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IBMaths: AI HLDPRevision Notes4. Statistics & Probability4.1 Statistics Toolkit4.1.5 Linear Transformations of Data

Linear Transformations of Data (DP IB Maths: AI HL)

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Linear Transformations of Data

Why are linear transformations of data used?

  • Sometimes data might be very large or very small
  • You can apply a linear transformation to the data to make the values more manageable
    • You may have heard this referred to as:
      • Effects of constant changes
      • Linear coding
  • Linear transformations of data can affect the statistical measures

How is the mean affected by a linear transformation of data?

  • Let x with bar on top be the mean of some data
  • If you multiply each value by a constant k then you will need to multiply the mean by k
    • Mean is k x with bar on top
  • If you add or subtract a constant a from all the values then you will need to add or subtract the constant a to the mean
    • Mean is x with bar on top plus-or-minus a

How is the variance and standard deviation affected by a linear transformation of data?

  • Let sigma squared be the variance of some data
    • sigma is the standard deviation
  • If you multiply each value by a constant k then you will need to multiply the variance by k²
    • Variance is k squared sigma squared
    • You will need to multiply the standard deviation by the absolute value of k
      • Standard deviation is open vertical bar k close vertical bar sigma
    • If you add or subtract a constant a from all the values then the variance and the standard deviation stay the same
      • Variance is sigma squared
      • Standard deviation is sigma

Examiner Tip

  • If you forget these results in an exam then you can look in the HL section of the formula booklet to see them written in a more algebraic way
    • Linear transformation of a single variable

table row cell straight E left parenthesis a X plus b right parenthesis end cell equals cell a straight E left parenthesis X right parenthesis plus b end cell row cell Var left parenthesis a X plus b right parenthesis end cell equals cell a squared Var left parenthesis X right parenthesis end cell end table

    • where E(...) means the mean and Var(...) means the variance

Worked example

A teacher marks his students’ tests. The raw mean score is 31 marks and the standard deviation is 5 marks. The teacher standardises the score by doubling the raw score and then adding 10.

a)
Calculate the mean standardised score.

4-1-4-ib-ai-aa-sl-linear-trans-data-a-we-solution

b)
Calculate the standard deviation of the standardised scores.

4-1-4-ib-ai-aa-sl-linear-trans-data-b-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.