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

Dan Finlay

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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 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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    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths 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.