Distributions

It is always fun to connect with old college friends … this week, I had the privilege of talking with one of my long time friend’s son, who just finished his freshmen year. He needed a bit of help in the summer statistics class he is taking. Now, I know the prof teaching the class … top rate, one of the best, and this young man is quite bright, which may be why he recognized that though he could calculate the values, and spit back definitions, he didn’t really understand what he was doing.

I love working with students one on one, as it gives me a chance to see … what they are thinking, and thus often what information are they missing. When working with this young man, who had no problem calculating the mean, median, standard deviation, and correlation coefficient, he had no idea what a distribution really was. The idea of a univariate vs. bivariate distribution was even more foreign to him, even though his notes were filled with illustrations that I’m quite certain were well articulated during class.

He is not alone. An interesting article on teaching the concepts of distributions can be found http://www.stat.auckland.ac.nz/~iase/publications/isi56/IPM37_Reading.pdf . Distributions and Graphs are identified on CAUSEweb.org as a critical concepts for students to understand, and are often the beginning of the “end” so to speak, as if students miss mastering this concept, they lack the foundation for truly learning statistics.

Though it may seem like distributions are obvious for us, often for the students they are not.

I’ve been thinking about Piaget, the cognitive developmentalist, for a while. And often when I seek to figure out how to help students master concepts in statistics, I think of Piaget, the constructivist, who said children learn by interacting with their environment. Well, college students may not be “children” but they are developing their cognitive functions, and thus Piaget’s idea that before we try to have a child work abstractly with a concept, we must aid them in properly mentally representing the concept, works as well in applied college statistics as it does in elementary school. Helping students develop a complete mental representation  involves starting with the concrete examples, going to the semi-concrete, and ending with the abstract. Sure, everyone says … start with the concrete, but our natural inclination is to keep it abstract. As we prepare for the start of a new semester, I encourage everyone to look at the concepts students struggle with the most, and look toward ways to make them a bit more concrete.

In fact, as I often listen to students talk about how they have struggled in statistics classes, it often seems that professors (myself included) are simply too quick to jump into the abstract without first making sure that the student has mentally  represented the concept. The best way to do this is to have students work directly with a concept, physically. With distributions, it becomes easy. Collect some form of data from the students, and turn it into a distribution.

Basic examples of data I collect from students include:

Nominal Data: Students intention after graduation.

Interval Data: In pulling upon a topic of local interest, our school has a mascot with an interesting history. Asking questions, and then graphing students attitudes often results in a bimodal distribution.

Bivariate Distribution:  How many hours they slept the night before and how alert they feel today, or on raining or snowy days, how much the like the weather and their mood.

From this point, it becomes easier to show students distributions of varying types, and I have them identify what kind of data is being graphed in the distribution.

Here is a link to several distributions from which to choose to use in class. http://www.google.com/search?q=graphs+of+populations&hl=en&rls=com.microsoft:en-us&rlz=1I7GPCK_en&prmd=ivns&tbm=isch&tbo=u&source=univ&sa=X&ei=4DZFTt67N8n00gHkw_2NCA&ved=0CDcQsAQ&biw=1024&bih=587

Another activity is to use Legos to have students make their own distributions. Sure it’s a bit cheesy, but students can make the distributions, then describe it, and try to come up with something that might be graphed in this fashion (obviously, it would work best for a univariate, frequency distribution).

These activities of making distributions and describing distributions needs to be replicated, and required of the students for homework. The use of a relevant example for the students, and providing the background, functions as a form of making the otherwise abstract concept of distributions more concrete

 A more “semi-concrete” method, which enables to the student to often do things like change the mu or the sigma on a distribution, and see how it impacts the distribution is through using visual demonstration software like Mathematica (see a prior blog entries on this topic: https://statisticalsage.wordpress.com/2011/05/24/using-mathematica-deomnstrations-to-visualize-statistical-concepts/ for a general overview of how Mathematica works or for specific demonstrations look to https://statisticalsage.wordpress.com/2011/01/08/before-the-semester-starts-im-playing-with-pictures/).

However, we can define distributions for students, and show them different types of distributions, discrete, continuous, univariate, bivariate, but by having students actually work, as directly as possible with the distributions will better assure students’ mastery of this seemingly simple concept that often gets students caught up.

Like most things in applied statistics, failure to truly understand distributions will inhibit a students’ mastery of the more specific normal distribution, and thus the z-score, and the sampling distribution of the means … and by this point, the student will have such a conceptual gap that success is not even possible.

I know, you’re probably saying … but I don’t have the time. In truth, well spent time on critical concepts and helping students to form those detailed and comprehensive mental representations will speed the teaching of additional concepts. In the end, I find, I don’t have the time to skip it!

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