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

Algebra 1 (Eureka Math/EngageNY)

Course: Algebra 1 (Eureka Math/EngageNY) > Unit 2

Lesson 14: Topic D: Lesson 19: Interpreting correlation

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.

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How can we prove that the value of r always lie between 1 and -1 ?
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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?
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  Theoretically, yes. The r-value you are referring to is specific to the linear correlation.
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johra914
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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?
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Calculating the correlation coefficient is complex, but is there a way to visually
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- Saivishnu Tulugu
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  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.
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Is the correlation coefficient a measure of the association between two random variables?
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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?
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- Bradley Reynolds
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  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
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What's spearman's correlation coefficient?
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