Optional derivations

These videos may help your understanding if you're someone who prefers proofs to intuitive explanations. The purpose is to explain/derive/prove some of the math that we have been skipping, because these modules have no math prerequisites. In particular, by the end of this lecture, you'll see why s^2 is an unbiased estimate of sigma^2. In other words, this is the mathematical reason that we use n-1 as the denominator for the sample variance.

A background in probability is helpful for understanding the explanations below.

The total length of these videos is approximately 36 minutes.

You can also view all the videos in this section at the YouTube playlist linked here.

Expected value

Derivation of variance

Variance as an expectation

Variance and covariance

This particular video has 25,000+ views! Why??

Moving toward showing that s^2 is unbiased

s^2 is unbiased

During this tutorial you learned:

• The definition and properties of expected value

• Variance as the expected value of the squared difference between X and its mean

• Properties of variance 

• The relationship between variance and the correlation between X & Y

• The definition of covariance

• To prove that s2 is an unbiased estimate of population variance, σ2

Terms and concepts:

expected value, variance, linearity of expectation, covariance, correlation, unbiased