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  1. 4. What is the difference between confounding and lurking variables?


  1. BulletPaul Velleman gives a great post about confounding and lurking variables here.


  1. Bullet The list had a great discussion about confounding and lurking variables on Nov. 13th and 12th.  Click here and also go back one day to see the full discussion.


  1. BulletJosh Zucker discusses the issue of extraneous variables and gives a list of links.


  1. 5. What is the difference between independence and mutually exclusive?


If two events are independent, the outcome of one will not affect the outcome of the other.  i.e., Whether or not it rains and whether or not a coin flips heads or tails.


If two events are mutually exclusive, if one happens the other event cannot happen.  For example, in picking one M&M from a bag, I can find the probability of drawing green or red.  But if I draw green, I cannot draw red.


As we saw on the '02 MC #23, it is useful to notice that if two events are mutually exclusive, they affect each other quite powerfully:  if one of them happens, the other CANNOT occur.  Thus they are dependent.


Independence vs. mutually exclusive has been discussed on the list.


  1. 6. How much probability do I have to teach?


Floyd Bullard has submitted an awesome post about probability that can be read in the archives. 


I would strongly encourage rookies to read this post BEFORE teaching probability!


  1. 7. Why doesn’t X + X = 2X?


Here's a great explanation from Dave Bock:


For a short answer, try a thought experiment.


Let X represent the outcome when you roll a die. the 2X represents

rolling one die and doubling the result. The possible outcomes are {2,

4, 6, 8, 10, 12}; they are equiprobable.


On the other hand, X+X represents rolling two dice (or one twice). Now

the possible outcomes are {2, 3, 4, 5, 6, 7, ..., 12}. Some are far

less likely than others. Clearly this is a very different situation.


You can actually calculate both variances, but first just think about

the distributions. It should be pretty obvious that X+X is unimodal and

symmetric, peaking around 7 with very low tails while 2X is uniform

across the same range. The two means are the same, but X+X has a

smaller variance than 2X.


When confronting these situations, students must learn to ask

themselves how many random values they are working with. One random

value multiplied by a constant behaves much differently from summing

several different random values.


I urge students to recognize that a random variable in Statistics is

not the same animal as a variable in algebra. In algebra what we call a

"variable" is really just an unspecified constant. With that

understand, no matter what number I use for X I'll always substitute

that same value every time I see an X, so it must be true that X+X+X =

3X.


Then I put my Statistics hat on, declare them "random variables", and

pick up a die. I substitute the results of the first roll for the first

X, roll again for the value of the second X, etc. It's pretty clear now

that this X+X+X = 3X equation that seems so obvious in algebra is false

for random variables in Statistics. (One time the four values I

randomly rolled actually worked! The kids thought that was hilarious.

Their laughter at my bad luck clearly showed they understood the

issue.)


Read a list discussion thread about adding random variables.


Peter Flannigan-Hyde as written an article for AP Central about adding random variable.