How can we prove that the value of r always lie between 1 and -1 ?

Here https://sebastiansauer.github.io/why-abs-correlation-is-max-1/

Weaker relationships have values of r closer to 0. But r = 0 doesn’t mean that there is no relation between the variables, right? I mean, if r = 0 then there is no *linear* correlation, but we still could have a *non linear* correlation?

Theoretically, yes. The r-value you are referring to is specific to the linear correlation.

How does the slope of r relate to the actual correlation coefficient? For example, if the points of Scatter Plot A form a perfect line y=0.25x, and the points of Scatter Plot B also form perfect line y=3x, would the correlation coefficient be r=1 for both?

Yes, the correlation coefficient measures two things, form and direction. If you have two lines that are both positive and perfectly linear, then they would both have the same correlation coefficient. The only way the slope of the regression line relates to the correlation coefficient is the direction

Main content

Statistics and probability

Course: Statistics and probability > Unit 5

Lesson 2: Correlation coefficients

Correlation coefficient review

Google Classroom

The correlation coefficient r measures the direction and strength of a linear relationship. Calculating r is pretty complex, so we usually rely on technology for the computations. We focus on understanding what r says about a scatterplot.

What is a correlation coefficient?

Here are some facts about r:

It always has a value between minus, 1 and 1.
Strong positive linear relationships have values of r closer to 1.
Strong negative linear relationships have values of r closer to minus, 1.
Weaker relationships have values of r closer to 0.

Let's look at a few examples:

Want to learn more about the correlation coefficient? Check out this video.

Practice problem

Match the correlation coefficients with the scatterplots shown below.

start color #1fab54, start text, S, c, a, t, t, e, r, p, l, o, t, space, A, end text, end color #1fab54	start color #ca337c, start text, S, c, a, t, t, e, r, p, l, o, t, space, B, end text, end color #ca337c

start color #e07d10, start text, S, c, a, t, t, e, r, p, l, o, t, space, C, end text, end color #e07d10	start color #11accd, start text, S, c, a, t, t, e, r, p, l, o, t, space, D, end text, end color #11accd
A scatterplot labeled Scatterplot C on an x y coordinate plane. Points fall diagonally in a relatively narrow pattern.	A scatterplot labeled Scatterplot B on an x y coordinate plane. Points fall diagonally in a weak pattern.

Want to practice more problems like this? Check out this exercise on correlation coefficient intuition.

Want to join the conversation?

Sort by:

hamadi aweyso
Posted 4 years ago. Direct link to hamadi aweyso's post “i dont know what im still...”
i dont know what im still doing here
Button navigates to signup pageButton navigates to signup page
(25 votes)
Answer
Shreyes M
Posted 6 years ago. Direct link to Shreyes M's post “How can we prove that the...”
How can we prove that the value of r always lie between 1 and -1 ?
Button navigates to signup pageComment on Shreyes M's post “How can we prove that the...”
(12 votes)
Answer
- Luis Fernando Hoyos Cogollo
  Posted 5 years ago. Direct link to Luis Fernando Hoyos Cogollo's post “Here https://sebastiansau...”
  Here https://sebastiansauer.github.io/why-abs-correlation-is-max-1/
  Button navigates to signup page
  (2 votes)
WeideVR
Posted 6 years ago. Direct link to WeideVR's post “Weaker relationships have...”
Weaker relationships have values of r closer to 0. But r = 0 doesn’t mean that there is no relation between the variables, right? I mean, if r = 0 then there is no linear correlation, but we still could have a non linear correlation?
Button navigates to signup pageButton navigates to signup page
(6 votes)
Answer
- dufrenekm
  Posted 6 years ago. Direct link to dufrenekm's post “Theoretically, yes. The r...”
  Theoretically, yes. The r-value you are referring to is specific to the linear correlation.
  Button navigates to signup page
  (8 votes)
Jake Kroesen
Posted 3 years ago. Direct link to Jake Kroesen's post “I am taking Algebra 1 not...”
I am taking Algebra 1 not whatever this is but I still chose to do this
Button navigates to signup pageComment on Jake Kroesen's post “I am taking Algebra 1 not...”
(5 votes)
Answer
johra914
Posted 6 years ago. Direct link to johra914's post “Calculating the correlati...”
Calculating the correlation coefficient is complex, but is there a way to visually "estimate" it by looking at a scatter plot? Or do we have to use computors for that?
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
jlopez1829
Posted 3 years ago. Direct link to jlopez1829's post “Calculating the correlati...”
Calculating the correlation coefficient is complex, but is there a way to visually
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Saivishnu Tulugu
  Posted 3 years ago. Direct link to Saivishnu Tulugu's post “Yes on a scatterplot if t...”
  Yes on a scatterplot if the dots seem close together it indicates the r is high. If points are from one another the r would be low.
  Button navigates to signup page
  (1 vote)
Cha Kaur
Posted 4 years ago. Direct link to Cha Kaur's post “Is the correlation coeffi...”
Is the correlation coefficient a measure of the association between two random variables?
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
False Shadow
Posted 3 years ago. Direct link to False Shadow's post “How does the slope of r r...”
How does the slope of r relate to the actual correlation coefficient?

For example, if the points of Scatter Plot A form a perfect line y=0.25x, and the points of Scatter Plot B also form perfect line y=3x, would the correlation coefficient be r=1 for both?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- Bradley Reynolds
  Posted 3 years ago. Direct link to Bradley Reynolds's post “Yes, the correlation coef...”
  Yes, the correlation coefficient measures two things, form and direction. If you have two lines that are both positive and perfectly linear, then they would both have the same correlation coefficient. The only way the slope of the regression line relates to the correlation coefficient is the direction
  Button navigates to signup page
  (2 votes)
ayooyedemi45
Posted 3 years ago. Direct link to ayooyedemi45's post “What's spearman's correla...”
What's spearman's correlation coefficient?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
Jeric JosephM
Posted a month ago. Direct link to Jeric JosephM's post “why is it called a oven w...”
why is it called a oven when you put in the cold food but hot food comes out?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer