**Property A**: For constant *a* and random variables *x*, y and *z*, the following are true both for the sample and population definitions of covariance

*cov*(x, y) =*cov*(y,*x*)*cov*(*x, x*) =*var*(*x*)*cov*(*a*, y) = 0*cov*(*ax*, y) =*a*·*cov*(x, y)*cov*(*x+z*, y) =*cov*(*x*, y)+*cov*(*z*, y)

Proof: We give the proofs for the population version of covariance. The proofs for the sample versions are similar.

a) Clear from the definition of covariance

b) Clear from the definitions of covariance and variance

c) Follows from

d) Follows from

e) Follows from

**Property B**: If *x* and y are random variables and* z = ax + b* where *a* and *b* are constants then the correlation coefficient between *x* and y is the same as the correlation coefficient between *z* and y.

Proof: By Property A

By Property 3 of Expectation

And so *stdev*(z) = *a · stdev*(*x*). Thus

**Property 1**

Proof: Let

Collecting together identical terms, together with a change in summation indices, we find:

and so

from which the result follows.

**Property 2**:

Proof: The proof is similar to the proof of Property 1.

**Property 3**:

Proof: The proof is similar to that of Property 2 of Expectation.

**Property 4**: The following is true for both for the sample and population definitions of covariance:

If *x* and y are independent then *cov*(*x, *y) = 0

Proof: The proof follows from Property 1d of Expectation and Property 3.

**Property 5: **The following are true both for samples and populations:

Proof: We give the proof of the first property for the population. The proof of the sample case is similar, as is the proof of the second property. By Property 2 of Expectation and Property 3

Sir

What is the meaning of property 1? How do you define t and e?

Colin,

The proof given is for Property 1 of http://www.real-statistics.com/reliability/. This is not the correct place for this proof. What I had intended to provide was the proof of Property 1 of http://www.real-statistics.com/correlation/basic-concepts-correlation/. I have now corrected this and so you should now see the proof of -1 <= r <= 1. Charles

I am confused by the term ” Average variance extracted (AVE)”. Dr. Charles, can you please explain this to me? Thank you.

Natalie,

I don’t see this term on the webpage that you are referencing. Are you referring to a term that I use on the website or is a term you found somewhere else? Generally this is a term used with Factor Analysis.

Charles