Concepts of Variance (CFA Video)

This CFA video is to clarify some concepts of variance. What is variance? In the variance equation, why sometimes probability is used? Why sometimes N is used? Why sometimes N-1 is used?

Answer:

Variance is the expected square deviation of that variable from its expected value.

Var(X) = Sum(P(X)(X – E(X))^2)

P(X) represents the probability of the occurrence of X. By multiplying P(X) to the square deviation (and then sum over every X), we get the expected square deviation.

However, if the population is countable, P(X) is just 1/N (assume every outcome have the same change to appear).

Var(X) = Sum((1/N)(X – E(X))^2)

But if we are dealing with the sample (instead of the population), we need to multiple the sample variance by N/(N-1) as a correction to get the population variance.

Var_population(X)

= Var_sampe(X) * N/N-1

= Sum((1/N)(X – E(X))^2 )* N/N- 1

= Sum((1/N-1) (X – E(X))^2)