Friday, April 5, 2013

How To Interpret Distributions (Histograms)

Here's a set of Y-Distributions (histograms) I saw on the data visualization sub-Reddit.

On the left side, we have Polish language scores. On the right, we have mathematics.

Each row is a year... 2010 through 2012.

According to the notes on the page, these are the high-school exit exam scores for which passing is to receive 30% of the total available points.

Most people know what a "bell-shaped" curve looks like and those Polish language scores don't look like bells. In fact, it looks like right around the 30% mark, someone took the non-passing scores that were "close enough" and just handed out the passing score.

We sometimes see this in biotech manufacturing... where in order to proceed to the next step, you need to take a sample and measure the result. If there is a specification, you'll see a lot of just-passing results. What is euphemistically called, "Wishful sampling."

The process is the process and if the sampling is random, you expect a bell-shaped curve. In the case of Polish high school students, their Polish skills are what they are. What you're seeing is an artifact of the people grading the tests. I would bet a fair amount of money that teachers or schools are rewarded according to the number of students who pass this test.

Let's look at the mathematics scores. This "wishful grading" is going on in mathematics, but is far less pronounced. What is crazy is how different the distributions look from year to year (compared to the language histograms).

It's hard for me to think that mathematics skills of students across Poland vary that much from year to year. Like the U.S. News and World Reports rankings of schools, it's more likely that the difficulty of the test changes significantly from year to year... in this case with 2011 tests with particularly difficult questions.

Histograms say quite a bit about your process. What they never tell you is that the histograms also tell you quite a bit about your process specifications and how truthful your measurement systems are.

If I were the FDA... and I wanted to be mean about it, I'd request a distribution of measurements for every single process specification, and if I saw something like this "Polish language" test, someone has some explaining to do.

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