Calculate Correlation Coefficient For Data In A Table
In this comprehensive article, we will delve into the concept of correlation coefficient, its significance in data analysis, and how to calculate it for a given dataset. Specifically, we will address the question: What is the correlation coefficient for the data shown in the table?
| x | y |
|---|---|
| 0 | 15 |
| 5 | 10 |
| 10 | 5 |
| 15 | 0 |
Before we jump into the solution, let's first establish a solid understanding of what the correlation coefficient represents.
Understanding the Correlation Coefficient
The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. It is denoted by 'r' and its value ranges from -1 to +1. Here's a breakdown of what different values of 'r' signify:
- r = +1: This indicates a perfect positive correlation. As one variable increases, the other variable increases proportionally. The data points will form a perfect straight line with a positive slope.
- r = -1: This indicates a perfect negative correlation. As one variable increases, the other variable decreases proportionally. The data points will form a perfect straight line with a negative slope.
- r = 0: This indicates no linear correlation. There is no discernible linear relationship between the two variables. The data points will appear scattered randomly.
- 0 < r < +1: This indicates a positive correlation, but not a perfect one. As one variable increases, the other variable tends to increase as well, but the relationship is not perfectly linear. The closer 'r' is to +1, the stronger the positive correlation.
- -1 < r < 0: This indicates a negative correlation, but not a perfect one. As one variable increases, the other variable tends to decrease, but the relationship is not perfectly linear. The closer 'r' is to -1, the stronger the negative correlation.
The correlation coefficient is a crucial tool in various fields, including finance, economics, and social sciences, as it helps to identify relationships between variables and make predictions. However, it is essential to remember that correlation does not imply causation. Just because two variables are correlated does not necessarily mean that one causes the other. There might be other underlying factors influencing the relationship.
Calculating the Correlation Coefficient: A Step-by-Step Guide
To calculate the correlation coefficient (r), we use the following formula:
r = [ Σ (xi - x̄)(yi - ȳ) ] / [ √Σ (xi - x̄)² * √Σ (yi - ȳ)² ]
Where:
- xi represents the individual values of the x-variable.
- yi represents the individual values of the y-variable.
- x̄ represents the mean (average) of the x-variable.
- ȳ represents the mean (average) of the y-variable.
- Σ represents the summation (sum) of the values.
Let's break down the calculation into several steps for clarity:
-
Calculate the means (averages) of x and y:
- x̄ = (Σ xi) / n
- ȳ = (Σ yi) / n
Where 'n' is the number of data points.
-
Calculate the deviations from the mean for both x and y:
- (xi - x̄) for each x-value
- (yi - ȳ) for each y-value
-
Multiply the deviations for each data point:
- (xi - x̄)(yi - ȳ)
-
Square the deviations for both x and y:
- (xi - x̄)² for each x-value
- (yi - ȳ)² for each y-value
-
Sum the results from steps 3 and 4:
- Σ (xi - x̄)(yi - ȳ)
- Σ (xi - x̄)²
- Σ (yi - ȳ)²
-
Plug the sums into the formula for 'r':
- r = [ Σ (xi - x̄)(yi - ȳ) ] / [ √Σ (xi - x̄)² * √Σ (yi - ȳ)² ]
Now that we have the formula and the steps, let's apply it to the given data.
Calculating the Correlation Coefficient for the Given Data
Let's revisit the data provided:
| x | y |
|---|---|
| 0 | 15 |
| 5 | 10 |
| 10 | 5 |
| 15 | 0 |
We have four data points (n = 4). Let's follow the steps outlined above to calculate the correlation coefficient.
-
Calculate the means (averages) of x and y:
- x̄ = (0 + 5 + 10 + 15) / 4 = 30 / 4 = 7.5
- ȳ = (15 + 10 + 5 + 0) / 4 = 30 / 4 = 7.5
-
Calculate the deviations from the mean for both x and y:
x y xi - x̄ yi - ȳ 0 15 -7.5 7.5 5 10 -2.5 2.5 10 5 2.5 -2.5 15 0 7.5 -7.5 -
Multiply the deviations for each data point:
x y (xi - x̄)(yi - ȳ) 0 15 -56.25 5 10 -6.25 10 5 -6.25 15 0 -56.25 -
Square the deviations for both x and y:
x y (xi - x̄)² (yi - ȳ)² 0 15 56.25 56.25 5 10 6.25 6.25 10 5 6.25 6.25 15 0 56.25 56.25 -
Sum the results from steps 3 and 4:
- Σ (xi - x̄)(yi - ȳ) = -56.25 - 6.25 - 6.25 - 56.25 = -125
- Σ (xi - x̄)² = 56.25 + 6.25 + 6.25 + 56.25 = 125
- Σ (yi - ȳ)² = 56.25 + 6.25 + 6.25 + 56.25 = 125
-
Plug the sums into the formula for 'r':
- r = [ -125 ] / [ √125 * √125 ]
- r = -125 / 125
- r = -1
Therefore, the correlation coefficient for the given data is -1.
Interpreting the Result
The correlation coefficient of -1 indicates a perfect negative correlation between the variables x and y. This means that as x increases, y decreases proportionally, and the data points form a perfect straight line with a negative slope. In this specific dataset, we can observe that for every 5-unit increase in x, y decreases by 5 units, confirming the perfect negative linear relationship.
Conclusion
In conclusion, the correlation coefficient for the data shown in the table is -1. This signifies a perfect negative correlation between the variables x and y. We have walked through the process of calculating the correlation coefficient step-by-step, providing a clear understanding of the underlying concepts and the application of the formula. Remember that the correlation coefficient is a powerful tool for analyzing relationships between variables, but it is essential to interpret it correctly and avoid the trap of assuming causation from correlation.
The correct answer is B. -1.
This detailed explanation provides a comprehensive understanding of the correlation coefficient and its calculation, making it a valuable resource for anyone interested in data analysis and statistics. By breaking down the process into manageable steps and providing clear interpretations, this article empowers readers to confidently calculate and interpret correlation coefficients in various contexts.
