Glossary

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.

accuracy

The condition or quality of being true, correct or exact; freedom from error or defect; precision or exactness; correctness; in science, the extent to which a measurement result represents the quantity it purports to measure; an accurate measurement result includes an estimate of the true value and an estimate of the uncertainty.

algorithm

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

association

A general term used to describe the relationship between two or more variables. The term association is often used interchangeably with the term correlation. The latter tends to be used when referring to the strength of a linear relationship between two numerical variables.

associative operations

Operations are associative if the order in which operations take place does not affect the result.

For example, addition of numbers is associative, since the order in which they are added does not change their sum. The corresponding associative law is:

`(a+b)+c=a+(b+c)` for all numbers `a`, `b` and `c`.

Multiplication is also associative, as the product of the numbers does not vary with the order of their multiplication. The corresponding associative law is:

`(ab)c=a(bc)` for all numbers `a`, `b` and `c`.

Subtraction and division are not associative, as the order of operations changes the value of the expression.

average speed

The total distance travelled divided by the total time taken.

array

An ordered collection of objects or numbers.

back-to-back stem-and-leaf plots

A method for comparing two data distributions attaching two sets of ‘leaves’ to the same ‘stem’ in a stem and leaf plot.

bias

Generally, refers to a systematic favouring of certain outcomes more than others, due to unfair influence, knowingly or otherwise.

bimodality

A dataset is bimodal if it has two modes; this means that there is not a single data value that occurs with the highest frequency, but two data values have the same and highest frequency.

break-even point

The point at which revenue begins to exceed the cost of production.

calculates

Determine or find; for example, a number or answer by using mathematical processes; obtain a numerical answer showing the relevant stages in the working; ascertain or determine from given facts, figures or information.

Cartesian plane

Two number lines intersect at right angles at their origins to form the axes of the coordinate system; the plane is divided into four quadrants by these perpendicular axes, called the x-axis, horizontal line, and the y-axis, vertical line; the position of any point in the plane can be represented by an ordered pair of numbers, x, y. These ordered pairs are called the coordinates of the point; this is called the Cartesian coordinate system. The plane is called the Cartesian plane.

categorical data

Data associated with a categorical variable is called categorical data.

categorical variable

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; for example, the numbers worn by players in a sporting team, but these labels have no numerical significance, they merely serve as labels.

census

A population is the complete set of individuals, objects, places etc. that we want information about. A census is an attempt to collect information about the entire population.

commutative operations

Operations are commutative if the order in which terms are given does not affect the result.

The commutative law for addition is:

`a+b=b+a`, for all numbers `a` and `b`.

For example, 3+5=5+3.

The commutative law for multiplication is: `ab=ba`, for all numbers `a` and `b`.

For example, 4×7=7×4.

Subtraction and division are not commutative because; for example, 5-3≠3-5 and 12÷4≠4÷12.

conversion

A change in the form or units of an expression.

compound interest

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

For example, 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.

correlation

A measure of the strength of the linear relationship between two variables.

correlation coefficient

The correlation coefficient (`r`) is a measure of the strength of the liner relationship between a pair of variables.

decile

Any of the nine values that divide a ranked dataset into ten equal parts.

distributive law

Multiplication of numbers is said to be ‘distributive over addition’, because the product of one number with the sum of two others equals the sum of the products of the first number with each of the others.

For example, the product of 3 with (4+5) gives the same result as the sum of 3×4 and 3×5:

3×(4+5)=3×9=27 and 3×4+3×5=12+15=27

This distributive law is expressed algebraically as follows:

`a(b+c)=ab+ac`, for all numbers `a`, `b` and `c`.

elevation views

Scale drawings showing what a building looks like from the front, back and sides.

equivalence

Two expressions are said to be equivalent if they are equal in value.

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.

face, shape

Any of the individual flat surfaces of a solid object.

five-number summary

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)

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.

histogram

A statistical graph for displaying the frequency distribution of continuous data.

A histogram is a graphical representation of the information contained in a frequency table. In a histogram, class frequencies are represented by the areas of rectangles centred on each class interval. The class frequency is proportional to the rectangle’s height when the class intervals are all of equal width.

integer

The integers are the “whole numbers” including those with negative sign ⋯-3, -2, -1, 0, 1, 2, 3⋯. In Latin, the word integer means “whole.” The set of integers is usually denoted by Z. Integers are basic building blocks in mathematics.

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 (`Q_3`) minus the lower quartile (`Q_1`);

that is, `IQR=Q_3-Q_1`

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.

kilowatt hour (kWh)

A unit of energy equal to 1000 watt hours or 3.6 megajoules. The kilowatt hour is most commonly known as a billing unit for energy delivered to consumers by electric utilities.

megajoule (MJ)

A joule is the SI unit of work. The megajoule, MJ, is equal to one million joules.

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.

measures of central tendency

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

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.

median

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

Is the most frequently occurring value is a data set.

order of operations

The order of performing mathematical operations:

- evaluate brackets or grouping symbols first
- evaluate any powers and roots
- working left to right, evaluate any multiplication and division
- working left to right, evaluate any addition or subtraction, may also be known as BODMAS, BIDMAS, BEDMAS, etc.

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.

parallel box plots

Used to visually compare the five-number summaries of two or more datasets.

partitioning

Dividing a quantity into parts. In the early years of schooling, it commonly refers to the ability to think about numbers as made up of two parts, such as, 10 is 8 and 2. In later years it refers to dividing both continuous and discrete quantities into equal parts.

percentage error

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

picture graph

A statistical graph for organising and displaying categorical data.

piecework

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

place value

The value of a digit as determined by its position in a number, relative to the ones, or units, place. For integers, the ones place is occupied by the rightmost digit in the number. The value of the next column, the first after the decimal point, represents tenths of ones and this continues with the value of each corresponding digit being representative of a value 10 times smaller than the previous.

For example, in the number 2 594.6 the 4 denotes 4 ones, the 9 denotes 90 ones or 9 tens, the 5 denotes 500 ones or 5 hundreds, the 2 denotes 2000 ones or 2 thousands, and the 6 denotes `6/10` of a one or 6 tenths.

precision

Refers to how close the measured values are to each other. Precision does not account for how close the measured values are to the actual, expected, value.

probability

The likelihood or chance of something; the relative frequency of the occurrence of an event as measured by the ratio of the number of cases or alternatives favourable to the event to the total number of cases or alternatives.

quartile

The quartiles of a ranked set of data values are the three points that divide the dataset into four equal groups.

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.

range

The difference between the largest and smallest observations in a data set.

rate

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 comparison of two quantities of the same kind; for example, if a recipe uses 2 cups of milk and 3 cups of flour, the ratio of milk to flour is 2 is to 3. This can also be written with a colon, 2:3, or as a fraction, `(2 )/3`

ray

The part of a line that starts at a point and continues in a particular direction to infinity. Rays are usually depicted with an arrowhead, which indicates the direction in which the line continues to infinity.

reaction time

The time a person takes to react to a situation. For example: time taken for a person to press the brake when a situation requires them to stop.

reasoned argument or conclusion

A reasoned argument or conclusion is logical, sound, considered and well grounded.

recurrence relation

An equation that recursively defines a sequence; that is, once one or more initial terms are given, each further term of the sequence is defined as a function of the preceding terms.

recurring decimal

Non-terminating decimals may be recurring; that is, contain a pattern of digits that repeats indefinitely after a certain number of places.

reducing balance loan

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 loan, after each payment is made.

reflection

To reflect the point `A` in an axis of reflection, a line is drawn at right angles to the axis of reflection and the point `A` is marked at the same distance from the axis of reflection as `A`, but on the other side.

The point `A` is called the reflection image of `A`.

A reflection is a transformation that moves each point to its reflection image.

relative frequency

The number of items of a certain type divided by the number of all the items considered.

sample

Part of a population; a subset of the population, often randomly selected for the purpose of estimating the value of a characteristic of the population as a whole.

sample space

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

sampling

The selection of a subset of data from a statistical population. Methods of sampling include:

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

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

scale

A graduated line, as on a map, representing proportionate size.

simple interest

The interest (`I`) accumulated when the interest payment in each period is a fixed fraction of the principal; for example, 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.

sketch

Execute a drawing or painting in simple form, giving essential features but not necessarily with detail or accuracy; in mathematics, represent by means of a diagram or graph; the sketch should give a general idea of the required shape or relationship and should include features.

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))`

stopping distances

The distance a car travels before it comes to rest after the driver has applied the brake given speed of the vehicle and conditions of the road which can be found using formulas or tables.

Stopping distance = braking distance + reaction time (seconds) × speed

straight-line method of depreciation

In the 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’.

symmetry

A plane figure `f` has line symmetry in a line `m`, if the image of `f` under the reflection in `m` is `f` itself. The line `m` is called the axis of symmetry.

A plane figure `f` has rotational symmetry about a point O if there is a rotation such that the image of `` under the rotation is `` itself.

terminating decimal

A decimal that contains a finite number of digits.

translation

Shifting a figure in the plane without turning it is called translation. To describe a translation in the plane, it is enough to say how far left or right and how far up or down the figure is moved.

A translation is a transformation that moves each point to its translation image.

trapezoidal rule

The trapezoidal rule uses trapezia to approximate the area of an irregular shape, often with a curved boundary. The rule for a single application is: `A≈h/2(x_1+x_(2 ))`

travel graph

Line graphs that are used to describe the motion of objects such as cars, trains, walkers and cyclists. The distance travelled is represented on the vertical axis and the time taken to travel that distance is represented on the horizontal axis.

tree diagram

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.

two-way table

Commonly used for displaying the two-way frequency distribution that arises when a group of individuals or objects are categorised according to two criteria.

Vertex, in shape

A point in which edges intersect.