Change of origin and scale
Topic: Economic Data: Census, NSS, Surveys and Statistical Tools · NCERT: Class 11, Ch 6 "Correlation"
Meaning
The correlation coefficient r does not change when we shift the origin or change the scale of X and Y. Shifting the origin means subtracting a constant. Changing the scale means dividing by a constant.
If U = (X − A)/B and V = (Y − C)/D, then r_uv = r_xy, provided B and D have the same sign. If they have opposite signs, the size of r stays the same but its sign flips.
This lets us work with small, simple numbers instead of large ones. It is the basis of the step-deviation shortcut.
Example
Suppose X values are 1,000, 1,100, 1,200 and so on. We can take U = (X − 1,000)/100, which gives 0, 1, 2 and so on. Then we work out r from these small numbers. The answer is exactly the r for the original X and Y.
Don't confuse with
- Covariance: this does change with scale, because it carries the units of X and Y. Only r, which has no units, stays the same.
Related concepts
- Measures of central tendency
- Arithmetic mean
- Weighted arithmetic mean
- Median
- Quartiles
- Decile
- Percentiles
- Mode
- Bimodal and multimodal distribution
- Modal class