The expectation of the sampling distribution of Pearson's correlation

In summary, the shape of the sampling distribution of the Pearson product moment correlation coefficient will be a perfect V-shaped curve with a maximum at the true population correlation coefficient. However, as the sample size increases, the shape of the distribution becomes more normal and the expectation of the sampling distribution may not always be equal to the population correlation coefficient. This is because the joint distribution of the variables plays a role in determining the shape of the distribution. This is discussed in more detail in the article mentioned.
  • #1
Ad VanderVen
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TL;DR Summary
The shape of the sampling distribution of the Pearson product moment correlation coefficient depends on the size of the sample. Is the expectation of the sampling distribution of the Pearson product moment correlation coefficient always equal to the population correlation coefficient, regardless of the sample size?
The shape of the sampling distribution of the Pearson product moment correlation coefficient depends on the size of the sample. Is the expectation of the sampling distribution of the Pearson product moment correlation coefficient always equal to the population correlation coefficient, regardless of the sample size?
 
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  • #2
Ad VanderVen said:


The shape of the sampling distribution of the Pearson product moment correlation coefficient depends on the size of the sample.

Doesn't it also depend on the joint distribution of the variables involved?

Ad VanderVen said:
Is the expectation of the sampling distribution of the Pearson product moment correlation coefficient always equal to the population correlation coefficient, regardless of the sample size?

No, not according to the first paragraphs of https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=2403&context=jmasm. However, I haven't read the entire article.
 
  • #3
Consider the case of a sample of 2 data points.
 

1. What is the expectation of the sampling distribution of Pearson's correlation?

The expectation of the sampling distribution of Pearson's correlation is the average value of the correlation coefficient that would be obtained if an infinite number of samples were drawn from a population. It represents the true underlying correlation between two variables and is denoted by the Greek letter rho (ρ).

2. How is the expectation of the sampling distribution of Pearson's correlation calculated?

The expectation of the sampling distribution of Pearson's correlation is calculated using the formula ρ = cov(X,Y) / (σX * σY), where cov(X,Y) is the covariance between the two variables and σX and σY are the standard deviations of X and Y, respectively.

3. Why is the expectation of the sampling distribution of Pearson's correlation important?

The expectation of the sampling distribution of Pearson's correlation is important because it allows us to make inferences about the true correlation between two variables based on a sample. It also helps us to understand the relationship between variables and make predictions about future observations.

4. How does sample size affect the expectation of the sampling distribution of Pearson's correlation?

The expectation of the sampling distribution of Pearson's correlation is not affected by sample size. It remains the same regardless of the size of the sample, as long as the sample is representative of the population and meets the assumptions of the Pearson's correlation test.

5. Can the expectation of the sampling distribution of Pearson's correlation be negative?

Yes, the expectation of the sampling distribution of Pearson's correlation can be negative. This indicates a negative correlation between the two variables, meaning that as one variable increases, the other decreases. A negative value for ρ indicates a negative linear relationship, while a positive value indicates a positive linear relationship.

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