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 SL)

Revision Note

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?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Exponents & Logarithms

1.1.3 Approximation & Estimation

1.1.4 GDC: Solving Equations

1.2 Sequences & Series

1.2.1 Language of Sequences & Series

1.2.2 Arithmetic Sequences & Series

1.2.3 Geometric Sequences & Series

1.2.4 Applications of Sequences & Series

1.3 Financial Applications

1.3.1 Compound Interest & Depreciation

1.3.2 Amortisation & Annuities

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear & Piecewise Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Voronoi Diagrams

3.4.1 Voronoi Diagrams

3.4.2 Toxic Waste Dump Problem

4. Statistics & Probability

4.1 Statistics Toolkit

4.1.1 Sampling & Data Collection

4.1.2 Statistical Measures

4.1.3 Frequency Tables

4.1.4 Linear Transformations of Data

4.1.5 Outliers

4.1.6 Univariate Data

4.1.7 Interpreting Data

4.2 Correlation & Regression

4.2.1 Bivariate data

4.2.2 Correlation Coefficients

4.2.3 Linear Regression

4.3 Probability

4.3.1 Probability & Types of Events

4.3.2 Conditional Probability

4.3.3 Sample Space Diagrams

4.4 Probability Distributions

4.4.1 Discrete Probability Distributions

4.4.2 Expected Values

4.5 Binomial Distribution

4.5.1 The Binomial Distribution

4.5.2 Calculating Binomial Probabilities

4.6 Normal Distribution

4.6.1 The Normal Distribution

4.6.2 Calculations with Normal Distribution

4.7 Hypothesis Testing

4.7.1 Hypothesis Testing

4.7.2 Chi-squared Test for Independence

4.7.3 Goodness of Fit Test