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.

When it is said that to calculate the correlation coefficient is complex, is this simply because there are a lot of data points at play, or is the math difficult to comprehend for the course level?

lots of data points. definition is easy to understand.

Main content

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?

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.

Here are some facts about

r

It always has a value between $- 1$ ‍ and $1$ ‍.
Strong positive linear relationships have values of $r$ ‍ closer to $1$ ‍.
Strong negative linear relationships have values of $r$ ‍ closer to $- 1$ ‍.
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.

1

$Scatterplot A$ ‍	$Scatterplot B$ ‍

$Scatterplot C$ ‍	$Scatterplot D$ ‍
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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hamadi aweyso
Posted 5 years ago. Direct link to hamadi aweyso's post “i dont know what im still...”
i dont know what im still doing here
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Shreyes M
Posted 7 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 ?
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- Luis Fernando Hoyos Cogollo
  Posted 6 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/
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WeideVR
Posted 7 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?
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- dufrenekm
  Posted 7 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.
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Jake Kroesen
Posted 4 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
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johra914
Posted 7 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?
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User Name
Posted 10 months ago. Direct link to User Name's post “When it is said that to c...”
When it is said that to calculate the correlation coefficient is complex, is this simply because there are a lot of data points at play, or is the math difficult to comprehend for the course level?
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- cossine
  Posted 10 months ago. Direct link to cossine's post “lots of data points. defi...”
  lots of data points. definition is easy to understand.
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ayooyedemi45
Posted 4 years ago. Direct link to ayooyedemi45's post “What's spearman's correla...”
What's spearman's correlation coefficient?
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jamari-wood
Posted 2 days ago. Direct link to jamari-wood's post “i dont know what im still...”
i dont know what im still doing here
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Eric Bandera
Posted 10 months ago. Direct link to Eric Bandera's post “Based on the formula, I t...”
Based on the formula, I think I can imagine examples that give high scores for correlation, but have values that are off. Likewise, I can image examples where the score is lower and everything is on the line. For example, any data point that has an x value near the x mean will be valued at 0, even if the y value is in outer space. Yes, it is still counted as a data point, and so reduces the correlation, but not by much. Likewise, if there is a line with (x,y) at the edges (driving up the standard deviation up), and then the mass of data points, towards the mean in both x and y, that could drive the correlation down. In other words it seems to favor record sets with similar z indexes in the extremities, which to me does not seem exactly like a clean measurement.
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- daniella
  Posted 2 months ago. Direct link to daniella's post “Your observations about t...”
  Your observations about the correlation coefficient are valid. The correlation coefficient is influenced by the distribution and spread of data points, as well as their deviations from the means of the variables. While the formula for the correlation coefficient accounts for deviations and standard deviations, it may not capture all aspects of the relationship between variables, especially in non-linear relationships. As you mentioned, extreme data points or outliers can affect the correlation coefficient, potentially leading to misleading interpretations. It's essential to consider the context of the data and interpret the correlation coefficient alongside other statistical measures and graphical representations to gain a comprehensive understanding of the relationship between variables.
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fancy.shuu
Posted 6 years ago. Direct link to fancy.shuu's post “is correlation can only ...”
is correlation can only used in two features instead of two clustering of features?
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