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"80% of Americans believe they are better-than-average drivers. Can't be true, right?" No, not "right". For any finite (> 1) set of people on nearly any meas

"80% of Americans believe they are better-than-average drivers. Can't be true, right?" No, not "right". For any finite (> 1) set of people on nearly any measure, it is easy for all but one to be "better than average". E.g., for any positive integer n > 1, everyone gets a 1 except the one person with a 0 so that the average is (n - 1)/n < 1. Done.

alextp 16y ago
Yes, abstractly, of course. On the other hand, this sort of distribution is usually gaussian (a whole bunch of average drivers, and a few specially good and bad ones), and it is true that 80% of the people can't be above average if they are samples from a well-behaved distribution (with controlled skewness, and in the correct scale, etc)
- HilbertSpace 16y ago
  "this sort of distribution is usually gaussian". No. That distributions are commonly Gaussian is some quasi-mystical nonsense left over from 'psychometrics' about 100 years ago. The most common way to get a Gaussian distribution is from the central limit theorem, and it's much worse than clumsy to argue its assumptions here. Gaussian needs MUCH more than your "a whole bunch of average drivers, and a few specially good and bad ones". Your "well-behaved" is asking far too much for practice or reality. Gaussian isn't "well-behaved"; from Melon in 'Back to School', it's "fantasy land". E.g., there's no reason to believe that can have your "controlled skewness". On your "scale", sure, if know the distribution and if it is absolutely continuous with respect to Lebesgue measure, then can pick a scale that maintains order and yields Gaussian. The educational testing people love to do this. It's quasi-religion. And we collect taxes to pay them for that? Yes, for stock market data can argue that changes in price each 15 seconds are independent and identically distributed with finite mean and variance so that the central limit theorem applies so that the change in price from noon today to noon next week will be Gaussian. Of course there are traders driving Ferraris laughing at that. Do yourself a favor: For real data, assume mean and variance exist and are finite, but without a really careful appeal the the central limit theorem dumpster Gaussian. For multivariate data, don't even think about Gaussian. Uh, some diffusion processes in physics might give you 3D Gaussian quite accurately.
  - alextp 16y ago
    To be pedantic, most people usually mean "median" (or a robust estimation of the mean without considering outliers) by average, and in samples from a continuous distribution it is necessarily true that ~50% are above/below the median.
    - HilbertSpace 16y ago
      You are not being "pedantic". You are being wrong. There's no reason to assume that commonly mean means median or that there are enough properties to make them equal. Besides a claim about "most people" without some solid data is poor support for anything.
- tbrownaw 16y ago
  Gaussians don't work very well when the typical value is near one of the bounds. For example say 3/4 of drivers don't eat, apply makeup, talk on the phone, etc, while driving, and generally pay attention to the road; that sort of attentiveness is probably most of what makes a good driver, so you'd have the majority of people being near one extreme of the possible range and get a rather one-sided distribution.
nudge 16y ago
I think it's fair to assume that, in this context, people are using the 'median' sense of average rather than the 'mean' sense. When someone thinks "I am a better driver than average", this probably means "If you were to rank all drivers by skill, I would be in the top half" rather than "If you take the mean of the quantified driving skill of all drivers, my quantified driving skill is above that."
- HilbertSpace 16y ago
  "I think it's fair to assume that, in this context, people are using the 'median' sense of average rather than the 'mean' sense." Not "fair" at all, and not at all appropriate. Still want the average, the mean, the expectation because of the law of large numbers and the best L^2 approximation property of the mean. Besides, often in practice, e.g., for even positive integer n independent, identically distributed samples from a distribution absolutely continuous with respect to Lebesgue measure, the median is not even well defined. The median is good for politically correct news stories about data from political economics.
  - nudge 16y ago
    Nobody's talking about which is the better average. I was merely suggesting that this is how you can interpret the quote in a way that escapes your attempt to show how it is incorrect.
    - HilbertSpace 16y ago
      "I was merely suggesting" No, as is clear on the screen in front of you, you wrote "I think it's fair to assume" No it's not "fair to assume" that.
      - shasta 16y ago
        I say the point goes to HilbertSpace here. I think Nudge wasn't merely suggesting that there's another way to interpret the quote, but was rather suggesting that was the intended way to interpret the quote. Of course, Nudge was right about that.
furtivefelon 16y ago
The assumption is that like many natural phenomenons (height, weight, etc), driving skills are distributed normally, which means mean = median.
- HilbertSpace 16y ago
  That assumption was unstated and, if stated, would be wildly unjustified. For what was stated and assumed, my example works. Sure, my example looks contrived and pathological, but so it commonly is with simple examples. Your claim that height, weight, etc. are normally distributed is also wildly unjustified. You have swallowed a common bit of old quasi-mystical nonsense from 100 years ago that real data is commonly normally distributed. That claim is nonsense. Read what else I wrote about Gaussian distributions in this thread. Grow up. Learn. In practice, Gaussian is rare and otherwise fantasy land.
  - ellyagg 16y ago
    Jesus, I don't know what you're on about in this thread. When people think to themselves that they are better than average drivers, they mean "I am better than the majority of people, for example most of my coworkers, at driving." No one had to swallow any quasi-mystical nonsense about Gaussian distributions to understand this. It's human nature. They are implicitly comparing themselves to other people and believing they are better.
    - HilbertSpace 16y ago
      You have it backwards. You are agreeing with me. Sorry 'bout that. I know you want to disagree, but you made a mistake, not nearly the first in this thread on my posts. I gave a simple, mathematical example of how all but one person could be better than the average in response to the claim that this could not happen. You gave a way it could happen from how people might think intuitively, and I gave a mathematical way. The people who wanted to argue with my math example kept claiming, based on quasi-mystical nonsense, that mean and median have to be the same due to various unstated, unjustified assumptions, that 'most people' mean median when they say average, etc. So, sorry 'bout that, but here you and I agree: Most people can be better than the average. You gave one justification, and I gave a mathematical one. The people who want to argue with me want to go way out in fantasy land and claim that mean and median have to be equal in practice, which is total BS. My example is dirt simple. It should be taken as funny. Alas, some people apparently got their Statistics 101 feathers and Gaussian religion in an uproar and wanted to argue. A claim in this tread is that there is nothing worth learning in college. Hmm .... Looks like this thread is mostly high school dropouts self-taught in programming. Thankfully for US national security, our DoD is far ahead of such nonsense. You guys are hopeless. Also, this Web site doesn't work with Firefox 3.0.1.8 on XP 3.0 and, instead, keeps giving "don't match" or some such. The programming here sucks. So, guys, on this thread, let's agree: The greatest 'business advantage' possible ever is to have some high gloss plastic case on some handheld consumer product from Apple aimed at teenage girls who want to gossip. Ah, of such stuff will be the future of technology! Gossip! That's the key to business! Who would have thought! SUCH progress!

alextp 16y ago

Yes, abstractly, of course. On the other hand, this sort of distribution is usually gaussian (a whole bunch of average drivers, and a few specially good and bad ones), and it is true that 80% of the people can't be above average if they are samples from a well-behaved distribution (with controlled skewness, and in the correct scale, etc)

HilbertSpace 16y ago
"this sort of distribution is usually gaussian". No. That distributions are commonly Gaussian is some quasi-mystical nonsense left over from 'psychometrics' about 100 years ago. The most common way to get a Gaussian distribution is from the central limit theorem, and it's much worse than clumsy to argue its assumptions here. Gaussian needs MUCH more than your "a whole bunch of average drivers, and a few specially good and bad ones". Your "well-behaved" is asking far too much for practice or reality. Gaussian isn't "well-behaved"; from Melon in 'Back to School', it's "fantasy land". E.g., there's no reason to believe that can have your "controlled skewness". On your "scale", sure, if know the distribution and if it is absolutely continuous with respect to Lebesgue measure, then can pick a scale that maintains order and yields Gaussian. The educational testing people love to do this. It's quasi-religion. And we collect taxes to pay them for that? Yes, for stock market data can argue that changes in price each 15 seconds are independent and identically distributed with finite mean and variance so that the central limit theorem applies so that the change in price from noon today to noon next week will be Gaussian. Of course there are traders driving Ferraris laughing at that. Do yourself a favor: For real data, assume mean and variance exist and are finite, but without a really careful appeal the the central limit theorem dumpster Gaussian. For multivariate data, don't even think about Gaussian. Uh, some diffusion processes in physics might give you 3D Gaussian quite accurately.
- alextp 16y ago
  To be pedantic, most people usually mean "median" (or a robust estimation of the mean without considering outliers) by average, and in samples from a continuous distribution it is necessarily true that ~50% are above/below the median.
  - HilbertSpace 16y ago
    You are not being "pedantic". You are being wrong. There's no reason to assume that commonly mean means median or that there are enough properties to make them equal. Besides a claim about "most people" without some solid data is poor support for anything.
tbrownaw 16y ago
Gaussians don't work very well when the typical value is near one of the bounds. For example say 3/4 of drivers don't eat, apply makeup, talk on the phone, etc, while driving, and generally pay attention to the road; that sort of attentiveness is probably most of what makes a good driver, so you'd have the majority of people being near one extreme of the possible range and get a rather one-sided distribution.

nudge 16y ago

I think it's fair to assume that, in this context, people are using the 'median' sense of average rather than the 'mean' sense. When someone thinks "I am a better driver than average", this probably means "If you were to rank all drivers by skill, I would be in the top half" rather than "If you take the mean of the quantified driving skill of all drivers, my quantified driving skill is above that."

HilbertSpace 16y ago
"I think it's fair to assume that, in this context, people are using the 'median' sense of average rather than the 'mean' sense." Not "fair" at all, and not at all appropriate. Still want the average, the mean, the expectation because of the law of large numbers and the best L^2 approximation property of the mean. Besides, often in practice, e.g., for even positive integer n independent, identically distributed samples from a distribution absolutely continuous with respect to Lebesgue measure, the median is not even well defined. The median is good for politically correct news stories about data from political economics.
- nudge 16y ago
  Nobody's talking about which is the better average. I was merely suggesting that this is how you can interpret the quote in a way that escapes your attempt to show how it is incorrect.
  - HilbertSpace 16y ago
    "I was merely suggesting" No, as is clear on the screen in front of you, you wrote "I think it's fair to assume" No it's not "fair to assume" that.
    - shasta 16y ago
      I say the point goes to HilbertSpace here. I think Nudge wasn't merely suggesting that there's another way to interpret the quote, but was rather suggesting that was the intended way to interpret the quote. Of course, Nudge was right about that.

furtivefelon 16y ago

The assumption is that like many natural phenomenons (height, weight, etc), driving skills are distributed normally, which means mean = median.

HilbertSpace 16y ago
That assumption was unstated and, if stated, would be wildly unjustified. For what was stated and assumed, my example works. Sure, my example looks contrived and pathological, but so it commonly is with simple examples. Your claim that height, weight, etc. are normally distributed is also wildly unjustified. You have swallowed a common bit of old quasi-mystical nonsense from 100 years ago that real data is commonly normally distributed. That claim is nonsense. Read what else I wrote about Gaussian distributions in this thread. Grow up. Learn. In practice, Gaussian is rare and otherwise fantasy land.
- ellyagg 16y ago
  Jesus, I don't know what you're on about in this thread. When people think to themselves that they are better than average drivers, they mean "I am better than the majority of people, for example most of my coworkers, at driving." No one had to swallow any quasi-mystical nonsense about Gaussian distributions to understand this. It's human nature. They are implicitly comparing themselves to other people and believing they are better.
  - HilbertSpace 16y ago
    You have it backwards. You are agreeing with me. Sorry 'bout that. I know you want to disagree, but you made a mistake, not nearly the first in this thread on my posts. I gave a simple, mathematical example of how all but one person could be better than the average in response to the claim that this could not happen. You gave a way it could happen from how people might think intuitively, and I gave a mathematical way. The people who wanted to argue with my math example kept claiming, based on quasi-mystical nonsense, that mean and median have to be the same due to various unstated, unjustified assumptions, that 'most people' mean median when they say average, etc. So, sorry 'bout that, but here you and I agree: Most people can be better than the average. You gave one justification, and I gave a mathematical one. The people who want to argue with me want to go way out in fantasy land and claim that mean and median have to be equal in practice, which is total BS. My example is dirt simple. It should be taken as funny. Alas, some people apparently got their Statistics 101 feathers and Gaussian religion in an uproar and wanted to argue. A claim in this tread is that there is nothing worth learning in college. Hmm .... Looks like this thread is mostly high school dropouts self-taught in programming. Thankfully for US national security, our DoD is far ahead of such nonsense. You guys are hopeless. Also, this Web site doesn't work with Firefox 3.0.1.8 on XP 3.0 and, instead, keeps giving "don't match" or some such. The programming here sucks. So, guys, on this thread, let's agree: The greatest 'business advantage' possible ever is to have some high gloss plastic case on some handheld consumer product from Apple aimed at teenage girls who want to gossip. Ah, of such stuff will be the future of technology! Gossip! That's the key to business! Who would have thought! SUCH progress!