## Building blocks

Success in mathematics and statistics often rely on expanding a previously mastered technique. Indeed, my father was always fond of saying, “Mathematics is like a pyramid. You need a good foundation to reach the summit!”. Although this is truism should be self-evident in a well-structured curriculum, I often felt "lost" in math classes at school, especially in late high-school. The syllabus always seemed so disjointed to me and it always seemed like I had so many unrelated formulae and disparate techniques to remember. What I learned in geometry appeared completely unrelated to algebra; the Calculus was some kind of black magic that had nothing to do with anything I’d learned back in primary school. Needless to say, I did not enjoy high-school math and did not do as well as one might expect from someone who later becomes a math teacher. Indeed, it was this experience in high-school that made me want to pursue a teaching career, especially after I learned (or, in some cases, discovered) the "missing" connections between various topics from primary and secondary mathematics that could have helped me back in high-school. Before we begin looking at some of the typical techniques used in statistics, it is worthwhile covering some of these connections to ensure a common understanding and language of the mathematics underpinning the statistical techniques.

## Things you should know after high-school mathematics…

### … but probably don’t:

- Elementary Glossary
- Basic Mathematical Symbology
- Numbers of different sorts
- The Order of Operations
- Properties of the Real Numbers
- Set Relationships
- Rules of Divisibility
- Prime Factorisation
- Long Division (and Polynomial Division)
- Pascal’s Triangle
- Basic Modular Arithmetic
- Quadratic Formula
- Logarithm and Index Laws
- Basic Geometry and Trigonometry
- Pythagora’s Theorem (and the Trigonometric Identity)
- Straight Lines and the geometric interpretation
- Simple and Compound Interest
- Exponential Growth/Decay
- The "Grand-daddy" Equation
- Introductory Calculus

## Statistical Techniques

Many students (and researchers) are often daunted by the multitude of available statistical techniques they could use to analyse their data. How do they know they’ve picked the right technique? And, how do they use and interpret the technique?

It may surprise you to know that the choice of statistical technique actually starts with the research design, hence, one should make a choice about the technique(s) you will use as part of the research design process. Formulating a research question begins the process of statistical analysis (it establishes the statistical hypotheses you will test) and deciding how you will collect data in an attempt to answer the research question (i.e. the research design) specifies the levels of measurement for each study variable (i.e. nominal, ordinal, interval or ratio) and therefore what statistical techniques are appropriate.

The following attempts to present the common statistical terminology and techniques that students (*should*) meet early in their undergraduate studies, ordered by the way I would typically introduce them to undergraduate students in first-year.

### Techniques from first-year:

- The Science of Uncertainty
- Levels of Measurement
- Descriptive Statistics
- Basic Probability Theory
- Research Design
- The Logic of Hypothesis Testing
- Chi-Square Goodness-of-fit test
- Chi-Square Test of Independence
- One-Sample Z test
- One-Proportion Z test
- Two-Proportion Z test
- One-Sample t test
- Paired t test / Repeated-Measures t test
- Two-Sample t test / Independent Samples t test
- One-way ANOVA
- Two-way ANOVA
- Simple Linear Regression
- Multiple Linear Regression

Last updated: 6 November 2024