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Correlation and Covariance for GATE Exam

Module - 1 Probability and Statistics
Correlation and Covariance for GATE Exam

Correlation and covariance are fundamental statistical concepts used to analyze the relationships between variables. They help us understand how two variables are related and whether they tend to change together. In this lesson, we'll explore these concepts, their significance, calculations, interpretation, and real-world applications.

1. Understanding Correlation:

Definition:

Correlation measures the strength and direction of the linear relationship between two variables. It provides insights into how one variable changes as the other changes.

Significance of Correlation:

  • Predictive Power: Correlation helps predict one variable's behavior based on the other.
  • Risk Management: In finance, it's crucial for diversifying investments.
  • Quality Control: In manufacturing, it can indicate relationships between process parameters and product quality.

Calculating Correlation:

  • Pearson's Correlation Coefficient (r) is commonly used.
  • Formula:

Screenshot 2023-10-13 at 6.43.08 PM.png

  • r: Pearson's correlation coefficient, a measure of the linear relationship between two variables.
  • X and Y: The two variables being correlated.
  • ˉ_X_ˉ and _Y_ˉ: The means (average values) of X and Y respectively.
  • ∑: The sum symbol, indicating summation over a set of values.

This formula quantifies the strength and direction of the linear relationship between two variables X and Y by comparing their deviations from their respective means and their joint deviations from their means.

Interpreting Correlation:

  • r > 0 indicates a positive correlation (both variables tend to increase together).
  • r < 0 indicates a negative correlation (one variable increases as the other decreases).
  • The closer r is to 1 or -1, the stronger the correlation.

Example:

  1. Suppose you have data on the monthly sales of ice cream (X) and the average temperature (Y) over a year in a beach town. You want to calculate the covariance between these two variables to understand how they change together. Here's the data:
MonthIce Cream Sales (X)Average Temperature (Y)
Jan10018°C
Feb15020°C
Mar20022°C
Apr25025°C
May35028°C
Jun40030°C
Jul60032°C
Aug75032°C
Sep55030°C
Oct35028°C
Nov20025°C
Dec15020°C

To calculate the correlation, we shall follow these steps:

1. Calculate the means (average) for both X and Y:

Screenshot 2023-10-13 at 6.43.08 PM.png

we have the Pearson’s correlation formula

Screenshot 2023-10-13 at 7.26.53 PM.png

Plugging in the values

Screenshot 2023-10-13 at 7.27.22 PM.png

2. True or False: The correlation coefficient between two variables is always between -1 and 1.

Answer

True

2. Understanding Covariance:

Definition:

Covariance measures how two variables change together. It quantifies whether the variables tend to increase or decrease simultaneously.

Significance of Covariance:

  • Portfolio Management: In finance, it helps assess the risk and return of a portfolio.
  • Scientific Research: In biology, it can show how two species' populations change together.
  • Quality Control: In manufacturing, it can indicate relationships between manufacturing parameters and product quality.

Calculating Covariance:

  • Formula:

Sample Covariance Formula:

The sample covariance is typically used when you have a sample of data points. The formula is:

Screenshot 2023-10-13 at 7.28.25 PM.png

Population Covariance Formula:

The population covariance is used when you have data for the entire population. The formula is:

Screenshot 2023-10-13 at 7.29.57 PM.png

  • n is the number of data points .
  • X and Y are the individual data points.
  • ˉ_X_ˉ and _Y_ˉ are the means of the X and Y data points, respectively.

Interpreting Covariance:

  • Positive covariance suggests both variables tend to increase or decrease together.
  • Negative covariance suggests one variable increases as the other decreases.
  • The magnitude of covariance is challenging to interpret without standardized units.

Example:

  1. Let us calculate the covariance for the same ice cream sales example

Screenshot 2023-10-13 at 7.31.39 PM.png

Calculating this, we find:

Screenshot 2023-10-13 at 7.31.39 PM.png

The result, approximately -1482.67, indicates a negative covariance between X and Y. This means that as X values tend to be lower than their mean, Y values tend to be higher than their mean, and vice versa. The magnitude of the covariance (1482.67) suggests a fairly strong relationship between the variables.

  1. True or False: If the covariance between two variables is zero, then the two variables are independent.

Answer

False

Explanation: The covariance between two variables is a measure of how much the two variables vary together. The covariance between two variables can be zero even if the two variables are not independent. For example, the covariance between two variables that are both normally distributed with a mean of 0 and a standard deviation of 1 will always be zero, even though the two variables are not independent.

3. The Relationship Between Correlation r and Covariance Cov(X,Y)

Formula for Pearson's Correlation Coefficient (r):

Screenshot 2023-10-13 at 7.32.50 PM.png

Where:

  • r is the Pearson correlation coefficient.
  • Cov(X,Y) is the covariance between variables X and Y.
  • σX is the standard deviation of variable X.
  • σY is the standard deviation of variable Y.

Interpretation:

  1. When r=1: It indicates a perfect positive linear relationship between X and Y.
  2. When r=−1: It indicates a perfect negative linear relationship between X and Y.
  3. When r=0: It indicates no linear relationship between X and Y.

Exercise

1. The standard deviation of a variable X is 5. The standard deviation of a variable Y is 10. The covariance between X and Y is 20. What is the correlation coefficient between X and Y?

Answer

The correlation coefficient between X and Y is equal to the covariance between X and Y divided by the product of the standard deviations of X and Y. In this case, the covariance between X and Y is 20 and the standard deviations of X and Y are 5 and 10, respectively. Therefore, the correlation coefficient between X and Y is equal to 20 / (5 * 10) = 0.4

2. The covariance between two variables X and Y is 2. What is the correlation coefficient between X and Y?

Answer

The correlation coefficient between X and Y cannot be determined.

Explanation: The covariance between two variables is a measure of how much the two variables vary together. The correlation coefficient between two variables is a measure of the strength of the linear relationship between the two variables. The covariance between X and Y can be any value, positive or negative. The correlation coefficient between X and Y can only be positive or negative, and it cannot be greater than 1 or less than -1. Therefore, the correlation coefficient between X and Y cannot be determined from the information given.

4. Limitations and Interpretation

Limitations of Correlation and Covariance:

  • They measure linear relationships and may miss non-linear associations.
  • Correlation does not imply causation; a strong correlation doesn't mean one variable causes the other.

Interpretation Tips:

  • Always consider context and domain knowledge when interpreting results.
  • Be cautious about drawing causal conclusions based solely on correlation or covariance.

Real-World Applications:

1. Finance:

  • Portfolio Management: Correlation and covariance help investors diversify portfolios to reduce risk.
  • Risk Assessment: They are used to assess the risk of assets and their potential returns.

2. Economics:

  • Economic Indicators: Covariance helps analyze the relationships between economic indicators (e.g., GDP and unemployment rates).

3. Medicine:

  • Clinical Trials: Correlation is used to study relationships between patient characteristics and treatment outcomes.

4. Quality Control:

  • Manufacturing: Covariance can indicate how manufacturing parameters affect product quality.

5. Environmental Science:

  • Climate Studies: Correlation is used to analyze the relationships between variables like temperature and CO2 levels.

Conclusion

Correlation and covariance are fundamental concepts in statistics and data analysis. Correlation allows us to assess the strength and direction of linear relationships between two variables, providing insights into how they move together. On the other hand, covariance quantifies how two variables change in relation to each other, indicating whether they tend to increase or decrease together. Both correlation and covariance play crucial roles in understanding and modeling data, making informed decisions, and identifying patterns and trends in various fields, from finance and economics to science and engineering.

Key Takeaways:

  • Understand that correlation and covariance indicate relationships, not causation.
  • Interpret results based on context and domain knowledge.
  • Apply these concepts in diverse fields, from finance to environmental science.
  • Understanding correlation and covariance helps in identifying patterns, making predictions, and assessing relationships in real-world scenarios.
  • They are applied in finance, economics, medicine, quality control, and environmental science, among others.

Correlation and covariance are essential concepts for anyone dealing with data analysis and decision-making, enabling a deeper understanding of the relationships between variables and their real-world applications.

Practice Questions

1. If σx = σy and x, y are related by u = x + y; v = x − y, what is the cov(u,v)?

Answer

The covariance between u and v is equal to zero.

cov(u, v) = E[(u - u')(v - v')] = E[(x + y - u')(x - y - v')] where u' and v' are the means of u and v respectively.

Expanding the terms, we get

cov(u, v) = E[x^2 - 2xy + y^2 - u'^2 + u'v' - v'^2] Since x and y are related by u = x + y and v = x - y, we can see that the terms x^2 - 2xy + y^2 and u'^2 - v'^2 cancel each other out.

Therefore, the covariance between u and v is equal to

cov(u, v) = E[u'v'] = 0 This is because the expected value of the product of two independent random variables is zero. In this case, u' and v' are independent because they are the means of u and v respectively.

2. What is the correlation between x and a−x?

Answer

The correlation coefficient between two variables is a measure of how strongly they are related. It is calculated by taking the covariance of the two variables and dividing it by the product of their standard deviations.

In this case, the covariance of x and a-x is zero. This is because the expected value of the product of x and a-x is zero. The standard deviations of x and a-x are also equal, since they are both equal to the absolute value of a.

Therefore, the correlation coefficient between x and a-x is equal to -1 This means that x and a-x are perfectly negatively correlated. In other words, as x increases, a-x decreases, and vice versa.

3. The variance of return on investment A is 0.68, while the variance of return on investment B is 0.55. If the correlation coefficient between the returns on A and B is -.50, the covariance of returns on A and B is:

Answer

To calculate the covariance between two assets (A and B), you can use the following formula:

Cov(A, B) = Corr(A, B) * σ(A) * σ(B)

Where:

  • Cov(A, B) is the covariance between A and B.
  • Corr(A, B) is the correlation coefficient between A and B.
  • σ(A) is the standard deviation (square root of the variance) of returns on asset A.
  • σ(B) is the standard deviation (square root of the variance) of returns on asset B.
  • Variance of return on investment A (σ²(A)) = 0.68
  • Variance of return on investment B (σ²(B)) = 0.55
  • Correlation coefficient between A and B (Corr(A, B)) = -0.50

4. If the correlation coefficient between x and y is 0.6, the covariance is 27, and the variance of y is 25, what is the variance of x?

Answer

Given:

  • Correlation coefficient between x and y (Corr(x, y)) = 0.6
  • Covariance between x and y (Cov(x, y)) = 27
  • Variance of y (Var(y)) = 25
  1. Let the correlation coefficient between X and Y be 0.6. Random variables Z and W are defined as Z = X + 5 and W = (Y) / (3). What is the correlation coefficient between Z and W?
  • Answer

Screenshot 2023-10-13 at 7.37.02 PM.png

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