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Linear Transformations of Data (DP IB Applications & Interpretation (AI)): Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Roger B

Updated on

DP Applications & Interpretation (AI): SLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

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 to or 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 Tips and Tricks

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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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Subject Lead

    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.

    Roger B

    Reviewer: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.