statistics

Math Rant: Screwy Stats

I have to say, I love me some statistics. Have I collected student scores and done a little analysis? Why, yes, I have. Have I collected and graphed data related to my writing? Oh, wait, you already know I have.

The thing is, I also know the limitations of statistics—what it takes for them to be meaningful, how far you can or can’t take the results. That data I analyze from my students? I use it to give me some direction as a teacher, figuring whether things are improving, whether a particular concept fell through the cracks, etc. Not much more than that.

As we all know, of course, statistics on education can get used for a lot more. I get the need for assessment (in some form) and accountability (in some form), but often when I see articles reporting school success/failure, I wonder if the people involved have the first clue about statistics.

Case in point: I recently saw an online report about the 50 best and 50 worst schools in the state, in reference to percentage of students achieving proficiency on the state’s high-stake testing. It reported results for Language Arts, Science, and Math.

The first thing that struck me was that whether looking at the 50 best or 50 worst, the percent passing math was WAY lower than the other two the majority of the time. That made me scratch my head, so I glanced down at the comments.

Several people noted that AP students didn’t take the state test.

I haven’t had a chance to dig into it yet, but if true, it makes those reported percentages almost meaningless. “We want to see how your school measures up … but we’re not going to count the top students.”

This is why when I see statistics reported, I have next-to-no reaction. Not until I know more about where the numbers are coming from. In broader situations, I ask myself questions like, who was included in the sample? How was the sample selected? How were questions worded?

Be careful when reporting statistics as part of an argument. They may or may not back you up as much as you think. Dig a little deeper to find the whole story.

ETA: Did my own digging-a-little-deeper, and it’s actually worse than I thought. The last math courses to participate in the state test are Algebra I and Geometry. The website was reporting on the results of high schools, many of which around here are only grades 10-12. By that time, even the “average” students are past those levels. So the published results only showed the proficiency of the lowest group. No wonder the math percentages were so much worse than the other subjects (which I believe test higher numbers of students in high school).

Have you run into questionable statistics? Any pet peeves on how you see them reported? Do you find yourself completely confuzzled when facing the numbers?

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Chances Are, We Don’t Understand Chances

Personally, I think probability is one of the most fun math concepts to teach. Break out the dice, the coins, the different-colored marbles, and the spinners. Do a bunch of trials to see how the experimental compares to the theoretical.

Despite the fun, I see a lot of students get all the way to high school without a solid understanding of what probabilities really mean. Take, for example, these two questions:

People often think these are asking the same thing. Our gut instinct for #1 is that if we’ve already gotten an uncommon five heads in a row, surely the chance of getting heads again isn’t that good. But the coin doesn’t know what it landed on before. The situation only has two choices: heads or tails. For that single sixth flip, it has a 50% chance of landing heads just like every other time.

The situation in #2 is completely different. You’re taking all six flips as one situation, so there are a lot more “choices” for the results. All heads, all tails, one tail and five heads (with six different configurations for this one alone), and so on. There is only a 1/64, or a little more than 1.5% chance, of this happening.

The difference in the two is that in #1, the five heads in a row have already happened, and cannot influence the sixth flip.

It’s also good to talk about what makes a game fair or unfair, and why gambling isn’t such a great idea.

The thing about probabilities is that they often make an assumption about all else being equal. The coin or dice being evenly weighted. Every individual outcome (like heads or tails) having an equal chance.

In life, we can’t always make that assumption. That’s where people sometimes confuse “probability” with “statistics.” For example, say we collect some data and find that 2% of writers querying a novel this year will secure representation with an agent. Does that mean any given querying writer this year has a 2% chance of getting an agent?

Not remotely.

Within that pool of querying writers, we can’t say “all things being equal,” because they aren’t. Some of the writers don’t have a clue what they’re doing. (You’ve seen Slushpile Hell, right?) Some aren’t making such egregious mistakes, but just aren’t ready yet. Some just don’t have the right timing with market trends. Some aren’t querying that aggressively, only sending out a few here and there. And then some are at the top of their game, do their homework, and go at it. The percentage of that last group getting representation is probably quite different.

So, strange as it is for a math teacher to say, don’t get caught up in the numbers when it comes to these subjective, highly variable, real life scenarios. Save thoughts of probability for when you’re deciding whether to walk into a casino, or figuring out whether you should take an umbrella when you leave for work.

When it comes to situations where all things aren’t equal, work to make sure you belong to the group that successes draw from. That’s the way to up your chances.

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Earth: By the Numbers

Yesterday was Earth Day, and today is Mathematical Monday, so I figured I’d put the two together with some stats on this planet of ours.

• The diameter of Earth is about 7,920 miles. That’s nearly 140,000 football fields. If it were possible to drive at highway speeds (say, 75 mph) through the planet, it would take nearly 4.5 days to reach the other side.
• The mass of Earth is about 6 septillion kilograms. (That’s 6 followed by 24 zeroes. I’m going by the U.S. definition of the word—apparently it’s different in the U.K.) That’s about the same as 165 quintillion empty 18-wheelers. Or the Empire State Building 16,000,000,000,000,000,000 times over.
• The volume of water on Earth is about 1.5 billion cubic kilometers. That’s about 600 trillion Olympic swimming pools. (I think. I don’t have my preferred calculators handy.) 97% of that is saltwater in the oceans.
• So, Earth is pretty big. But if our sun were empty, it could fit over 1,000,000 Earths inside.

Now, while you ponder that massiveness, check out this very cool view of the planet from orbit.

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From the Department of Made-Up Statistics

I admit it—I’m a data geek. (Shocking, right?) Give me some data, and I can’t help but analyze it at least a little. I’ve even made graphs to analyze my writing.

I’ve often heard people claim you can make statistics say anything you want. That’s not entirely true, but you can usually frame them in a way that leans in a certain direction, even if that direction is misleading. Some easy ways to do this are asking your question in a particular way, choosing a biased sample, and setting up a graph with an inappropriate axis. (All of these will get you labeled a bad statistician, though.)

Sometimes, it’s easier just to skip all the technical steps and just make up results. So here are some claims that are entirely made up based only on my gut instinct. If anyone finds hard data on any of them, feel free to let me know.

• 47% of cancer survivors will (attempt to) write a memoir on the topic.
• 72% of drivers don’t know how to merge properly.
• 96% of literate Americans don’t know how to use a semicolon.
• 83% of teenagers declare something boring a minimum of twice every school day.
• Chocolate makes everything better 99.9% of the time.
• I remember 74% of the useless minutiae I come across, but only 51% of the information I actually need.
• 2% of the people who protest that a given book should be banned have actually read the book.

Go ahead and make up some of your own statistics, or let me know if you think my percentages are off on any of the above. It’s fun and makes you sound knowledgeable. 😉

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Primer #1 on Deaf Can/Can’t

Every so often, I’ll get a particular comment about Fingerprints on critique sites—something about Tasmin (the Deaf character) displaying unrealistic English skills.

These commentators mean well and undoubtedly speak from their personal experience, so I don’t mind.  I see it as opportunity to spread a little knowledge.

When I was in grad school, we frequently discussed the hated statistic: Most deaf people read at a fourth grade level.  Please note that the statistic on that website is actually that the median reading level among 17- and 18-year-olds in the sample was 4.0, so there’s one inaccuracy that creeps into the discussion.  Generalizing that, half of the individuals in the sample read at or below that level … and half read at that level or above.

Another thing to note: The literacy statistics among the general U.S. population aren’t too great, either.  Check here for some stats that those in medical fields should keep in mind.  There are a lot of reasons for this, including school performance, education level of parents, and language access.

That last point—language access—is likely the biggest hurdle for deaf kids.  The most accessible language is most likely not one that’s used in the home when the deaf kid comes along.  An exception is when there is a Deaf parent (or two), which does happen, but overall isn’t that likely.  Some hearing parents dive right into signing classes and/or take other steps, working their tails off to help their kids succeed.

Regardless, a huge number of variables are involved … enough to make generalizations pretty useless.

What I do know is that I’ve worked with deaf students on both ends of the spectrum.  I’ve known deaf kids who read above grade level.  I know several others in high school who read and write at or very close to their grade level.  It happens, and if I see it at our tiny little school, it happens everywhere to one degree or another.

So do I stand by Tasmin’s skills?  Absolutely, and not just because the character is meant to be unusually intelligent.  I chose to focus on the “can” … and the only thing Tasmin can’t do is hear.

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