Modelling, Statistics & Application (DP IB Theory of Knowledge): Revision Note

Modelling, statistics & application

• Mathematics is often used to produce knowledge about the world by creating models and analysing data

• This raises ToK questions about how far mathematical results depend on assumptions, measurement choices and uncertainty

Mathematical modelling

• Mathematical modelling uses mathematical structures to represent a real-world situation to make predictions or give explanations

  • A model simplifies reality to focus on selected features

    • This means conclusions depend on what is included, ignored or treated as constant

• Modelling requires assumptions that act like “starting points” for the knowledge claim

  • If assumptions change, the model’s conclusions may change

• A model can be internally consistent but still produce unreliable knowledge about the world

  • Mathematical validity does not guarantee real-world accuracy

• Models often create a trade-off between simplicity and realism

  • Simple models are easier to use and test

  • More realistic models can become complex and sensitive to small changes in input

• The usefulness of a model depends on its purpose and context

  • A model can be good enough for one aim (rough prediction) but unsuitable for another (high-stakes decision)

Data representation

• Data representation shapes how mathematical information is interpreted and judged

  • The same data can suggest different conclusions depending on how it is displayed

• Representations can highlight patterns but also hide limitations

  • Aggregated data can make trends seem clearer

    • It can also conceal important variation within the dataset

• Choices in data processing affect knowledge claims

  • Deciding what counts as an outlier changes results

  • Choosing which average to use (mean, median, or mode) changes what the “typical” value seems to be

• Visual representations can influence how convincing a claim appears

  • Graph scale and axis choice can exaggerate or minimise differences

• Data categories and definitions affect what is counted in the first place

  • If terms are defined differently, comparisons can become misleading

• Interpreting data requires judgement, not just calculation

  • Deciding whether a pattern is meaningful depends on context and explanation, not only numbers

Probability and uncertainty

• Probability is used to express uncertainty when outcomes cannot be predicted with complete certainty

  • It changes a knowledge claim from “this will happen” to “this is likely to happen”

• Statistical conclusions depend on the reliability of the data and the method used

  • Biased samples produce biased probabilities

  • Small samples increase uncertainty and reduce confidence in generalisations

• Probability supports knowledge by making risk and uncertainty explicit

  • This allows decision-making to be justified using degrees of confidence

• Probability can be misinterpreted as certainty if it is treated as a prediction about individuals rather than populations

  • A probability statement describes the likelihood across many cases

    • It does not guarantee what happens in one case

• Uncertainty can come from randomness, incomplete information or measurement error

  • Different sources of uncertainty require different kinds of justification

• Confidence in a statistical claim depends on both calculation and interpretation

  • Statistical significance alone does not show that a difference is important in real terms

Real-world limitations

• Applying mathematics to the world introduces limitations that do not exist in purely abstract mathematics

  • Real-world quantities can be hard to measure accurately

  • Data collection methods can introduce systematic error

• Mathematical conclusions in applied contexts depend on the quality and relevance of evidence

  • Weak evidence can make a precise calculation produce a weak knowledge claim

• Real-world systems can be too complex for models to capture fully

  • Simplification can leave out causes that later become important

• Mathematical results can create an illusion of certainty because they look exact

  • Precision in the output can hide uncertainty in the inputs

• Knowledge claims based on application must consider scope and context

  • A model may work well in one setting but fail when conditions change

• Disagreement in applied mathematics often comes from different modelling choices rather than different calculations

  • Different assumptions or selecting different variables can lead to different interpretations and justifications