Here is a bivariate data set.


xy

83

24

68

26

67

27

71

3

68

17

71

23

74

32

79

38


Find the correlation coefficient and report it accurate to four decimal places. 
*r* =

To calculate the correlation coefficient \( r \), we can use the formula:

\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]

where:

- \( n \) is the number of pairs
- \( \sum x \) 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

First, let's organize the dataTo calculate the correlation coefficient \( r \), we can use the formula:

\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}
\]

where:

- \( n \) is the number of pairs
- \( \sum x \) 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

First, let's organize the 

\[
x = [8, 6, 7, 1, 6, 7, 4, 9]
\]
\[
y = [3, 8, 7, 3, 7, 1, 2, 8]
\]

Now I'll calculate the necessary sums and plug them into the formula.The correlation coefficient \( r \) is approximately \( 0.3733 \), accurate to four decimal places.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How is the correlation coefficient interpreted in terms of strength and direction?
2. What does a correlation coefficient of 0.3733 imply about the relationship between the variables?
3. How would the correlation coefficient change if an outlier were introduced?
4. What other methods are available to determine the strength of a relationship between two variables?
5. Can correlation coefficients be used for non-linear relationships?

**Tip:** Always visualize your data with a scatter plot to better understand the relationship between variables before relying on the correlation coefficient alone.

Here is a bivariate data set. xy 83 24 68 26 67 27 71 3 68 17 71 23 74 32 79 38 Find the correlation coefficient and report it accurate to four decimal places. *r* =

Learn how to find the correlation coefficient for a bivariate data set, including a step-by-step solution using the Pearson correlation formula. This example includes data points and calculates the result to four decimal places, suitable for high school students in Grades 11-12.

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