The other day I attended my first Ph.D. course, Quantitative Reasoning I. Sure, I’m not actually enrolled in the program or even the course but you can’t stop my edu-ma-cation! Now before I get into my academic-y post I need to preface it with Johnny T-Bird’s (the course instructor’s flashy name I just now gave him) great learning philosophy based on root vegetables. The analogy follows that you when you do constant drill activities, such as addition, it is akin to taking a carrot, writing down the addition algorithm on the carrot and placing it back in the ground. Then, later on, when you have to add two numbers you simply find your carrot, pull it out of the ground, read it, say “yep, that’s addition” and finish the problem. As time passes, however, if you don’t keep drilling a particular procedure you’ll eventually forget where you’re addition carrot lives. Sure, its still there but you can’t remember how to get to it. On the other hand, if you force students to understand a new concept based on what they already know its like a potato. The potato starts off as a single potato but then grows connections to other potatoes in the ground. Therefore, later on when you’re trying to figure out some crazy stats you pick up your crazy stats potato and up pops a string of knowledge used in connection. The theory of learning based on carrots and potatoes…very interesting.
So, during the first lecture we touched upon distributions of the normal and skewed type. First we reviewed the basics of mean, median and mode then we got into the definitions of positively and negatively skewed distributions. Since Johnny T-Bird enjoys growing potatoes and not carrots he told us to work out the relationship between mean and median in skewed distributions. My first instinct was that the mean would fall away from the median towards the tail of the skew (right of the median in a positive skew, left of the median in a negative skew) but the more I thought about it the concept didn’t mesh right in my mind. Of course, as some may remember from all those glorious math courses you took, my first instinct was correct and I had successfully created some potatoes…but were they rotten?
It seemed to me that two conditions could occur which would reverse the mean/median order:
- The tail was long enough that it contained more samples than that of the bulge of the skew
- The bulge was “heavy” enough (in terms of density and number of samples) so that it would pull the mean towards itself.
But this couldn’t possibly be right, the textbooks were all telling me the mean/median relationship rule is infallible…am I taking crazy pills? Well, turns out the mathematics community regards the infalliability (is that even a word? sure) of this rule to be a liiiiitle white lie. In fact, there are three instances where this rule fails with a surprising frequency:
- Multimodal distributions
- Discrete distributions
- The kind of distribution like I described above