Correlation Coefficient: Understanding the Basics

The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It is often denoted by '$r$' and ranges from -1 to +1.

• A value close to +1 indicates a strong positive linear relationship (as one variable increases, the other tends to increase).
• A value close to -1 indicates a strong negative linear relationship (as one variable increases, the other tends to decrease).
• A value close to 0 indicates a weak or no linear relationship.

Pearson Correlation Coefficient Formula

For calculating the correlation coefficient between two sets of values, say X and Y, we typically use Pearson's product-moment correlation coefficient. The formula for Pearson's $r$ is given by:

$$r = \frac{n\sum XY - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}}$$

Where:

• $n$ is the number of data points.
• $\sum X$ is the sum of all X values.
• $\sum Y$ is the sum of all Y values.
• $\sum XY$ is the sum of the products of corresponding X and Y values.
• $\sum X^2$ is the sum of the squares of all X values.
• $\sum Y^2$ is the sum of the squares of all Y values.

Data Preparation and Summation for Correlation

Let's list the given values for X and Y and perform the necessary preliminary calculations. We are provided with the following data points:

• X: 3, 5, 1, 7, 5
• Y: 4, 3, 0, 8, 2

The number of data points, $n$, is 5. To use the formula, we need to calculate $\sum X$, $\sum Y$, $\sum XY$, $\sum X^2$, and $\sum Y^2$. We can organize these calculations in a table:

From the table, we have the following sums:

• $\sum X = 21$
• $\sum Y = 17$
• $\sum XY = 93$
• $\sum X^2 = 109$
• $\sum Y^2 = 93$

Calculating the Correlation Coefficient

Now, we will substitute these values into the Pearson correlation coefficient formula:

$$r = \frac{n\sum XY - (\sum X)(\sum Y)}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}}$$

Step 1: Calculate the Numerator

The numerator of the formula is $n\sum XY - (\sum X)(\sum Y)$.

$$ \text{Numerator} = (5 \times 93) - (21 \times 17) $$ $$ \text{Numerator} = 465 - 357 $$ $$ \text{Numerator} = 108 $$

Step 2: Calculate the Denominator's First Part ($X$ values)

The first part under the square root in the denominator is $n\sum X^2 - (\sum X)^2$.

$$ \text{Denominator Part 1} = (5 \times 109) - (21)^2 $$ $$ \text{Denominator Part 1} = 545 - 441 $$ $$ \text{Denominator Part 1} = 104 $$

Step 3: Calculate the Denominator's Second Part ($Y$ values)

The second part under the square root in the denominator is $n\sum Y^2 - (\sum Y)^2$.

$$ \text{Denominator Part 2} = (5 \times 93) - (17)^2 $$ $$ \text{Denominator Part 2} = 465 - 289 $$ $$ \text{Denominator Part 2} = 176 $$

Step 4: Calculate the Full Denominator

Now, multiply the two parts of the denominator and take the square root:

$$ \text{Denominator} = \sqrt{\text{Denominator Part 1} \times \text{Denominator Part 2}} $$ $$ \text{Denominator} = \sqrt{104 \times 176} $$ $$ \text{Denominator} = \sqrt{18224} $$ $$ \text{Denominator} \approx 134.996296 $$

Step 5: Calculate the Correlation Coefficient ($r$)

Finally, divide the numerator by the denominator:

$$ r = \frac{108}{134.996296} $$ $$ r \approx 0.8000207 $$

Rounding to one decimal place, the correlation coefficient is approximately 0.8.