# What is the meaning of a quadratic relation when r = 0?

A website (on page 4) says:

The correlation coefficient is a measure of linear relationship and thus a value of r = 0 does not imply there is no relationship between the variables. For example in the following scatterplot which implies no (linear) correlation however there is a perfect quadratic relationship.

What is the meaning of a perfect quadratic relationship?

• I just wanna mention if the distribution is gaussian, that, $r=0$, does imply independence. Apr 23 '20 at 4:03
• What @Media means is when the joint distribution is Gaussian. Uncorrelated univariate Gaussians can be dependent; they just won’t be jointly Gaussian.
– Dave
Apr 23 '20 at 4:11
• There also is a good description on Wikipedia: en.wikipedia.org/wiki/Pearson_correlation_coefficient Apr 23 '20 at 10:34

Below I have graphed a linear and quadratic function, $$y = 50x + 3$$ and $$y=5x^2 - 1000$$, respectively.

Before calculating anything, we can observe that $$x$$ and $$y$$ are related to one another in some form. If we describe in words, we can say for the left plot that as $$x$$ increases so does $$y$$.

Similarly, for the right plot, we can say that as $$x$$ moves towards 0, from the left, $$y$$ decreases towards 0, and as $$x$$ moves away from 0 towards the right, $$y$$ increases.

In fact, $$x$$ and $$y$$ have a perfect linear relationship in the left plot, while $$x$$ and $$y$$ has a perfect quadratic relationship in the right plot. We can say this because the points, in both plots, lie on the red line

If we take it a step further and calculate the correlation between x and y.

We observe for the linear relationship that $$x$$ and $$y$$ has a correlation of 1. In contrast, for the quadratic relation, $$x$$ and $$y$$ has a correlation of 0.

Below you will find R code for the plot above, in addition, code to calculate the correlation coefficients

set.seed(1)
library(ggplot2)
library(dplyr)
library(gridExtra)

# Linear Function
linear = function(x){(50)*x + 3 }

# Create Data Frame
df = data.frame(x = c(-20:20),

# Plot Functions
lp = ggplot(subset(df,type == 'linear'), aes(x,y)) + geom_point() +
stat_function(fun=linear, colour="red") + ggtitle('Linear Relationship')

qp = ggplot(subset(df,type == 'quadratic'), aes(x,y)) + geom_point() +

grid.arrange(lp, qp, ncol=2)

# Calculate correlation coefficients
df %>% filter(type =='linear') %>% select(x,y) %>% cor %>% (function(x){x[1,2]})
df %>% filter(type =='quadratic') %>% select(x,y) %>% cor %>% (function(x){x[1,2]})



The perfect quadratic relationship means that the point lie perfectly on some parabola.

Consider a perfect linear relationship: points lying in a straight line, such as $$(1,2)$$, $$(2,5)$$, and $$(5, 14)$$ that are on $$y=3x-1$$.

Ditto for points lying perfectly on some parabola giving a perfect quadratic relationship. Consider $$(-2,4)$$, $$(1,1)$$, $$(0,0)$$, $$(1,1)$$, and $$(2,4)$$ that lie on $$y=x^2$$.