If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Covariance and the regression line

Covariance, Variance and the Slope of the Regression Line. Created by Sal Khan.

## Want to join the conversation?

• Wouldn't this just be 0? Isn't the mean of the product of two variables the same thing as the product of the means of those two variables? Or am I wrong about this.
• The mean of the product is not the same as the product of the means. For example if x = [1,2,3] and y = [4,5,6] then the mean of the product of [x,y] would be (1 * 4 + 2 * 5 + 3 *6)/3 or (4 + 10 + 18)/3 = 32/3 = 10.666... Alternatively, the product of the means would be ((1+2+3)/3) * ((4+5+6)/3) = 2*5 = 10 So they are not equal. Hope this helps!
• Why did we assume the expected value of (XY) and can be approximated to sample mean of product of XY? Is this a standard rule? (This was discussed on the 11th minute into the video)
• Since X and Y are both random variables, the product of X and Y can be viewed as another random variable. With a "large enough" sample size, we can then use the Central Limit Theorem to approximate the expected value of XY with the sample mean of paired XY products.
• Can you make the connection between Pearsons Coefficient correlation (R) and the Coefficient of determination (R2). I'd like to know the difference. thanks
• The coefficient of determination is the PCC squared.
• This equation only works for the covariance of a population not a sample. How would you modify this equation to work for a sample?
• Anytime the notation "X-bar" is used (X with a line above it), this means we are dealing with a sample. So X-bar is a sample statistic that approximates the population parameter, i.e. E[X].
• Somebody plz tell me what is the practical usage of Covariance? Thanks! An Illustration would be very aprappreciated!
• Sal Khan, you made me fall in love with Statistics (the logically and step by step explained concepts, accompanying examples etc). God bless you much. How can I donate to this cause please?
• At to about , Sal says that E[E[x]] is just E[X]. I just can't intuitively see why this is true. I don't understand.
• So in other words, it's like we expect the expected value of X to never change with a particular X. Say the expected value of X is 'N' when X is 3. N will always be N as long as X = 3 . So we know that the expected value of X will always equal N if X = 3. EACH X HAS ONLY ONE EXPECTED VALUE... NOW I SEE, X can have more than one value, but each X can only have 1 expected value. That's why the expected value is a constant. Unlike X that might be 3 or 5, N will never change for a particular X value, so we can expect it to be N. The expected value of the expected value of X is a constant that changes as X changes. Thanks a lot guys!
• what is meant by expected value is it different from normal value of x and y