Calculate The Coefficient Of Correlation Of The Following Data:${ \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline X X X & 10 & 6 & 9 & 10 & 12 & 13 & 11 & 9 \ \hline Y Y Y & 9 & 4 & 6 & 9 & 11 & 13 & 8 & 4 \ \hline \end{tabular} }$(Ans.
Introduction
In statistics, the coefficient of correlation 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 calculate the coefficient of correlation for the given data set.
What is the Coefficient of Correlation?
The coefficient of correlation, denoted by the symbol 'r', is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It ranges from -1 to 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 close to 0 indicates no linear relationship between the two variables.
Calculating the Coefficient of Correlation
To calculate the coefficient of correlation, we need to follow these steps:
Step 1: Calculate the Mean of X and Y
The mean of X is calculated by summing up all the values of X and dividing by the number of values.
| X | 10 | 6 | 9 | 10 | 12 | 13 | 11 | 9 |
|---|---|---|---|---|---|---|---|---|
| Sum | 10 + 6 + 9 + 10 + 12 + 13 + 11 + 9 = 80 | |||||||
| Mean | 80 / 8 = 10 |
The mean of Y is calculated by summing up all the values of Y and dividing by the number of values.
| Y | 9 | 4 | 6 | 9 | 11 | 13 | 8 | 4 |
|---|---|---|---|---|---|---|---|---|
| Sum | 9 + 4 + 6 + 9 + 11 + 13 + 8 + 4 = 64 | |||||||
| Mean | 64 / 8 = 8 |
Step 2: Calculate the Deviations from the Mean
The deviations from the mean are calculated by subtracting the mean from each value.
| X | 10 | 6 | 9 | 10 | 12 | 13 | 11 | 9 |
|---|---|---|---|---|---|---|---|---|
| Deviation | 10 - 10 = 0 | 6 - 10 = -4 | 9 - 10 = -1 | 10 - 10 = 0 | 12 - 10 = 2 | 13 - 10 = 3 | 11 - 10 = 1 | 9 - 10 = -1 |
| Y | 9 | 4 | 6 | 9 | 11 | 13 | 8 | 4 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Deviation | 9 - 8 = 1 | 4 - 8 = -4 | 6 - 8 = -2 | 9 - 8 = 1 | 11 - 8 = 3 | 13 - 8 = 5 | 8 - 8 = 0 | 4 - 8 = -4 |
Step 3: Calculate the Sum of the Products of the Deviations
The sum of the products of the deviations is calculated by multiplying the deviations of X and Y and summing them up.
| X | 10 | 6 | 9 | 10 | 12 | 13 | 11 | 9 |
|---|---|---|---|---|---|---|---|---|
| Deviation | 0 | -4 | -1 | 0 | 2 | 3 | 1 | -1 |
| Y | 9 | 4 | 6 | 9 | 11 | 13 | 8 | 4 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Deviation | 1 | -4 | -2 | 1 | 3 | 5 | 0 | -4 |
| Product | 0 | 16 | 2 | 0 | 6 | 15 | 0 | 4 |
| Sum | 0 + 16 + 2 + 0 + 6 + 15 + 0 + 4 = 43 |
Step 4: Calculate the Sum of the Squares of the Deviations of X and Y
The sum of the squares of the deviations of X is calculated by squaring each deviation and summing them up.
| X | 10 | 6 | 9 | 10 | 12 | 13 | 11 | 9 |
|---|---|---|---|---|---|---|---|---|
| Deviation | 0 | -4 | -1 | 0 | 2 | 3 | 1 | -1 |
| Square | 0 | 16 | 1 | 0 | 4 | 9 | 1 | 1 |
| Sum | 0 + 16 + 1 + 0 + 4 + 9 + 1 + 1 = 32 |
The sum of the squares of the deviations of Y is calculated by squaring each deviation and summing them up.
| Y | 9 | 4 | 6 | 9 | 11 | 13 | 8 | 4 |
|---|---|---|---|---|---|---|---|---|
| Deviation | 1 | -4 | -2 | 1 | 3 | 5 | 0 | -4 |
| Square | 1 | 16 | 4 | 1 | 9 | 25 | 0 | 16 |
| Sum | 1 + 16 + 4 + 1 + 9 + 25 + 0 + 16 = 72 |
Step 5: Calculate the Coefficient of Correlation
The coefficient of correlation is calculated using the following formula:
r = Σ[(X - X̄)(Y - Ȳ)] / sqrt[Σ(X - X̄)² * Σ(Y - Ȳ)²]
where r is the coefficient of correlation, X̄ is the mean of X, Ȳ is the mean of Y, Σ is the sum, and (X - X̄) and (Y - Ȳ) are the deviations from the mean.
Plugging in the values, we get:
r = 43 / sqrt[32 * 72] r = 43 / sqrt[2304] r = 43 / 48 r = 0.896
Conclusion
In this article, we calculated the coefficient of correlation for the given data set. We used the formula for the coefficient of correlation and calculated the sum of the products of the deviations, the sum of the squares of the deviations of X and Y, and finally, the coefficient of correlation. The coefficient of correlation is a measure of the strength and direction of the linear relationship between two variables. In this case, the coefficient of correlation is 0.896, indicating a strong positive linear relationship between the two variables.
Discussion
The coefficient of correlation is a widely used statistical measure in data analysis. It helps us understand the relationship between two variables and make predictions about future data points. In this article, we calculated the coefficient of correlation for the given data set and found a strong positive linear relationship between the two variables. This indicates that as one variable increases, the other variable also tends to increase.
Limitations
One of the limitations of the coefficient of correlation is that it only measures the linear relationship between two variables. It does not account for non-linear relationships. Additionally, the coefficient of correlation is sensitive to outliers and can be affected by the presence of extreme values.
Future Research
Future research can focus on developing new statistical measures that can account for non-linear relationships and outliers. Additionally, researchers can explore the use of machine learning algorithms to predict the relationship between two variables.
References
- [1] Pearson, K. (1895). "Note on regression and inheritance in the case of two parents." Proceedings of the Royal Society of London, 58, 240-242.
- [2] Spearman, C. (1904). "The proof and measurement of association between two things." American Journal of Psychology, 15(1), 72-101.
Frequently Asked Questions: Coefficient of Correlation ===========================================================
Q: What is the coefficient of correlation?
A: The coefficient of correlation, denoted by the symbol 'r', is a statistical measure that calculates the strength and direction of the linear relationship between two variables.
Q: What is the range of the coefficient of correlation?
A: The coefficient of correlation ranges from -1 to 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 close to 0 indicates no linear relationship between the two variables.
Q: How is the coefficient of correlation calculated?
A: The coefficient of correlation is calculated using the following formula:
r = Σ[(X - X̄)(Y - Ȳ)] / sqrt[Σ(X - X̄)² * Σ(Y - Ȳ)²]
where r is the coefficient of correlation, X̄ is the mean of X, Ȳ is the mean of Y, Σ is the sum, and (X - X̄) and (Y - Ȳ) are the deviations from the mean.
Q: What is the difference between correlation and causation?
A: Correlation does not imply causation. Just because two variables are related, it does not mean that one variable causes the other. There may be other factors at play that are influencing the relationship between the two variables.
Q: What is the significance of the coefficient of correlation?
A: The coefficient of correlation is a widely used statistical measure in data analysis. It helps us understand the relationship between two variables and make predictions about future data points. It is also used to determine the strength and direction of the linear relationship between two variables.
Q: Can the coefficient of correlation be used to predict future values?
A: Yes, the coefficient of correlation can be used to predict future values. By using the coefficient of correlation, we can make predictions about the relationship between two variables and use this information to make informed decisions.
Q: What are some common applications of the coefficient of correlation?
A: The coefficient of correlation has many applications in various fields, including:
- Economics: to study the relationship between economic variables such as GDP and inflation.
- Finance: to study the relationship between stock prices and other financial variables.
- Medicine: to study the relationship between disease outcomes and treatment variables.
- Social sciences: to study the relationship between social variables such as education and income.
Q: What are some limitations of the coefficient of correlation?
A: Some limitations of the coefficient of correlation include:
- It only measures the linear relationship between two variables.
- It does not account for non-linear relationships.
- It is sensitive to outliers and can be affected by the presence of extreme values.
Q: Can the coefficient of correlation be used with non-normal data?
A: Yes, the coefficient of correlation can be used with non-normal data. However, it is recommended to use non-parametric tests or transformations to normalize the data before calculating the coefficient of correlation.
Q: What is the difference between the coefficient of correlation and the coefficient of determination?
A: The coefficient of correlation 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.
Conclusion
In this article, we have answered some frequently asked questions about the coefficient of correlation. We have discussed its definition, calculation, significance, and limitations. We have also explored its applications in various fields and its relationship with other statistical measures.