected together, but not necessarily all the edges of the original network diagram. A network can have many different spanning trees.

standard deviation

Standard deviation is a measure of the spread of a dataset. It gives an indication of how far, on average, individual data values are spread from the mean.

standard drink

A drink that contains 10 grams of alcohol is called a standard drink.

standard form

A real number is expressed in standard form when it is written in the form \(a\times10^n\) where \(1\le a<10\) and \(n\) is an integer. Also known as scientific notation.

straight-line method

In straight-line method of depreciation, the value of the depreciating asset decreases by the same amount during each time period.

Also known as the ‘Prime Cost method’.

summary statistics

Summary statistics refers to numbers that summarise a given dataset. For example a five-number summary.

target heart rate

The target heart rate is defined as the minimum number of heartbeats in a given amount of time in order to reach the level of exertion necessary for cardiovascular fitness and is specific to a person's age, gender or physical fitness. An example of a target heart rate is 150 bpm to burn fat for a woman in her 30s.

tax return

A tax return is an annual statement of all income, allowable deductions, PAYG tax paid and other personal financial information so as to allow the Australian Taxation Office to calculate the amount of income tax an individual should pay for the financial year.

taxable income

Taxable income is the amount of yearly income that is used to calculate an individual’s payable income tax equal to gross income less allowable tax deductions.

Trapezoidal rule ⚐

The Trapezoidal rule uses trapezia to approximate the area of an irregular shape, often with a curved boundary. Given a transverse line of length \(h\) and two perpendicular offset lengths \(d_f\) and \(d_l\), one application of the Trapezoidal rule is given by: \({Area}≈\frac{h}{2}(d_f+d_l)\)

tree diagram (probability)

A tree diagram is a diagram that can be used to determine the outcomes of a multistep random experiment. A probability tree diagram has the probability for each stage written on the branches.

tree (networks) ⚐

A tree is an undirected network in which any two vertices are connected by exactly one path.

true bearing

True bearings are measured in degrees clockwise from true north and are written with three digits being used to specify the direction.

For example, the direction of north is specified \(000^\circ\), east is specified as \(090^\circ\), south is specified as \(180^\circ\) and north-west is specified as \(315^\circ\).

vertex (in networks)

A vertex is a point in a network diagram at which lines of pathways (called edges) intersect or branch. Also called a node.

weighted edge

A weighted edge is an edge of a network diagram that has a number assigned to it which implies some numerical value such as cost, distance or time.

\(\boldsymbol z\)-score

A \(z\)-score is a statistical measure of how many standard deviations a raw score is above or below the mean. A \(z\)-score can be positive or negative, indicating whether it is above or below the mean, or zero. Also known as a standardised score.