Deviations from the mean of x

Oct 13, 2025 by talynt.org 110 views

Introduction

In statistics, the linear correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It is a widely used statistical tool in various fields, including mathematics, economics, and social sciences. In this article, we will explore the concept of linear correlation coefficient and its application to a given data set.

Data Set

The given data set consists of two variables, x and y, with five observations each.

x	y
2	1.5
4	1.9
8	2.6
8	2.4
9	3.0

Part (a) - Calculate the Mean of x and y

To calculate the linear correlation coefficient, we need to first calculate the mean of x and y.

Mean of x: $\bar{x} = \frac{\sum x_i}{n} = \frac{2 + 4 + 8 + 8 + 9}{5} = 6.2$
Mean of y: $\bar{y} = \frac{\sum y_i}{n} = \frac{1.5 + 1.9 + 2.6 + 2.4 + 3.0}{5} = 2.36$

Part (b) - Calculate the Deviations from the Mean

Next, we need to calculate the deviations from the mean for both x and y.

Deviations from the mean of x: $(x_i - \bar{x}) = (2 - 6.2), (4 - 6.2), (8 - 6.2), (8 - 6.2), (9 - 6.2)$
Deviations from the mean of y: $(y_i - \bar{y}) = (1.5 - 2.36), (1.9 - 2.36), (2.6 - 2.36), (2.4 - 2.36), (3.0 - 2.36)$

Part (c) - Calculate the Sum of the Products of Deviations

Now, we need to calculate the sum of the products of deviations for both x and y.

Sum of the products of deviations: $\sum (x_i - \bar{x})(y_i - \bar{y}) = (2 - 6.2)(1.5 - 2.36) + (4 - 6.2)(1.9 - 2.36) + (8 - 6.2)(2.6 - 2.36) + (8 - 6.2)(2.4 - 2.36) + (9 - 6.2)(3.0 - 2.36)$

Part (d) - Calculate the Sum of the Squares of Deviations for x and y

Next, we need to calculate the sum of the squares of deviations for both x and y.

Sum of the squares of deviations for x: $\sum (x_i - \bar{x})^2 = (2 - 6.2)^2 + (4 - 6.2)^2 + (8 - 6.2)^2 + (8 - 6.2)^2 + (9 - 6.2)^2$
Sum of the squares of deviations for y: $\sum (y_i - \bar{y})^2 = (1.5 - 2.36)^2 + (1.9 - 2.36)^2 + (2.6 - 2.36)^2 + (2.4 - 2.36)^2 + (3.0 - 2.36)^2$

Part (e) - Calculate the Linear Correlation Coefficient

Now, we can calculate the linear correlation coefficient using the following formula:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Part (f) - Calculate the Numerator of the Linear Correlation Coefficient

First, we need to calculate the numerator of the linear correlation coefficient.

Numerator of the linear correlation coefficient: $\sum (x_i - \bar{x})(y_i - \bar{y}) = (2 - 6.2)(1.5 - 2.36) + (4 - 6.2)(1.9 - 2.36) + (8 - 6.2)(2.6 - 2.36) + (8 - 6.2)(2.4 - 2.36) + (9 - 6.2)(3.0 - 2.36)$

Part (g) - Calculate the Denominator of the Linear Correlation Coefficient

Next, we need to calculate the denominator of the linear correlation coefficient.

Denominator of the linear correlation coefficient: $\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2} = \sqrt{(2 - 6.2)^2 + (4 - 6.2)^2 + (8 - 6.2)^2 + (8 - 6.2)^2 + (9 - 6.2)^2} \sqrt{(1.5 - 2.36)^2 + (1.9 - 2.36)^2 + (2.6 - 2.36)^2 + (2.4 - 2.36)^2 + (3.0 - 2.36)^2}$

Part (h) - Calculate the Linear Correlation Coefficient

Now, we can calculate the linear correlation coefficient using the following formula:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Part (i) - Interpret the Linear Correlation Coefficient

Finally, we need to interpret the linear correlation coefficient.

Linear correlation coefficient: $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Conclusion

Introduction

In our previous article, we explored the concept of linear correlation coefficient and its application to a given data set. In this article, we will provide a comprehensive Q&A guide to help you understand the linear correlation coefficient and its interpretation.

Q: What is the linear correlation coefficient?

A: The linear correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It is a widely used statistical tool in various fields, including mathematics, economics, and social sciences.

Q: What is the formula for the linear correlation coefficient?

A: The formula for the linear correlation coefficient is:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Q: What is the range of the linear correlation coefficient?

A: The range of the linear correlation coefficient is -1 to 1. A value of 1 indicates a perfect positive linear relationship, while a value of -1 indicates a perfect negative linear relationship.

Q: How do I interpret the linear correlation coefficient?

A: To interpret the linear correlation coefficient, you need to consider the following:

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 no linear relationship.

Q: What is the difference between the linear correlation coefficient and the coefficient of determination?

A: The linear correlation coefficient measures the strength and direction of the linear relationship between two variables, while the coefficient of determination measures the proportion of the variance in one variable that is explained by the other variable.

Q: Can the linear correlation coefficient be used to predict the value of one variable based on the value of the other variable?

A: No, the linear correlation coefficient cannot be used to predict the value of one variable based on the value of the other variable. It only measures the strength and direction of the linear relationship between the two variables.

Q: What are the limitations of the linear correlation coefficient?

A: The linear correlation coefficient has several limitations, including:

It assumes a linear relationship between the two variables.
It is sensitive to outliers and non-normal data.
It does not account for non-linear relationships.

Q: How do I calculate the linear correlation coefficient using a calculator or software?

A: To calculate the linear correlation coefficient using a calculator or software, you need to follow these steps:

Enter the data into the calculator or software.
Select the linear correlation coefficient function.
Enter the variables and the data range.
Calculate the linear correlation coefficient.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand the linear correlation coefficient and its interpretation. We have covered topics such as the formula, range, interpretation, and limitations of the linear correlation coefficient. We hope this guide has been helpful in your understanding of the linear correlation coefficient.

Frequently Asked Questions

Q: What is the linear correlation coefficient? A: The linear correlation coefficient is a measure of the strength and direction of the linear relationship between two variables.
Q: What is the formula for the linear correlation coefficient? A: The formula for the linear correlation coefficient is:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Q: How do I interpret the linear correlation coefficient? A: To interpret the linear correlation coefficient, you need to consider the following:
- 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 no linear relationship.

Additional Resources

Linear Correlation Coefficient Calculator: A calculator that calculates the linear correlation coefficient.
Linear Correlation Coefficient Software: Software that calculates the linear correlation coefficient.
Linear Correlation Coefficient Tutorial: A tutorial that explains the linear correlation coefficient in detail.