Central Tendency in Skewed Distributions: A Lesson in Social Justice

I think I'm one of the few academic bloggers in the world who blogs in a discipline that has nothing to do with what I teach at the university. Theology and this blog are my hobby. By profession I'm an experimental psychologist. Which means that my day job is largely about teaching undergraduate and graduate statistics for psychology students. That's what pays the rent.

In short, nothing I write about in this blog is a part of my daily classroom teaching. I've never taught a class in theology. I don't teach in our College of Biblical Studies. I have no contact with our MDiv students. In my entire career at ACU I've guest lectured in a graduate bible class exactly...once.

Basically, I'm living two lives.

But my interest in theology does, from time to time, leak into my statistics lectures. A recent example.

Earlier this semester in my undergraduate statistics class we were talking about measures of Central Tendency and how they behave in skewed frequency distributions. Let me explain this.

A measure of Central Tendency is a number helping you ballpark the "middle" or "center" of a distribution. The most commonly used estimate of Central Tendency is the mean, the arithmetic average. You calculate the mean by adding up all the scores and then dividing by the total number of scores.

The second most common measure of Central Tendency is the median. The median marks the 50th percentile. Fifty percent of the scores are above the median and fifty percent fall below the median.

When a distribution of scores is bell-shaped and balanced (a normal distribution) both the mean and the median sit in the exact center splitting the distribution right down the middle. That is, the mean and median are equal. See the center distribution in the picture below.

Well, if that's the case, if the mean and median have the same value, why have two different measures?

Because this only happens in perfectly symmetrical distributions. When the distribution is skewed and asymmetrical the mean and median take on different values. Which is to say when a distribution is unbalanced there's no consensus on where the "middle" might be located. You could say the middle is where the mean sits. Or you could say the middle is where the median sits.

Okay, so what issues might affect that choice? Well, the key thing to note is that the mean is the most sensitive to the effects of skew. That is, the mean is very sensitive to extreme scores and, thus, is "tugged" more rightward or leftward compared to the median. This can be seen in the left and right distributions of the picture below (Note: the Mode is a third measure of central tendency and is the most frequently occurring score, thus it always sits at the top/highest point of the distribution):

Note how in the left picture (an example of negative skew) the mean is the most leftward measure of central tendency. That is, the mean is the most affected by the extreme scores on the left and is, thus, pulled furthest away from where the scores are piling up to the right. A similar thing is observed in the right picture (an example of positive skew) where the mean has been tugged the furthest rightward.

What is the implication of all this? Basically the following. When a distribution is "normal" people usually report the mean. But when the distribution is skewed we tend to report the median as the median is less affected by the extreme scores.

So where does theology fit into this?

Well, as I was describing all this to my students a month ago I asked the following question:

"When you hear people report the average family income of American households do you hear people say 'mean family income' or 'median family income'?"

A few students respond, "I think I hear people say 'median family income'."

"That's right. The measure of Central Tendency we tend to use in reporting family income is the median. Okay, so what does that tell you about the distribution of family incomes?"

"That it's skewed?"

"Right. When you hear people using the median that's often a clue that the distribution they are trying to describe is skewed. And the distribution of family incomes in America is skewed."

I follow up with another question. "Can you guess if the distribution of American incomes is positively or negatively skewed?" (Refer to the picture above to make your own guess.)

"Is it positively skewed?"

"Yes, it's positively skewed. The great majority of American incomes pile up on the left, on the low end. But there are a few extreme scores--the millionaires and billionaires--that pull the distribution to the right."

I draw this distribution on the board. For you, here is the distribution of American family incomes based on 2005 data (H/T to Visualizing Economics):

Note the positive skew. Note also the behavior of the median and mean (you may need to click on the graph for a closer look). The median is $46,326. The mean is $64,344. Again, the mean is more affected by the presence of the extreme scores, being pulled more rightward by those millionaires and billionaires (who are actually so rightward they are literally off the chart).

Okay, again where is the theology in all this? Well, it has to do with issues related to social justice. I made this point in class a month ago in the following way:

"Note how American family incomes are all piled up on the left. What does that mean? What are the practical implications of that?

Think about it this way. What is the income that officially marks poverty? It's around $20,000. Okay, now imagine a solidly middle class person, someone who makes, say, $50,000.

Given that, what is the distance between the middle class and poverty? About $30,000. Is that a lot of money?

What if someone in the family has a catastrophic illness? Can the hospital bills from a catastrophic or chronic illness run over $30,000 in a year? Oh my yes. Hospital bills from illnesses like that can run in the hundreds of thousands of dollars annually.

And what about disability or injury to the breadwinner? Or divorce? Or layoffs?

Shoot, if your car breaks down you're screwed. Many new cars are well over $30,000. And a good used car can deplete that $30,000 buffer pretty quickly.

The point being, the lesson of the positive skew, is that the distance between being middle class and being poor is very, very small. We're all piled up on the left of the distribution. So a little bit of bad luck--illness, injury, layoffs, something going wrong with the house or car--and a solidly middle class family can fall below the poverty line. Can even become homeless. And if not that, can struggle mightily and will have to forgo things like sending their kids to college. A little bit of bad luck and a family might suffer generational consequences.

Now consider this. If the distance between the middle class and poverty is about $30,000 what is the distance between being middle class and, say, being Donald Trump or Bill Gates?

If the distance between middle class and poverty is $30,000 the distance between middle class and being a millionaire is $950,000. See the difference? There's not really a difference, a few thousand dollars, between the working poor and the middle class. We are all piled up, the great majority of Americans, on the left. And the difference between all those folks and the rich is, well, measured in the millions if not billions of dollars. It's a distance that is hard to compute in your mind.

In short, to be middle class is to live with chronic vulnerability and uncertainty. A real day to day anxiety about waiting for the other shoe to drop. Which is why access to things like universal health care and unemployment benefits are so important, a social safety net for those at or near the bottom. Which, the positive skew tells us, is basically everyone."