Analysing Critically (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Comparing a Model to Real Data

How do I compare results from a model to real data?

• In the modelling cycle one of the stages is to evaluate the model by comparing the results to real life data

• To do this, you may have to:

  • Substitute values into a formula used in a given model

    • The output can then be compared with any real life data provided

  • Draw a graph to represent a given model

    • A graph can show many inputs and outputs for the model at once

    • It can then be used to identify where the model is accurate, or where there are issues or limitations

• You may be asked to evaluate the efficacy (effectiveness) of the model using the real data

  • You should consider:

    • Is the model accurate for all values?

      • Does it need to be accurate for all values, or just some?

    • How accurate does the model need to be?

      • This will depend on its purpose and context

      • E.g. Models for diagnosing medical conditions to assist doctors must have a higher level of accuracy than a model for the motion of a character in a video game

Critical Analysis of Quoted Data

Why is critical analysis important?

• In the media data is quoted and claims are made, which may not be mathematically thorough

• This could be because:

  • The author is trying to persuade the viewer or reader about an issue

  • The author has an agenda, which is not simply informing the viewer or reader and being objective

    • E.g. More dramatic stories sell more newspapers, and receive more clicks online

  • Or it can simply be due to not having a deep enough understanding of the underlying data or methodology

• This is why it is important to critically analyse data that is presented to you

• In an exam, you may be asked to critically analyse, this means you must:

  • Look for and identify errors, suggest improvements and evaluate

    whether a claim is justified

How might statistics and data be misleading?

• There are numerous ways in which data can be presented to be misleading or unrepresentative

  • Selectivity of data or "Cherry picking"

    • Only a portion of the data may be presented

      • This portion could be biased to support a particular viewpoint rather than showing the entire picture

  • Data visualisation issues

    • Graphs and charts can be manipulated to exaggerate trends or obscure details

    • Misleading scales and axes are very common

    • Scatter graphs may be shown which show a correlation

      • A correlation does not necessarily imply causation

      • I.e. Two variables changing at similar rates may not be linked at all

  • Use of averages

    • Whilst averages are not inherently bad, data with a large range can be obscured by using an average

    • The mean will be more affected by outliers than the median

      • This can be used to represent the data in a way that suits the author's agenda

      • E.g. A company could advertise their mean wage in a jo advert, which has been affected by a single extremely high wage of the CEO, making it seem as though the wages are higher than they are

  • Headlines

    • There may be a lack of context in a headline

      • The headline without the full story may be worrying

      • E.g. "crime rate at an all-time high",

      • This could be true but only by 0.2%

      • Deeper investigation could show it's because of an increase in shoplifting, whereas violent crime has dramatically decreased

    • If a dramatic headline is presented first, before the data, it can influence how the reader or viewer consumes the information

    • E.g. If a TV show stated "chocolate hugely increases your life expectancy" and then showed a graph with a weak positive correlation

      • Viewers may feel differently about the data compared to if they were shown the graph first

  • Bias in the source

    • If the source of the data is not neutral on the matter being investigated, there may be bias

      • E.g. If a local swimming pool was being closed down and the owner and staff carried out a survey to gauge opinion, there may be bias in their methodology due to their personal views

    • Sampling techniques can also be biased

      • E.g. Surveying only people from a specific area, or a specific age category may produce biased results

  • Extrapolating from data inappropriately

    • Extrapolation is when values are predicted beyond the range of the data

    • Data that is extrapolated may be less reliable as there is no evidence to support it

How should I answer a critical analysis question?

• You may be asked to consider the validity of or critically analyse a claim made about given data

• Make sure you understand the data that is shown

  • What does it represent?

  • On a graph, pay close attention to the scale

  • What is the scope of the data?

  • What are the limitations of the data?

• Read the claim carefully

  • How general or specific is it?

  • Is the claim extrapolating or interpolating from the data?

    • Interpolation is when values are predicted within the range of the data

    • Extrapolation is when values are predicted beyond the range of the data

• Consider if the claim is valid or not, or to what extent it is valid

  • There is usually some truth in the claim, but it may be exaggerated or some aspects of the data may have been ignored

• Explain your reasoning clearly, quoting data points to evidence your argument