Linear transformations
Big Idea: Transform data when applying a rule like temperature conversion, unit changes, or adding/multiplying by constant.
Linear transformation: newx = a·x + b where a and b are constants.
Effect on mean
Mean transforms the same way: If old mean = 10 and newx = 3x + 5: New mean = 3(10) + 5 = 35
Mean always transforms by the same rule as the data.
Worked example - Effect on mean
Apply the core method for Effect of Transformations in this section context.
Step by step
- Write the relevant formula or rule first to secure method marks.
- Substitute values from the question carefully and keep units/labels clear.
- Simplify and check whether the result is reasonable in context.
Final answer
Final answer should be clearly stated and interpreted for Effect of Transformations.
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Effect on spread
Only multiplier affects spread: Adding b shifts data but does not change spread.
Multiplying by a changes spread by factor |a|.
Worked example
If old SD = 5 and new data is y = 3x + 5, find new SD.
Step by step
- Use transformation rule for SD
- Substitute a=3 and old SD=5
Final answer
New SD is 15. The +5 shift does not affect spread.
Examples
| Transformation | Effect on mean | Effect on SD |
|---|---|---|
| Add 10 to all x | new mean = old mean + 10 | SD unchanged |
| Multiply all x by 2 | new mean = 2 × old mean | new SD = 2 × old SD |
| Convert cm to m (÷100) | new mean = old mean ÷ 100 | new SD = old SD ÷ 100 |
| Convert Celsius to Fahrenheit (×9/5 + 32) | new mean = 9/5 × old mean + 32 | new SD = 9/5 × old SD |