Page 1
,Contents:
Univariate Data ……………………….……………………….……………………….……………………….……………………..
Categorical Data (Nominal & Ordinal data) ………………………………………………………………………….……Page 4
Numerical Data (Discrete & Continuous data) ………………………………………………………………………….. Page 5
Segmented Bar Charts, Frequency tables, Bar Charts ………………………………………………………………Page 6-9
Histograms (incl. shape & outliers, centre & spread, CAS Instructions) ……………………………………. Page 10-14
Dot plots & Stem and Leaf Plots ………………………………………………………………………….……………...Page 14-15
Log scales (logarithm, conversion table) ………………………………………………………………………….……… Page 16
Measures of centre (mean, median & mode), and spread (range, IQR & standard deviation) ……….. Page 17-21
5 Number Summary (box plots, 5 number summary, shape of boxplots, outliers) ……………………… Page 22-27
Significant figures ………………………………………………………………………….………….………………………… Page 28
The 68-95-99.7% Rule ………………………………………………………………………….………….…………………… Page 29
Z-scores & Standardised Scores ………………………………………………………………………….………………… Page 30
Bivariate Data ……………………….……………………….……………………….……………………….………………………..
Explanatory & Response Variables ………………………………………………………………………….……………… Page 31
Associations between Categorical Variables & Numerical Variables ……………………………………… Page 32 - 35
Interpreting Scatterplots (direction, form/shape, strength) …………………………………………………… Page 36 - 37
Pearson’s Correlation Coefficient ………………………………………………………………………….………… Page 38 - 39
The Coefficient of Determination (r2) ………………………………………………………………………….…………… Page 40
Linear Associations ……………………….……………………….……………………….……………………….………………..
Residuals & Linear Modelling ………………………………………………………………………….……………… Page 41 - 42
Sketching Least Squares Regression Line on CAS …………………………………………………………………… Page 44
Analysing & Interpreting Least Squares Regression Line (Slope & Vertical “Y” Intercept) ………………… Page 44
Interpolation & Extrapolation ………………………………………………………………………….……………………… Page 45
The Residual Plot (Positive & Negative) ………………………………………………………………………….……….. Page 45
Linear & Non-linear relationships ………………………………………………………………………….………………. Page 46
Data Transformation ……………………….……………………….……………………….……………………….………………
The circle of Transformations ………………………………………………………………………….……………………. Page 48
Squared, Log & Reciprocal Transformations ………………………………………………………………………….… Page 49
CAS INSTRUCTIONS ………………………………………………………………………….…………………………… Page 50 - 51
Chapter Summary ………………………………………………………………………….…………………………………… Page 52
Times Series Data ……………………….……………………….……………………….……………………….………………….
Time series Data (increasing/decreasing trend pattern) ……………………………………………………………… Page 53
Seasonality ………………………………………………………………………….……………………….……………………. Page 53
Structural change, outliers & Irregular Fluctuations ………………………………………………………………….. Page 54
Smoothing time series using moving means (three-moving mean, five-moving mean) …………………… Page 55
Moving mean smoothing with centring & Smoothing time series using moving means ……………………. Page 56
Seasonal Indices ………………………………………………………………………….……………………………………… Page 57
Deseasonalising Data ………………………………………………………………………….………………………… Page 57 - 58
Correcting for Seasonality ………………………………………………………………………….………………………… Page 58
Calculating Seasonal Indices ………………………………………………………………………….……………… Page 59 - 60
Fitting trend line & forecasting ………………………………………………………………………….…………………… Page 61
Finance ……………………….……………………….……………………….……………………….………………………………..
SUMMARY – Linear Growth & Decay, Geometric Growth & Decay ………………………………………………. Page 62
Recurrence Relation …………………………………………………………………………………………………………… Page 63
Linear Growth & Decay & Simple Interest …………………………………………………………………………… Page 64 - 65
Depreciation (flat-rate & unit-cost depreciation) & Rate of Depreciation ………………………………… Page 65 - 67
Geometric Growth & Decay …………………………………………………………………………………………………… Page 68
Compound Interest Investments & Loans ………………………………………………………………………………… Page 69
Reducing Balance Depreciation ……………………………………………………………………………………………… Page 70
Page 2
,Nominal Interest Rate & Effective Interest Rate …………………………………………………………………… Page 71 - 72
Reducing Balance Loans ………………………………………………………………………………………………… Page 74
Annuities ……………………………………………………………………………………………………………………………. Page 74
Amortisation Tables ………………………………………………………………………………………………………… Page 76 - 77
Finance Solver Fields ……………………………………………………………………………………………………… Page 78 - 79
Using Finance Solver to find “Final Payments” & “cost of loan” ……………………………………………… Page 80 - 81
Interest only loans ………………………………………………………………………………………………………… Page 82 - 83
Perpetuities ………………………………………………………………………………………………………………… Page 84 - 85
FINANCE SUMMARY …………………………………………………………………………………………………………… Page 86
Networks ……………………….……………………….……………………….……………………….………………………………
Graphs & Networks (graph & loop) ………………………………………………………………………………………… Page 87
Connected graphs (undirected graph, simple graph, complete graph, degenerate graph, subgraph, isolated
vertex) ………………………………………………………………………………………………………………………… Page 88 - 89
Planar graphs (Euler’s formula) ……………………………………………………………………………………………… Page 89
Adjacency Matrix & Exploring & Travelling (walk, trail, path, circuit, cycle) ……………………………………… Page 90
Eulerian Trail & Eulerian Circuit ……………………………………………………………………………………………… Page 91
Hamiltonian Paths & Hamiltonian Cycle ………………………………………………………………………………… Page 91
Weighted Graphs and Networks & Dijkstra’s Algorithm ……………………………………………………………… Page 92
Trees (Spanning trees & Minimum spanning trees) & Prim’s Algorithm & Kruskal’s Algorithm …………… Page 93
Flow Problems (Maximum flow, Tracking flow method, Minimum cut method & Minimum flow) Page 94 - 95
Matching & Allocation problems (Bipartite graphs & The Hungarian Algorithm) ……………………………… Page 96
Drawing activity networks for precedence tables & Guidelines when drawing network diagrams ……… Page 97
Dummy Activities ………………………………………………………………………………………………………………… Page 97
Forward Scanning to determine EST, Backwards Scanning to determine LST, Float Time ………………… Page 98
Critical Path ………………………………………………………………………………………………………………………… Page 99
Crashing …………………………………………………………………………………………………………………………… Page 100
Matrices ……………………….……………………….……………………….……………………….………………………………
Matrices (rows & columns, naming elements, order of matrix) …………………………………………………… Page 101
Types of matrices (identity matrix) ………………………………………………………………………………………… Page 102
Transpose of a matrix ………………………………………………………………………………………………………… Page 103
Constructing a matrix given a rule of ij th term ………………………………………………………………………… Page 104
CAS Instructions ………………………………………………………………………………………………………… Page 105 – 106
Using a matrix to represent network diagrams & Equal matrices (matrix addition & subtraction) …… Page 107
Multiplying matrices (scalar multiplication) & percentage scalars ……………………………………………… Page 108
Multiplying matrices …………………………………………………………………………………………………… Page 109 - 110
Summing matrices, Matrix powers ………………………………………………………………………………………… Page 111
Matric inverse, determinant & matrix equations, singular matrix ………………………………………………… Page 112
Solving equations ……………………………………………………………………………………………………………… Page 113
Binary Matrices, Permutation Matrices & inverse of permutation matrices ………………………………… Page 114
Communication matrices & networks (multi-step, total & redundant communication) …………………. Page 115
Dominance Matrices (one-step, two-step & total dominances) ………………………………………… Page 116 - 117
Transition matrices & diagram ……………………………………………………………………………………………… Page 118
Transition matrices using recursion (recurrence relation, state matrix) ………………………………………. Page 119
Steady-state solution & Transition matrices – using the rule, using inverse of transition matrix ………. Page 120
Leslie Matrices (birth rates, age groups & survival rates) …………………………………………………… Page 121 - 122
Life Cycle transition diagram & population state matrix …………………………………………………………… Page 122
Recursion using Leslie Matrices …………………………………………………………………………………… Page 123 - 124
SUMMARY OF LESLIE MATRICES …………………………………………………………………………………………… Page 125
Page 3
, Univariate Data
1A – Classifying Data
Variables: are the quantities or qualities about which we record information.
When a data set contains only one variable that can it is called univariate.
e.g., the number of cars sold in a week.
Types of Data
When collecting data, it can be spilt into two main categories: Categorical or Numerical
Categorical Data:
Represents qualitative data, e.g., male/female, small/medium/large
(Fits or sorts into categories)
There are two types of categorical data: nominal and ordinal
1. Nominal data – Data that cannot be ordered into a logical sequence. The categories are simply names.
Examples:
• Gender / Race
• Hair Colour
• Country / Postcode
• Animals
• No / Yes
+ anything where order doesn’t matter (no natural order)
2. Ordinal data – Data that can be logically ordered or ranked. These categories can be ranked higher or lower
than others but doesn’t necessarily establish a numeric difference between each category. (Data has a
natural order)
Examples:
• House Number
• Age categories
• Rating scales / Likert scale
• Low – Medium – High
• Test Grade (A, B, C, D, E)
• Education level, e.g., High school, bachelor’s degree, Masters, PhD
Page 4
,Contents:
Univariate Data ……………………….……………………….……………………….……………………….……………………..
Categorical Data (Nominal & Ordinal data) ………………………………………………………………………….……Page 4
Numerical Data (Discrete & Continuous data) ………………………………………………………………………….. Page 5
Segmented Bar Charts, Frequency tables, Bar Charts ………………………………………………………………Page 6-9
Histograms (incl. shape & outliers, centre & spread, CAS Instructions) ……………………………………. Page 10-14
Dot plots & Stem and Leaf Plots ………………………………………………………………………….……………...Page 14-15
Log scales (logarithm, conversion table) ………………………………………………………………………….……… Page 16
Measures of centre (mean, median & mode), and spread (range, IQR & standard deviation) ……….. Page 17-21
5 Number Summary (box plots, 5 number summary, shape of boxplots, outliers) ……………………… Page 22-27
Significant figures ………………………………………………………………………….………….………………………… Page 28
The 68-95-99.7% Rule ………………………………………………………………………….………….…………………… Page 29
Z-scores & Standardised Scores ………………………………………………………………………….………………… Page 30
Bivariate Data ……………………….……………………….……………………….……………………….………………………..
Explanatory & Response Variables ………………………………………………………………………….……………… Page 31
Associations between Categorical Variables & Numerical Variables ……………………………………… Page 32 - 35
Interpreting Scatterplots (direction, form/shape, strength) …………………………………………………… Page 36 - 37
Pearson’s Correlation Coefficient ………………………………………………………………………….………… Page 38 - 39
The Coefficient of Determination (r2) ………………………………………………………………………….…………… Page 40
Linear Associations ……………………….……………………….……………………….……………………….………………..
Residuals & Linear Modelling ………………………………………………………………………….……………… Page 41 - 42
Sketching Least Squares Regression Line on CAS …………………………………………………………………… Page 44
Analysing & Interpreting Least Squares Regression Line (Slope & Vertical “Y” Intercept) ………………… Page 44
Interpolation & Extrapolation ………………………………………………………………………….……………………… Page 45
The Residual Plot (Positive & Negative) ………………………………………………………………………….……….. Page 45
Linear & Non-linear relationships ………………………………………………………………………….………………. Page 46
Data Transformation ……………………….……………………….……………………….……………………….………………
The circle of Transformations ………………………………………………………………………….……………………. Page 48
Squared, Log & Reciprocal Transformations ………………………………………………………………………….… Page 49
CAS INSTRUCTIONS ………………………………………………………………………….…………………………… Page 50 - 51
Chapter Summary ………………………………………………………………………….…………………………………… Page 52
Times Series Data ……………………….……………………….……………………….……………………….………………….
Time series Data (increasing/decreasing trend pattern) ……………………………………………………………… Page 53
Seasonality ………………………………………………………………………….……………………….……………………. Page 53
Structural change, outliers & Irregular Fluctuations ………………………………………………………………….. Page 54
Smoothing time series using moving means (three-moving mean, five-moving mean) …………………… Page 55
Moving mean smoothing with centring & Smoothing time series using moving means ……………………. Page 56
Seasonal Indices ………………………………………………………………………….……………………………………… Page 57
Deseasonalising Data ………………………………………………………………………….………………………… Page 57 - 58
Correcting for Seasonality ………………………………………………………………………….………………………… Page 58
Calculating Seasonal Indices ………………………………………………………………………….……………… Page 59 - 60
Fitting trend line & forecasting ………………………………………………………………………….…………………… Page 61
Finance ……………………….……………………….……………………….……………………….………………………………..
SUMMARY – Linear Growth & Decay, Geometric Growth & Decay ………………………………………………. Page 62
Recurrence Relation …………………………………………………………………………………………………………… Page 63
Linear Growth & Decay & Simple Interest …………………………………………………………………………… Page 64 - 65
Depreciation (flat-rate & unit-cost depreciation) & Rate of Depreciation ………………………………… Page 65 - 67
Geometric Growth & Decay …………………………………………………………………………………………………… Page 68
Compound Interest Investments & Loans ………………………………………………………………………………… Page 69
Reducing Balance Depreciation ……………………………………………………………………………………………… Page 70
Page 2
,Nominal Interest Rate & Effective Interest Rate …………………………………………………………………… Page 71 - 72
Reducing Balance Loans ………………………………………………………………………………………………… Page 74
Annuities ……………………………………………………………………………………………………………………………. Page 74
Amortisation Tables ………………………………………………………………………………………………………… Page 76 - 77
Finance Solver Fields ……………………………………………………………………………………………………… Page 78 - 79
Using Finance Solver to find “Final Payments” & “cost of loan” ……………………………………………… Page 80 - 81
Interest only loans ………………………………………………………………………………………………………… Page 82 - 83
Perpetuities ………………………………………………………………………………………………………………… Page 84 - 85
FINANCE SUMMARY …………………………………………………………………………………………………………… Page 86
Networks ……………………….……………………….……………………….……………………….………………………………
Graphs & Networks (graph & loop) ………………………………………………………………………………………… Page 87
Connected graphs (undirected graph, simple graph, complete graph, degenerate graph, subgraph, isolated
vertex) ………………………………………………………………………………………………………………………… Page 88 - 89
Planar graphs (Euler’s formula) ……………………………………………………………………………………………… Page 89
Adjacency Matrix & Exploring & Travelling (walk, trail, path, circuit, cycle) ……………………………………… Page 90
Eulerian Trail & Eulerian Circuit ……………………………………………………………………………………………… Page 91
Hamiltonian Paths & Hamiltonian Cycle ………………………………………………………………………………… Page 91
Weighted Graphs and Networks & Dijkstra’s Algorithm ……………………………………………………………… Page 92
Trees (Spanning trees & Minimum spanning trees) & Prim’s Algorithm & Kruskal’s Algorithm …………… Page 93
Flow Problems (Maximum flow, Tracking flow method, Minimum cut method & Minimum flow) Page 94 - 95
Matching & Allocation problems (Bipartite graphs & The Hungarian Algorithm) ……………………………… Page 96
Drawing activity networks for precedence tables & Guidelines when drawing network diagrams ……… Page 97
Dummy Activities ………………………………………………………………………………………………………………… Page 97
Forward Scanning to determine EST, Backwards Scanning to determine LST, Float Time ………………… Page 98
Critical Path ………………………………………………………………………………………………………………………… Page 99
Crashing …………………………………………………………………………………………………………………………… Page 100
Matrices ……………………….……………………….……………………….……………………….………………………………
Matrices (rows & columns, naming elements, order of matrix) …………………………………………………… Page 101
Types of matrices (identity matrix) ………………………………………………………………………………………… Page 102
Transpose of a matrix ………………………………………………………………………………………………………… Page 103
Constructing a matrix given a rule of ij th term ………………………………………………………………………… Page 104
CAS Instructions ………………………………………………………………………………………………………… Page 105 – 106
Using a matrix to represent network diagrams & Equal matrices (matrix addition & subtraction) …… Page 107
Multiplying matrices (scalar multiplication) & percentage scalars ……………………………………………… Page 108
Multiplying matrices …………………………………………………………………………………………………… Page 109 - 110
Summing matrices, Matrix powers ………………………………………………………………………………………… Page 111
Matric inverse, determinant & matrix equations, singular matrix ………………………………………………… Page 112
Solving equations ……………………………………………………………………………………………………………… Page 113
Binary Matrices, Permutation Matrices & inverse of permutation matrices ………………………………… Page 114
Communication matrices & networks (multi-step, total & redundant communication) …………………. Page 115
Dominance Matrices (one-step, two-step & total dominances) ………………………………………… Page 116 - 117
Transition matrices & diagram ……………………………………………………………………………………………… Page 118
Transition matrices using recursion (recurrence relation, state matrix) ………………………………………. Page 119
Steady-state solution & Transition matrices – using the rule, using inverse of transition matrix ………. Page 120
Leslie Matrices (birth rates, age groups & survival rates) …………………………………………………… Page 121 - 122
Life Cycle transition diagram & population state matrix …………………………………………………………… Page 122
Recursion using Leslie Matrices …………………………………………………………………………………… Page 123 - 124
SUMMARY OF LESLIE MATRICES …………………………………………………………………………………………… Page 125
Page 3
, Univariate Data
1A – Classifying Data
Variables: are the quantities or qualities about which we record information.
When a data set contains only one variable that can it is called univariate.
e.g., the number of cars sold in a week.
Types of Data
When collecting data, it can be spilt into two main categories: Categorical or Numerical
Categorical Data:
Represents qualitative data, e.g., male/female, small/medium/large
(Fits or sorts into categories)
There are two types of categorical data: nominal and ordinal
1. Nominal data – Data that cannot be ordered into a logical sequence. The categories are simply names.
Examples:
• Gender / Race
• Hair Colour
• Country / Postcode
• Animals
• No / Yes
+ anything where order doesn’t matter (no natural order)
2. Ordinal data – Data that can be logically ordered or ranked. These categories can be ranked higher or lower
than others but doesn’t necessarily establish a numeric difference between each category. (Data has a
natural order)
Examples:
• House Number
• Age categories
• Rating scales / Likert scale
• Low – Medium – High
• Test Grade (A, B, C, D, E)
• Education level, e.g., High school, bachelor’s degree, Masters, PhD
Page 4
