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Practising Mathematics: Developing the mathematician as well as the mathematics

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Big ideas

Interrelated big ideas underlying statistics.

Variation is the term that describes the differences we observe around us in every aspect of life, such as age, height, rainfall, temperature and prices. Variation is first mentioned in year 3 where students are asked to recognise variation in chance outcomes.

Expectation arises when we wish to harness variation and summarise data. For example,

• What is the typical price?
• What is the average temperature?
• What is the chance of tossing a head?

It arises throughout the curriculum but is referred to explicitly in year 6.

Distribution is the lens through which we look at variation, enabling us to identify and describe variation, and look for and confirm expectations. Distribution is the underlying concept for data representation at all levels of the curriculum.

Randomness describes a phenomenon in which the outcome of a single repetition is uncertain, but there is nonetheless a regular distribution of relative frequencies in a large number of repetitions. It is explicitly mentioned in year 8.

Informal inference is an evidence-based process balancing the variation and expectation found in sample data when answering a meaningful population-based question. It is implicit throughout the curriculum.

Variation

It is the variation in data that attracts our attention and prompts our questions. Without variation there would be no statistics!

Expectation

Expectation narrows the focus of data from its overall variation to a summary statistic, such as the arithmetic mean or most common value (mode), or to a theoretical value such as the probability of an event.

Distribution

Distribution provides the mechanism for studying variation and expectation by promoting visual representations of data.

Randomness

Randomness arises in processes that have unpredictable individual outcomes but that display patterns over the long term. The unpredictability creates variation and the pattern produces expectation.

Informal inference

Informal inference occurs when, from the evidence collected (perhaps heights of year 8 students in our class), we make a generalised claim (perhaps about the heights of year 8 students across Australia). We cannot be absolutely certain about our claim.