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
- You may have heard this referred to as:
- Linear transformations of data can affect the statistical measures
How is the mean affected by a linear transformation of data?
- Let 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
- 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
How is the variance and standard deviation affected by a linear transformation of data?
- Let be the variance of some data
- 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
- You will need to multiply the standard deviation by the absolute value of k
- Standard deviation is
- If you add or subtract a constant a from all the values then the variance and the standard deviation stay the same
- Variance is
- Standard deviation is
Exam 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
-
- 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.
b)
Calculate the standard deviation of the standardised scores.