Math | Graded Assignment | Unit Test, Part 2 | Bivariate Data(Score For Question 2: _____ Of 5 Points)2. The Table Shows The Soil Density Of Several Soil Samples At Different Depths.$\[ \begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|} \hline Depth $(m)$ &
Math | Graded Assignment | Unit Test, Part 2 | Bivariate Data
Understanding Bivariate Data: A Comprehensive Analysis
Bivariate data is a type of data that involves two variables, which are often related to each other in some way. In this context, we are given a table that shows the soil density of several soil samples at different depths. The table provides us with a wealth of information, and our task is to analyze it and extract meaningful insights.
The Table: A Closer Look
| Depth (m) | Soil Density (g/cm³) |
|---|---|
| 0.5 | 1.2 |
| 1.0 | 1.3 |
| 1.5 | 1.4 |
| 2.0 | 1.5 |
| 2.5 | 1.6 |
| 3.0 | 1.7 |
| 3.5 | 1.8 |
| 4.0 | 1.9 |
| 4.5 | 2.0 |
| 5.0 | 2.1 |
Analyzing the Data
To begin our analysis, let's first examine the relationship between the depth of the soil sample and its density. We can see that as the depth increases, the soil density also increases. This suggests a positive correlation between the two variables.
Calculating the Correlation Coefficient
To quantify this relationship, we can calculate the correlation coefficient (r) using the following formula:
r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])
where xi and yi are the individual data points, x̄ and ȳ are the means of the two variables, and Σ denotes the sum.
Performing the Calculation
Using the data from the table, we can calculate the correlation coefficient as follows:
| Depth (m) | Soil Density (g/cm³) |
|---|---|
| 0.5 | 1.2 |
| 1.0 | 1.3 |
| 1.5 | 1.4 |
| 2.0 | 1.5 |
| 2.5 | 1.6 |
| 3.0 | 1.7 |
| 3.5 | 1.8 |
| 4.0 | 1.9 |
| 4.5 | 2.0 |
| 5.0 | 2.1 |
x̄ = (0.5 + 1.0 + 1.5 + 2.0 + 2.5 + 3.0 + 3.5 + 4.0 + 4.5 + 5.0) / 10 = 2.7 ȳ = (1.2 + 1.3 + 1.4 + 1.5 + 1.6 + 1.7 + 1.8 + 1.9 + 2.0 + 2.1) / 10 = 1.65
Σ(xi - x̄)² = (0.5 - 2.7)² + (1.0 - 2.7)² + (1.5 - 2.7)² + (2.0 - 2.7)² + (2.5 - 2.7)² + (3.0 - 2.7)² + (3.5 - 2.7)² + (4.0 - 2.7)² + (4.5 - 2.7)² + (5.0 - 2.7)² = 5.29 + 2.89 + 2.89 + 0.49 + 0.04 + 0.09 + 0.49 + 2.89 + 2.89 + 5.29 = 22.1
Σ(yi - ȳ)² = (1.2 - 1.65)² + (1.3 - 1.65)² + (1.4 - 1.65)² + (1.5 - 1.65)² + (1.6 - 1.65)² + (1.7 - 1.65)² + (1.8 - 1.65)² + (1.9 - 1.65)² + (2.0 - 1.65)² + (2.1 - 1.65)² = 0.22 + 0.22 + 0.22 + 0.22 + 0.01 + 0.01 + 0.01 + 0.22 + 0.22 + 0.22 = 1.22
r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²]) = (0.5 - 2.7)(1.2 - 1.65) + (1.0 - 2.7)(1.3 - 1.65) + (1.5 - 2.7)(1.4 - 1.65) + (2.0 - 2.7)(1.5 - 1.65) + (2.5 - 2.7)(1.6 - 1.65) + (3.0 - 2.7)(1.7 - 1.65) + (3.5 - 2.7)(1.8 - 1.65) + (4.0 - 2.7)(1.9 - 1.65) + (4.5 - 2.7)(2.0 - 1.65) + (5.0 - 2.7)(2.1 - 1.65) = -2.2 * -0.45 + -1.7 * -0.35 + -1.2 * -0.25 + -0.7 * -0.15 + -0.2 * -0.05 + 0.3 * 0.05 + 0.8 * 0.15 + 1.3 * 0.25 + 1.8 * 0.35 + 2.3 * 0.45 = 0.99 + 0.595 + 0.3 + 0.105 + 0.01 + 0.015 + 0.12 + 0.325 + 0.63 + 1.035 = 3.55
√[Σ(xi - x̄)²] = √22.1 ≈ 4.69 √[Σ(yi - ȳ)²] = √1.22 ≈ 1.1
r = 3.55 / (4.69 * 1.1) ≈ 0.73
Interpreting the Results
The correlation coefficient (r) is approximately 0.73, which indicates a strong positive correlation between the depth of the soil sample and its density. This suggests that as the depth of the soil sample increases, its density also increases.
Conclusion
In this analysis, we have examined the relationship between the depth of the soil sample and its density using bivariate data. We have calculated the correlation coefficient (r) and found that it is approximately 0.73, indicating a strong positive correlation between the two variables. This suggests that as the depth of the soil sample increases, its density also increases.
Graded Assignment
Score for Question 2: _____ of 5 points
Note
This is a sample graded assignment and the score is not actual. The score will be determined by the instructor based on the student's performance.
Math | Graded Assignment | Unit Test, Part 2 | Bivariate Data: Q&A
Understanding Bivariate Data: A Comprehensive Analysis
In our previous article, we explored the concept of bivariate data and analyzed a table that showed the soil density of several soil samples at different depths. We calculated the correlation coefficient (r) and found that it is approximately 0.73, indicating a strong positive correlation between the depth of the soil sample and its density.
Q&A: Bivariate Data and Correlation Coefficient
Q: What is bivariate data?
A: Bivariate data is a type of data that involves two variables, which are often related to each other in some way. In this context, we are given a table that shows the soil density of several soil samples at different depths.
Q: What is the correlation coefficient (r)?
A: The correlation coefficient (r) is a statistical measure that indicates the strength and direction of the linear relationship between two variables. In this case, we calculated the correlation coefficient (r) and found that it is approximately 0.73, indicating a strong positive correlation between the depth of the soil sample and its density.
Q: What does a positive correlation coefficient (r) indicate?
A: A positive correlation coefficient (r) indicates that as one variable increases, the other variable also increases. In this case, as the depth of the soil sample increases, its density also increases.
Q: What does a strong correlation coefficient (r) indicate?
A: A strong correlation coefficient (r) indicates that the relationship between the two variables is very consistent and predictable. In this case, the correlation coefficient (r) is approximately 0.73, which indicates a strong positive correlation between the depth of the soil sample and its density.
Q: How is the correlation coefficient (r) calculated?
A: The correlation coefficient (r) is calculated using the following formula:
r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])
where xi and yi are the individual data points, x̄ and ȳ are the means of the two variables, and Σ denotes the sum.
Q: What is the significance of the correlation coefficient (r) in real-world applications?
A: The correlation coefficient (r) is a valuable tool in many real-world applications, including finance, economics, and social sciences. It helps to identify the strength and direction of the relationship between two variables, which can inform decision-making and policy development.
Q: Can the correlation coefficient (r) be used to predict the future behavior of a variable?
A: While the correlation coefficient (r) can provide valuable insights into the relationship between two variables, it should not be used to predict the future behavior of a variable. Correlation does not imply causation, and there may be other factors at play that can affect the behavior of the variable.
Conclusion
In this Q&A article, we have explored the concept of bivariate data and the correlation coefficient (r). We have answered questions about the significance of the correlation coefficient (r) in real-world applications and the limitations of using it to predict the future behavior of a variable.
Graded Assignment
Score for Question 2: _____ of 5 points
Note
This is a sample graded assignment and the score is not actual. The score will be determined by the instructor based on the student's performance.