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Monday, March 13, 2006

Fun with Statistics

I've been contemplating the prevelance of vague or misleading statistics in the media and in advertising lately. My first example is food products that sport a label claiming, for instance, that the contents are "99% fat free." I think what they mean is that 1% of the product's weight is fat, but it sure is an odd way to say it. Usually, if we heard a general claim that "99% of Iraq is now free of insurgents," we wouldn't interpret him as saying that 1% of the country is solidly covered in shoulder-to-shoulder resistance fighters. No, we'd imagine that in only 1% of Iraq's area are insurgents still regularly a problem. But that reading would make no sense applied to, say, yogurt -- is all the fat just causing trouble in a tiny section of the container?

While the above is stated in a weird way, at least I believe I know what it really says. Not so with a common type of assertion made by weather forecasters, e.g., "There is a 10% chance of rain in Boston today." Are they saying it is likely to be raining 10% of the time all across the city? Or that 10% of the area is likely to see at least some rain? That there is a 10% chance that any rain at all will fall anywhere in Boston? A 10% chance that rain will fall everywhere in the city? That 10% of the time a few drops will be falling on at least one neighborhood?

The last case I present I ran across in the sports pages of the NY Daily News. The reporter mentioned that while Vince Carter of the Nets had scored 29 points in the game he was reporting on, he had scored "only 7 points" in the 4th quarter. I stopped reading to wonder if the writer had tried dividing 29 by 7. Vince was only .25 points below his average for the game in the closing period -- you could hardly expect him to be any nearer.

While I find these sorts of things amusing in themselves, I also think they are signs of a more serious phenomenon: all of us are to ready to accept statements made by these "authoratative" sources as sensible and clear even though, upon examination, they turn out to be quite funky.

NEW EXAMPLE: On ESPN today, the anouncer was backing the assertion by the coach of the Tennessee women's team, that her quartile of the NCAA tournament was too hard, by pointing out that "Five of the team's in the bracket were in the top 25!" Of course, given that all of the top-25 teams are likely to make the 64-team tournament, and there are four brackets, the average number of top-25 teams in a bracket is always 6.25 -- so he was actually presenting evidence that Tennessee's bracket was too easy!

20 comments:

  1. I thought 33.33333% of your examples were bad.

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  2. I think you're 100.0000000000000000000% wrong about that.

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  3. In case you care, it was the weather one. Your quibble has nothing to do with probability but rather with, "What do we mean when we say it's raining in a city?" I mean, suppose the weatherman said, "It's raining right now in Boston." You could say, "What does that mean? That every single person in Boston is getting wet?" Etc.

    Where I _thought_ you were going with that one was statements like, "There's a 10% that Hillary Clinton will win the next election." As Mises would argue, that really makes no sense.

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  4. Yes, I agree -- but that doesn't mean the meaning of the "10% chance of rain" is clear! My broad point is that we accept these statements too casually.

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  5. Lately, I've had little tolerance for comparative statements. "The tornado was like a bomb" or "the bomb was like a tornado". Can't a tornado ever just be like a tornado and a bomb a bomb?

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  6. Anonymous9:57 PM

    Self-referential things are, um, self-referential, an artistic mode of fantasy.

    Some similes are empty forms, like "tastes like chicken" or "smells like fish". Everything tastes like chicken, and there are three things that smell like fish. Everyone knows this.

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  7. ""There's a 10% that Hillary Clinton will win the next election." As Mises would argue, that really makes no sense."

    Well it rather depends what you mean by probability. At which point it all gets rather heavy .....

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  8. As I've pondered the weather percentage thing before, I still want to know what the forcast percipitation percentage refers to. Is percipitation that evaporates in the atmosphere before reaching the surface of the forecasted geographic area included? If so, would that rain be exclusively tabulated in the overall equation of time and area? If it rained say 5% of the time for 20% of the area and 60% of the rain evaporated before reaching the ground but then recondensed ane fell to the surface on 15% of another part of the forecast area... No, I don't take "authoratative" statements at face value.
    For instance I recently read something that said "blame global warming" instead of the usual "because of global warming" which I appreciated because it offered the reader the option to blame something as compared to presenting a vastly complex hypothesis as a proven statement of fact.

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  9. Rain that does not reach the ground is called "virga". It is not counted as precipitation. Percentages wrok for any point in the area during the time period. A 50% chance of rain means that wherever you are in the area, there's a 50/50 chance you will get wet.

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  10. On the Vince Carter thing, I think the statement carries with it the implied background knowledge that, as the team's star, Carter plays disproportionately more minutes in the fourth quarter and is the focal point of a disproportionately large number of plays. That being the case, one would expect his average to go up "down the stretch."

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  11. A guy on the tele said that Cyclone Larry was like an atomic bomb going off. I wondered if there were scores of nauseated people with burned skin wandering the streets. Argh.

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  12. Woody7:23 PM

    From the New York Times today"
    "The flow of the Colorado River, which supplies water to millions of people across the West, is also projected to be below average, though just slightly. The river has had only one year above average in the last seven, according to the Upper Colorado River Commission, an interstate agency that administers it."

    While reading this I thought I could hear S. Jr saying, "That makes me go wow! It's super and terrific! All those below average years have dropped the average, so it will be much easier to have an above average year! Spend more and save!"

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  13. Yes, the "law" of the reversion to the mean seems to me to be just an implication of the definition of the mean.

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