AP® Statistics · Unit 1 · Topic 1.7
Changing Units and Linear Transformations
A linear transformation changes every value by adding a constant, multiplying by a constant, or both. Adding shifts center measures but leaves spread alone; multiplying stretches or shrinks both center and spread.
A linear transformation applies the same rule to every number in a data set: new value = a + bx. Adding a constant (a) slides the whole distribution left or right on the number line. Every value moves the same distance, so the gaps between values stay identical. That means the mean and median shift by that constant, but the standard deviation, which measures spread, does not change at all. Multiplying by a constant (b) is different: it stretches or compresses the distribution, pulling values apart or squeezing them together. That changes the mean, the median, and the standard deviation, all scaled by b (use the absolute value of b for standard deviation, since spread can't be negative).
Celsius to Fahrenheit is a full linear transformation: F = 1.8C + 32. The multiplication by 1.8 stretches the distribution, so the standard deviation grows by a factor of 1.8. The addition of 32 then just slides everything over, with no further effect on spread. This is why the conversion does both jobs at once: stretch, then slide.
This topic connects directly to z-scores, which are themselves built from a linear transformation: subtract the mean, divide by the standard deviation. Because z-scores are standardized this way, converting an entire data set to new units never changes anyone's z-score. A value that was 1.5 standard deviations above the mean in inches is still 1.5 standard deviations above the mean in centimeters. That invariance is exactly why z-scores let you compare values measured in different units.
Key terms
- Linear transformation
- A rule of the form new value = a + bx applied to every value in a data set.
- Standard deviation
- A measure of how spread out values are around the mean.
- z-score
- The number of standard deviations a value lies above or below the mean.
- Shift
- Moving every value by the same additive amount, which changes center but not spread.
- Scale
- Multiplying every value by the same constant, which changes both center and spread.
Watch it explained
- How it worksHow do the mean, the median, and the standard deviation each react to adding a constant?
- Worked exampleFind the new mean and standard deviation when 12 heights in inches become centimeters.
- Common mistakeWhy doesn't adding 5 to every value change the standard deviation at all?
- Going furtherWhat happens to a z-score when the whole data set is converted to different units?
Check yourself
End of topic.