What Is The Correlation Coefficient For The Data Shown In The Table?${ \begin{tabular}{|c|c|} \hline X X X & Y Y Y \ \hline 0 & 15 \ \hline 5 & 10 \ \hline 10 & 5 \ \hline 15 & 0 \ \hline \end{tabular} }$A. 1 B. -1 C. 5 D. 10
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Introduction
In statistics, the correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It is a crucial concept in data analysis, as it helps us understand the relationship between two variables and make predictions about future data points. In this article, we will explore the concept of the correlation coefficient and how to calculate it using a given dataset.
What is the Correlation Coefficient?
The correlation coefficient, denoted by the symbol r, is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It is a value between -1 and 1, where:
- A value of 1 indicates a perfect positive linear relationship between the two variables.
- A value of -1 indicates a perfect negative linear relationship between the two variables.
- A value of 0 indicates no linear relationship between the two variables.
Calculating the Correlation Coefficient
To calculate the correlation coefficient, we need to use 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 x and y variables, respectively
- Σ denotes the sum of the values
Calculating the Means
To calculate the means, we need to add up all the values and divide by the number of data points.
For the x variable:
x̄ = (0 + 5 + 10 + 15) / 4 = 10
For the y variable:
ȳ = (15 + 10 + 5 + 0) / 4 = 7.5
Calculating the Deviations
To calculate the deviations, we need to subtract the mean from each data point.
For the x variable:
(xi - x̄) = (0 - 10), (5 - 10), (10 - 10), (15 - 10)
(xi - x̄) = -10, -5, 0, 5
For the y variable:
(yi - ȳ) = (15 - 7.5), (10 - 7.5), (5 - 7.5), (0 - 7.5)
(yi - ȳ) = 7.5, 2.5, -2.5, -7.5
Calculating the Products
To calculate the products, we need to multiply the deviations.
(xi - x̄)(yi - ȳ) = (-10)(7.5), (-5)(2.5), (0)(-2.5), (5)(-7.5)
(xi - x̄)(yi - ȳ) = -75, -12.5, 0, -37.5
Calculating the Sums
To calculate the sums, we need to add up the products.
Σ[(xi - x̄)(yi - ȳ)] = -75 - 12.5 + 0 - 37.5
Σ[(xi - x̄)(yi - ȳ)] = -125
Calculating the Squares
To calculate the squares, we need to square the deviations.
(xi - x̄)² = (-10)², (-5)², (0)², (5)²
(xi - x̄)² = 100, 25, 0, 25
(yi - ȳ)² = (7.5)², (2.5)², (-2.5)², (-7.5)²
(yi - ȳ)² = 56.25, 6.25, 6.25, 56.25
Calculating the Sums of Squares
To calculate the sums of squares, we need to add up the squared deviations.
Σ(xi - x̄)² = 100 + 25 + 0 + 25
Σ(xi - x̄)² = 150
Σ(yi - ȳ)² = 56.25 + 6.25 + 6.25 + 56.25
Σ(yi - ȳ)² = 125
Calculating the Correlation Coefficient
Now that we have all the necessary values, we can calculate the correlation coefficient.
r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])
r = -125 / (√150 * √125)
r = -125 / (12.25 * 11.18)
r = -125 / 137.31
r = -0.91
Conclusion
In this article, we explored the concept of the correlation coefficient and how to calculate it using a given dataset. We calculated the means, deviations, products, sums, squares, and sums of squares, and finally, we calculated the correlation coefficient. The correlation coefficient is a powerful tool in data analysis, and it helps us understand the relationship between two variables. In this case, the correlation coefficient is -0.91, which indicates a strong negative linear relationship between the two variables.
Answer
The correct answer is B. -1.
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Q: What is the correlation coefficient?
A: The correlation coefficient is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It is a value between -1 and 1, where:
- A value of 1 indicates a perfect positive linear relationship between the two variables.
- A value of -1 indicates a perfect negative linear relationship between the two variables.
- A value of 0 indicates no linear relationship between the two variables.
Q: How is the correlation coefficient calculated?
A: The correlation coefficient 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 x and y variables, respectively
- Σ denotes the sum of the values
Q: What is the difference between a positive and negative correlation coefficient?
A: A positive correlation coefficient indicates a positive linear relationship between the two variables, meaning that as one variable increases, the other variable also increases. A negative correlation coefficient indicates a negative linear relationship between the two variables, meaning that as one variable increases, the other variable decreases.
Q: What is the significance of a correlation coefficient of 0?
A: A correlation coefficient of 0 indicates that there is no linear relationship between the two variables. This means that the variables are not related in a way that can be described by a straight line.
Q: Can a correlation coefficient be greater than 1 or less than -1?
A: No, a correlation coefficient cannot be greater than 1 or less than -1. The correlation coefficient is always a value between -1 and 1, inclusive.
Q: How is the correlation coefficient used in real-world applications?
A: The correlation coefficient is used in a variety of real-world applications, including:
- Predicting stock prices
- Analyzing the relationship between variables in a dataset
- Identifying patterns in data
- Making predictions about future data points
Q: What are some common mistakes to avoid when calculating the correlation coefficient?
A: Some common mistakes to avoid when calculating the correlation coefficient include:
- Not checking for outliers in the data
- Not transforming the data if necessary
- Not using the correct formula for the correlation coefficient
- Not interpreting the results correctly
Q: Can the correlation coefficient be used to determine causality between variables?
A: No, the correlation coefficient cannot be used to determine causality between variables. Correlation does not imply causation, and there may be other factors at play that are influencing the relationship between the variables.
Q: How can the correlation coefficient be used to improve decision-making?
A: The correlation coefficient can be used to improve decision-making by:
- Identifying patterns in data
- Making predictions about future data points
- Analyzing the relationship between variables in a dataset
- Identifying areas for further research or investigation
Q: What are some common applications of the correlation coefficient in business?
A: Some common applications of the correlation coefficient in business include:
- Analyzing the relationship between sales and marketing efforts
- Identifying patterns in customer behavior
- Making predictions about future sales or revenue
- Identifying areas for cost savings or efficiency improvements