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 $\bar{x}$ 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 $k \bar{x}$
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 $\bar{x} \pm a$

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

Let $σ^{2}$ be the variance of some data
- $σ$ 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 $k^{2} σ^{2}$
- You will need to multiply the standard deviation by the absolute value of k
  - Standard deviation is $|k| σ$
- If you add or subtract a constant a from all the values then the variance and the standard deviation stay the same
  - Variance is $σ^{2}$
  - Standard deviation is $σ$

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

$\begin{array}{rcl} E (a X + b) & = & a E (X) + b \\ Var (a X + b) & = & a^{2} Var (X) \end{array}$

- 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.

Calculate the mean standardised score.

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

Calculate the standard deviation of the standardised scores.

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

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

Revision Note

Linear Transformations of Data

Why are linear transformations of data used?

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

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

Examiner Tip

Worked example

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Author: Dan