Based On The Data Shown Below, Calculate The Correlation Coefficient (rounded To Three Decimal Places). \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline 5 & 13.1 \ \hline 6 & 13.54 \ \hline 7 & 13.08 \ \hline 8 & 17.72 \ \hline 9 & 17.76
Introduction
In statistics, the 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 calculate the correlation coefficient using the given data and discuss its significance.
Understanding the Data
The given data consists of two variables, x and y, with five observations each. The data is presented in the following table:
| x | y |
|---|---|
| 5 | 13.1 |
| 6 | 13.54 |
| 7 | 13.08 |
| 8 | 17.72 |
| 9 | 17.76 |
Calculating the Correlation Coefficient
To calculate the correlation coefficient, we will use the following formula:
r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²])
where r is the correlation coefficient, xi and yi are individual data points, x̄ and ȳ are the means of the x and y variables, respectively.
Step 1: Calculate the Means
First, we need to calculate the means of the x and y variables.
x̄ = (5 + 6 + 7 + 8 + 9) / 5 = 35 / 5 = 7 ȳ = (13.1 + 13.54 + 13.08 + 17.72 + 17.76) / 5 = 74.2 / 5 = 14.84
Step 2: Calculate the Deviations
Next, we need to calculate the deviations of each data point from the mean.
| x | x - x̄ | y | y - ȳ |
|---|---|---|---|
| 5 | -2 | 13.1 | -1.74 |
| 6 | -1 | 13.54 | -1.3 |
| 7 | 0 | 13.08 | -1.76 |
| 8 | 1 | 17.72 | 2.88 |
| 9 | 2 | 17.76 | 2.92 |
Step 3: Calculate the Products
Now, we need to calculate the products of the deviations.
| x | x - x̄ | y | y - ȳ | (x - x̄)(y - ȳ) |
|---|---|---|---|---|
| 5 | -2 | 13.1 | -1.74 | 3.48 |
| 6 | -1 | 13.54 | -1.3 | 1.69 |
| 7 | 0 | 13.08 | -1.76 | 0 |
| 8 | 1 | 17.72 | 2.88 | 2.88 |
| 9 | 2 | 17.76 | 2.92 | 5.84 |
Step 4: Calculate the Sum of Products
Next, we need to calculate the sum of the products.
Σ[(xi - x̄)(yi - ȳ)] = 3.48 + 1.69 + 0 + 2.88 + 5.84 = 14.89
Step 5: Calculate the Sum of Squares
Now, we need to calculate the sum of squares of the deviations.
Σ(xi - x̄)² = (-2)² + (-1)² + 0² + 1² + 2² = 4 + 1 + 0 + 1 + 4 = 10 Σ(yi - ȳ)² = (-1.74)² + (-1.3)² + (-1.76)² + 2.88² + 2.92² = 3.02 + 1.69 + 3.10 + 8.32 + 8.53 = 25.66
Step 6: Calculate the Correlation Coefficient
Finally, we can calculate the correlation coefficient using the formula.
r = Σ[(xi - x̄)(yi - ȳ)] / (√[Σ(xi - x̄)²] * √[Σ(yi - ȳ)²]) = 14.89 / (√10 * √25.66) = 14.89 / (3.162 * 5.065) = 14.89 / 16.005 = 0.929
Conclusion
In this article, we calculated the correlation coefficient using the given data. The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient is 0.929, which indicates a strong positive linear relationship between the x and y variables.
Discussion
The correlation coefficient is a widely used statistical tool in various fields. It is used to measure the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient is 0.929, which indicates a strong positive linear relationship between the x and y variables.
The correlation coefficient can be used to make predictions and forecasts. For example, if we know the value of x, we can use the correlation coefficient to predict the value of y. This can be useful in various fields, such as economics, finance, and social sciences.
Limitations
The correlation coefficient has some limitations. It only measures the linear relationship between two variables and does not account for non-linear relationships. It also assumes that the data is normally distributed and that the variables are independent.
Future Research
In future research, it would be interesting to explore the use of the correlation coefficient in various fields. For example, we could use the correlation coefficient to analyze the relationship between economic indicators and stock prices. We could also use the correlation coefficient to analyze the relationship between social media usage and consumer behavior.
References
- Pearson, K. (1895). Note on regression and inheritance in the case of two parents. Proceedings of the Royal Society of London, 58, 240-242.
- Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15(1), 72-101.
- Kendall, M. G. (1948). Rank correlation methods. Griffin.
Correlation Coefficient Q&A =============================
Introduction
In our previous article, we calculated the correlation coefficient using the given data. In this article, we will answer some frequently asked questions about the correlation coefficient.
Q: What is the correlation coefficient?
A: The 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 economics, finance, and social sciences.
Q: What is the range of the correlation coefficient?
A: The correlation coefficient ranges from -1 to 1. A value of 1 indicates a perfect positive linear relationship, while a value of -1 indicates a perfect negative linear relationship. A value of 0 indicates no linear relationship.
Q: What does a positive correlation coefficient indicate?
A: A positive correlation coefficient indicates a positive linear relationship between the two variables. This means that as one variable increases, the other variable also tends to increase.
Q: What does a negative correlation coefficient indicate?
A: A negative correlation coefficient indicates a negative linear relationship between the two variables. This means that as one variable increases, the other variable tends to decrease.
Q: What is the difference between correlation and causation?
A: Correlation does not imply causation. This means that just because two variables are correlated, it does not mean that one variable causes the other. There may be other factors at play that are driving the correlation.
Q: How is the correlation coefficient used in real-world applications?
A: The correlation coefficient is used in various real-world applications, including:
- Economics: To analyze the relationship between economic indicators, such as GDP and inflation.
- Finance: To analyze the relationship between stock prices and economic indicators.
- Social sciences: To analyze the relationship between social media usage and consumer behavior.
- Marketing: To analyze the relationship between advertising spend and sales.
Q: What are some common mistakes to avoid when using the correlation coefficient?
A: Some common mistakes to avoid when using the correlation coefficient include:
- Assuming causation: Correlation does not imply causation. Be careful not to assume that one variable causes the other.
- Ignoring non-linear relationships: The correlation coefficient only measures linear relationships. Be careful not to ignore non-linear relationships.
- Using the correlation coefficient with non-normal data: The correlation coefficient assumes that the data is normally distributed. Be careful not to use the correlation coefficient with non-normal data.
Q: What are some alternative measures of correlation?
A: Some alternative measures of correlation include:
- Spearman's rank correlation coefficient: This is a non-parametric measure of correlation that is used when the data is not normally distributed.
- Kendall's tau: This is a non-parametric measure of correlation that is used when the data is not normally distributed.
- Partial correlation coefficient: This is a measure of correlation that is used to control for the effect of a third variable.
Conclusion
In this article, we answered some frequently asked questions about the correlation coefficient. The correlation coefficient is a widely used statistical tool that is used to measure the strength and direction of the linear relationship between two variables. It is used in various real-world applications, including economics, finance, and social sciences. However, it is essential to be careful not to assume causation, ignore non-linear relationships, and use the correlation coefficient with non-normal data.