# How can correlation change without slope changing

For a least regression line how can the correlation increase without the slope increasing or decreasing I am having some trouble understanding it. For the slope formula m = (r) * (sy/sx). How is it possible for the value of r to change without slope changing

One way is for the variability around the line to change.

For example imagine you have some scatter of points around a linear relationship, giving some particular correlation and slope. Now move the points closer to the line. The slope is the same but the correlation is higher.

Or instead move the points twice as far from the line. The slope is the same but the correlation is lower.

What's going on? Just rearrange your formula so it's $$r = m \frac{s_x}{s_y}$$. Now while you hold $$m$$ constant, how can you change $$r$$? By changing the other two terms. Increasing or decreasing $$s_y$$ will move $$r$$ in the opposite direction, just as described above.

(The other way would be to change $$s_x$$.)

Correlation measures the strength of the linear relationship between two variables, which is a measure of the tendency for the two variables to "move together" (or "move opposite"). It does not measure the rate at which the variables move together (or opposite). In particular, the correlation is a measure that is invariant to scale, meaning that if you change the scale of one of the variables, the correlation coefficient will remain the same.

To understand this better, consider the case of perfect positive or negative correlation (i.e., when you have a correlation coefficient of one or negative one). In this case the variables all lie exactly on a line, but that line can have any slope.$$^\dagger$$ If you change the scale of one of the variables (e.g., measuring a length in metres instead of centimetres) then you will change the slope of the line, but the correlation coefficient remains the same.

How can correlation change without slope changing?

Since correlation does not measure the slope of the line relating the data, it is unsurprising that it can change without the slope of the line-of-best-fit changing. Generally speaking, if the data become more tightly clustered around the line-of-best-fit then the correlation coefficient will increase in absolute magnitude (i.e., if it is positive it goes closer to one and if it is negative it goes closer to negative-one). Contrarily, if the data become less clustered around the line-of-best-fit then the correlation coefficient will decrease in absolute magnitude (i.e., go closer to zero).

$$^\dagger$$ ...except zero or infinity - if the data occur along a line with slope zero or infinity then one of the variables is constant, and the correlation is zero.