# Critical Concept: Making Sense of Variability

Last week I introduced the five most critical topics I felt a professor of applied statistics could impart upon her students http://statisticalsage.wordpress.com/2012/06/10/the-most-critical-concepts-in-applied-statistics-treating-students-like-family/.

1. Making Sense of Variability
2. Capturing Variability
3. Normal Distribution
4. Sampling Distribution of the Means and Standard Error
5. Understanding Hypothesis Testing

This week, I would like to go into detail with the first of those five, “Making Sense of Variability.”

As this is the first critical topic, and I typically start teaching on the very first day of class, this is an issue I typically start to lay the foundation for  on the first day of class. I begin by  conduct activities that help students to think about variability.  Thus, it is typically the second day of class when I hit the foundational concept of variability, but before I enter into this, I provide a framework for students about knowledge. How do we know what we know? This is basic epistemology.

What is statistics? It is a way of us to gather knowledge. We are making sense out of variability. That is all that statistics is. Yet, for a student to understand that, he or she must first understand how do we gather knowledge? What are the “Ways of Knowing,” that is foundational epistemology? http://www.acperesearch.net/waysofknowing.pdf

• Empiricism – we learn through our observations/ perceptions (i.e., what are the data revealing?)
• Rationalism – we learn through the application of logic (i.e., if no one is in the woods when a tree falls, does it still make a sound?)
• Authority – we learn so much from what others tell us (i.e., what is your name? How do you know? Someone must have told you.)
• Intuition – gut instinct often reveals to us knowledge of a different type. (i.e., My dog’s love me. Science doesn’t tell me this, I just know.)

There are strengths and weaknesses to each way of knowing (follow the link above for additional details). By discussing this with students, we can start to understand why science has become a dance between empiricism and rationalism, as by combining the two … we first start with a logically deduced (or induced) hypothesis (rationalism) then we seek out data to test it (empiricism).  This goes so far beyond the steps of the scientific method discussed in middle schools throughout the US, as this progression is far from linear. I refer to it as a dance because there is give and take between these two approaches. Understanding how we know helps frame the context for statistics … it’s making sense of the observations. It’s just one small component of how we know anything.

In class assignment: This is fairly basic and should take no more than 10 – 15 minutes of class.

1. Start with asking students to make a list of what they know to be true.  (4 – 8 statements will work)
2. Have them form into groups of 4, and start to classify each of the statements into different ways of knowing. Any form of classification is fine.
3. Now, have them try to characterize each classification.
4. Bring the groups back together and see what they have found (in common/ different)
5. Now, introduce to them the four ways of knowing, providing an example and definition of each.
6. Have them put their examples into the four categories, and discuss.

Homework: Have a list of around 12 statements people “know,” and have them place them into one of the four categories.

From the four ways of knowing, we enter in into the foundational explanation of the Four Uses of Statistics. This topic is covered in detail in chapter 1 of Kiess and Green’s (2010) Statistical Concept for the Behavioral Sciences, 4/e. (http://www.pearsonhighered.com/educator/product/Statistical-Concepts-for-the-Behavioral-Sciences/9780205626243.page)

It helps to provide the basis for which most statistics are used, while providing them with concrete examples.

1. Describe samples: what is the number of people who live in dormitories in the classroom?
2. Draw inferences from samples to populations: If I wanted to get a sense of everyone in the class, but didn’t have the time to ask, I could just ask a sample of five students … how many hours per week to you expect to study for this class, and infer that the mean of the sample will be similar to the mean of the class (the population). Of course, issue’s of sampling error can start to be discussed.
3. Test hypotheses, about the relationship between two or more variables: I typically have students form a hypothesis, and then we get a sample of data to see if it’s correct. If you aren’t comfortable going with a class created option (which often requires you to tweak the hypothesis on the spot), a good hypothesis is … when it rains, people feel down.
4. Find associations among variables: Does where you sit in class impact how well you are going to do in the class?

Incredibly, within a few examples, many concepts that underlie the use of statistics can be introduced to students, and then, as you progress through the class, they can be reinforced. Now, you may say … how do the uses of statistics relate to making sense of variability?

1. Descriptive statistics capture the variability a sample
2. Inferential statistics capture the variability in the sample and use it to infer what is going on with the population
3. The critical variability in hypothesis testing is variability due to individual differences of the subjects who just happen to be in your sample, that is variability due to sampling error. The statistic estimates the sampling error, pulls it off, and enables us to test the hypothesis.
4. Of course, statistics used to find associations are looking for how variables are covarying (varying together).

Thus, all four uses of statistics are making sense of variability in different ways. They also have different statistics that they use to make sense of the variability.

This is a very term rich lesson, and I often encourage students to make use of flash cards. There also flash card apps and electric flashcards, but my students have told me … writing out their own flashcards on index cards works the best.

I will admit … this foundation is a bit awkward … I liken it to visiting a lot where a huge, never seen before building is going to be built. You are starting from scratch. Students don’t know where they are going, any more than you know what that new building is going to look like, but helping students to think about what they know and how they know it, how statistics are used, and helping them to get used to terms used in research and statistics, they begin to get comfortable with the thought of learning statistics … and it starts with the Critical Concept of Making Sense of Variability.