Math trial

Term

Definition

Categorical Data

Data that describes qualities/categories (nominal = no order, ordinal = has order). E.g. favourite colour (nominal), school grade A-E (ordinal)

Numerical Data

Data that is counted or measured (discrete = countable, continuous = measured on a scale). E.g. number of siblings (discrete), height in cm (continuous)

Frequency Table

A table showing how often each value or class interval occurs. E.g. tally the number of students scoring in each mark range

Histogram

A graph of grouped numerical data using bars with no gaps, area proportional to frequency. E.g. display distribution of exam scores across mark ranges

Precision

The smallest unit of measurement an instrument can measure to. E.g. a ruler marked in mm has a precision of 1mm

Accuracy

How close a measurement is to the true value. E.g. a scale reading 50.1kg for a true 50kg mass is accurate

Absolute Error

Half the precision of a measuring instrument; = 1/2 x precision. E.g. a ruler measures to the nearest mm, so absolute error = 0.5mm

Upper/Lower Bound

Measurement ± absolute error; the range the true value could lie in. E.g. a length measured as 12cm (nearest cm) has bounds 11.5-12.5cm

Converting Units

Changing between metric units (length, area, volume, mass) using conversion factors. E.g. convert 2.5m² to cm² (multiply by 10,000)

Formula Substitution

Replacing pronumerals in a formula with given values to calculate a result. E.g. substitute r=5 into V=(4/3)πr³ to find volume

Rearranging Formulae

Changing the subject of a formula to solve for a different variable. E.g. rearrange FV=PV(1+r)^n to make PV the subject

Gross Income

Total income earned before deductions (tax, super, etc.). E.g. calculate gross weekly income from an annual salary

Net Income

Income remaining after deductions like tax are taken out. E.g. calculate net pay after PAYG tax is withheld

Wage vs Salary

Wage = paid per hour/piece worked; Salary = fixed annual amount regardless of hours. E.g. a casual retail worker earns a wage; a teacher earns a salary

Overtime Pay

Extra pay (often time-and-a-half or double-time) for hours beyond normal shift. E.g. calculate pay for 3 hours overtime at time-and-a-half

Commission

Payment based on a percentage of sales made. E.g. calculate a salesperson's pay earning 5% commission on $10,000 sales

PAYG Tax / Tax Brackets

Progressive tax system where different portions of income are taxed at different rates. E.g. calculate tax payable using the ATO tax bracket table

Mean

The average of a data set; sum of values divided by number of values. E.g. find the mean test score of a class

Median

The middle value of an ordered data set. E.g. find the median house price in a suburb

Mode

The most frequently occurring value in a data set. E.g. find the mode shoe size sold in a store

Range

Highest value minus lowest value. E.g. find the range of daily temperatures over a week

Interquartile Range (IQR)

Q3 − Q1; the spread of the middle 50% of data, less affected by outliers than range. E.g. compare spread of two data sets using IQR

Standard Deviation

A measure of how spread out data is from the mean. E.g. compare consistency of two students' test scores using standard deviation

Outlier

A score less than Q1 − 1.5xIQR or more than Q3 + 1.5xIQR. E.g. identify an unusually high test score in a data set

Box Plot (Five-Number Summary)

Displays minimum, Q1, median, Q3, maximum of a data set. E.g. compare distributions of two classes' test results using box plots

Budget

A financial plan tracking income and expenses over a period to manage money. E.g. create a monthly budget balancing income against rent, food, bills

Fixed vs Variable Expenses

Fixed = same amount each period (rent); Variable = changes each period (groceries). E.g. classify household expenses as fixed or variable when budgeting

Relative Frequency

Frequency of an event divided by total number of trials; estimates probability. E.g. estimate probability of rolling a 6 based on 100 trial results

Theoretical Probability

P(event) = number of favourable outcomes / total possible outcomes, based on reasoning not trials. E.g. calculate probability of drawing an ace from a standard deck

Complementary Events

P(not A) = 1 − P(A). E.g. find probability it does NOT rain given 30% chance of rain

Two-Way Tables

A table showing frequencies for two categorical variables to calculate probabilities. E.g. find probability a student is both male and plays sport from survey data

Simultaneous Equations (Substitution)

Solving two equations by substituting one into the other to find a common solution. E.g. find where two cost/revenue lines intersect (break-even point)

Simultaneous Equations (Graphical)

Solving two equations by finding the intersection point of their graphs. E.g. graph supply and demand lines to find equilibrium price

Non-Right-Angled Trig: Sine Rule

a/sinA = b/sinB = c/sinC; finds unknown sides/angles in any triangle. E.g. find a distance across a lake using two angles and one side

Non-Right-Angled Trig: Cosine Rule (Side)

c² = a² + b² − 2ab cosC; finds a side given two sides and included angle. E.g. find distance between two ships given their bearings and distances travelled

Non-Right-Angled Trig: Cosine Rule (Angle)

cosC = (a²+b²−c²)/2ab; finds an angle given all three sides. E.g. find the angle of a triangular paddock given all three side lengths

Area of a Triangle (Trig)

A = (1/2)ab sinC; area using two sides and the included angle. E.g. find area of a triangular block of land given two sides and an angle

Bearings

Direction measured clockwise from true north, given as a 3-digit angle. E.g. describe the direction a hiker walks from camp to a lookout

Non-Linear Relationships: Parabola

y = ax² + bx + c; a U-shaped graph, used to model quantities like projectile height. E.g. model the height of a ball thrown in the air over time

Non-Linear Relationships: Exponential

y = a(b)^x; models growth or decay that changes by a constant percentage. E.g. model population growth or radioactive decay

Non-Linear Relationships: Hyperbola

y = a/x (inverse relationship); one variable increases as the other decreases. E.g. model the relationship between speed and travel time for fixed distance

Normal Distribution

A symmetric bell-shaped distribution of continuous data around the mean. E.g. model heights of adults in a population

68-95-99.7 Rule

~68% of scores within 1 SD of mean, ~95% within 2 SD, ~99.7% within 3 SD. E.g. estimate percentage of population within a height range

Z-Score

z = (x−μ)/σ; measures how many standard deviations a score is from the mean. E.g. compare a student's result to the class average using standard deviation

Rate

A comparison of two quantities with different units. E.g. calculate speed in km/h from distance and time

Ratio

A comparison of two quantities with the same units, simplified to whole numbers. E.g. divide a mixture of concrete in a 1:2:3 ratio of cement, sand, gravel

Unit Rate

A rate expressed with a denominator of 1, used to compare prices/value. E.g. compare unit price ($/100g) of two different sized grocery items

Network (Graph Theory)

A diagram of vertices (points) connected by edges (lines), representing a system. E.g. model a road network connecting towns as vertices and edges

Weighted Network

A network where each edge has a value (weight) representing cost, time, or distance. E.g. model travel times between cities on a road network

Shortest Path

The minimum total weight path between two vertices in a network. E.g. find the fastest delivery route between a warehouse and store

Minimum Spanning Tree

A subgraph connecting all vertices with minimum total edge weight and no cycles. E.g. find cheapest way to connect all houses in an estate to power

Critical Path Analysis

A project planning method identifying the sequence of tasks that determines minimum completion time. E.g. determine the minimum time to complete a building project

Activity Chart / Network Diagram

Shows tasks, their durations, and dependencies (which tasks must finish before others start). E.g. draw a network diagram for stages of manufacturing a product

Float / Slack Time

The amount of time a non-critical task can be delayed without affecting the project finish date. E.g. determine how much a task can be delayed without delaying the whole project

Critical Path

The longest path through a network diagram; determines the minimum project completion time. E.g. identify which tasks cannot be delayed without delaying the whole project

Simple Interest

I = PRN; interest calculated only on the original principal. E.g. calculate interest earned on a simple interest savings account

Compound Interest / Future Value

FV = PV(1+r)^n; interest calculated on principal plus previously earned interest. E.g. calculate the value of an investment compounding annually over 10 years

Flat Rate Loan Interest

Interest calculated on the original loan amount for the entire loan term. E.g. calculate total interest on a personal loan charged at a flat rate

Reducing Balance Loan

Interest calculated on the remaining loan balance, which decreases with each repayment. E.g. calculate interest on a home loan where repayments reduce the balance owed

Annuity

A series of equal regular payments made or received at regular intervals, often earning compound interest. E.g. calculate the future value of regular monthly contributions to a retirement fund

Future Value of an Annuity

The total value of a series of regular payments plus interest at the end of the term. E.g. calculate total savings after making $200 monthly deposits for 5 years

Present Value of an Annuity

The lump sum today that is equivalent to a series of future regular payments. E.g. calculate the lump sum needed today to fund fixed withdrawals over 10 years