Find the coefficient of correlation if given, cov(x, y) = 420, ${\sigma }_x^2=484$ , and ${\sigma }_y^2=441$

Coefficient of Correlation Overview

The coefficient of correlation, often denoted as $r$, is a statistical measure that quantifies the strength and direction of a linear relationship between two variables, typically $x$ and $y$. Its value ranges from -1 to +1.

- A value close to +1 indicates a strong positive linear relationship.
- A value close to -1 indicates a strong negative linear relationship.
- A value close to 0 indicates a weak or no linear relationship.

Correlation Formula Explained

To calculate the coefficient of correlation, we use the following formula, which involves the covariance between the two variables and their respective standard deviations:

$$r=\frac{\text{cov}(x,y)}{{\sigma }_x{\sigma }_y}$$

Where:

- $\text{cov}(x,y)$ represents the covariance between variables $x$ and $y$. It measures how two variables change together.
- ${\sigma }_x$ represents the standard deviation of variable $x$. It is the square root of the variance of $x$.
- ${\sigma }_y$ represents the standard deviation of variable $y$. It is the square root of the variance of $y$.

Given Values Analysis

From the question, we are provided with the following values:Standard Deviation Calculation

Before we can calculate the coefficient of correlation, we need to find the standard deviations (${\sigma }_x$ and ${\sigma }_y$) from the given variances. The standard deviation is simply the square root of the variance.

- Standard Deviation of $x$: $${\sigma }_x=\sqrt{{\sigma }_x^2}=\sqrt{484}$$ $${\sigma }_x=22$$
- Standard Deviation of $y$: $${\sigma }_y=\sqrt{{\sigma }_y^2}=\sqrt{441}$$ $${\sigma }_y=21$$

Coefficient Calculation Steps

Now that we have all the necessary values—covariance, standard deviation of $x$, and standard deviation of $y$ —we can substitute them into the coefficient of correlation formula.

Given:

- $\text{cov}(x,y)=420$
- ${\sigma }_x=22$
- ${\sigma }_y=21$

Substitute these values into the formula:

$$r=\frac{\text{cov}(x,y)}{{\sigma }_x{\sigma }_y}$$

$$r=\frac{420}{22\times 21}$$

$$r=\frac{420}{462}$$

Performing the division:

$$r\approx \mathrm{0.909090...}$$

Rounding to three decimal places, the coefficient of correlation is approximately $0.909$.

Result Interpretation

A coefficient of correlation of approximately $0.909$ indicates a strong positive linear relationship between variables $x$ and $y$. This means that as $x$ increases, $y$ also tends to increase significantly.