US Trends

what can we say about the relationship between the correlation r and the slope b of the least-squares line for the same set of data?

The correlation rrr and the slope bbb of the least-squares regression line for the same data always have the same sign (both positive or both negative), but generally different magnitudes.

What can we say about the relationship between the correlation r and the

slope b of the least-squares line?

Quick Scoop

For a single straight-line fit of yyy on xxx:

  • If r>0r>0r>0, then b>0b>0b>0 (line slopes upward).
  • If r<0r<0r<0, then b<0b<0b<0 (line slopes downward).
  • If r=0r=0r=0, then b=0b=0b=0 (horizontal line, no linear relationship).

So the key correct statement is:

r rr and bbb always have the same sign (+ or −).

They are not always equal, neither is always larger than the other, and bbb is not restricted to lie between −1-1−1 and 111 even though rrr is.

How are r and b mathematically connected?

In simple linear regression of yyy on xxx, the least-squares slope bbb (often written β^1\hat\beta_1 β^​1​) and the correlation rrr satisfy

b=r⋅sysxandr=sxsy⋅bb=r\cdot \frac{s_y}{s_x}\quad \text{and}\quad r=\frac{s_x}{s_y}\cdot bb=r⋅sx​sy​​andr=sy​sx​​⋅b

where sxs_xsx​ and sys_ysy​ are the sample standard deviations of xxx and yyy.

From this formula you can see:

  • The sign of bbb is the same as the sign of rrr, because sxs_xsx​ and sys_ysy​ are always positive.
  • The size of bbb depends on the units and spread of xxx and yyy through sysx\frac{s_y}{s_x}sx​sy​​.

Intuitive story: what’s really going on?

Imagine plotting height (x) vs. weight (y) for a group of people.

  • If taller people tend to weigh more, the cloud of points tilts upward , so r>0r>0r>0 and the regression line has b>0b>0b>0.
  • If taller people somehow tended to weigh less (a weird dataset!), the cloud would tilt downward , so r<0r<0r<0 and b<0b<0b<0.

Now change the units:

  • Switch height from inches to centimeters (multiply all x’s by a constant).
    • The correlation rrr stays the same.
* The **slope bbb** changes size, because the same change in height is now a different number of units, but its sign stays the same.

This is why rrr is a unit-free measure of strength and direction, while bbb has units (“change in y per one unit of x”) and can be any real number, not limited between −1-1−1 and 111.

Typical multiple-choice framing

When this question appears in homework or exams, you often see options like:

  • A. bbb is always larger than rrr.
  • B. rrr is always larger than bbb.
  • C. b=r2b=r^2b=r2.
  • D. Both rrr and bbb are always between −1-1−1 and 111.
  • E. rrr and bbb have the same sign (+ or −).

The correct choice is:

E. rrr and bbb have the same sign (+ or −).

Because:

  • A and B are false: the relative sizes depend on sy/sxs_y/s_xsy​/sx​.
  • C is false: r2r^2r2 is the coefficient of determination, not the slope.
  • D is false: only rrr is constrained between −1-1−1 and 111; bbb can be any real number.

SEO-style meta note

  • Focus keyword: what can we say about the relationship between the correlation r and the slope b of the least-squares line for the same set of data?
  • Core answer: They always share the same sign, and are linked by b=r⋅(sy/sx)b=r\cdot (s_y/s_x)b=r⋅(sy​/sx​), but only rrr is bounded between −1 and 1.

Information gathered from public forums or data available on the internet and portrayed here.