4 ms·

> And the standard deviation is a number, just a number, just as I defined it as σ = E[(X - E[X])^2]^(1/2) which clearly is just a number and not an area. Yes,

by lutusp 13y ago

> And the standard deviation is a number, just a number, just as I defined it as σ = E[(X - E[X])^2]^(1/2) which clearly is just a number and not an area. Yes, and if I graph a function in terms of X and Y, where X is a function's argument and Y is the value returned by a particular function, one can argue that X and Y are just placeholders for numbers without any intrinsic meaning by themselves. If I then say that Y represents the sine of X, surely someone will say, as you have said, "No, not at all, X is just a placeholder for a number, it's not what you say. And Y is just a placeholder for a number, it's not tied to any particular function." In point of fact, the standard deviation is more than a particular number, it's an idea, and its statistical purpose is met when it's associated with a context in which that idea is expressed. That context is the normal distribution. In the present context, a particular sigma value refers to a specific area under a normal distribution, and in turn, to the probability that a particular result might have arisen by chance. Without reference to a normal distribution, a standard deviation (a sigma) loses its conventional meaning. Variances are acquired by statistical tests of data sets, standard deviations (sigmas) are acquired from variances, and conclusions are drawn from sigmas only to the degree that they are applied to normal distributions. Link: http://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule http://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule Quote: "In statistics, the 68–95–99.7 rule, also known as the three-sigma rule or empirical rule, states that nearly all values lie within three standard deviations of the mean in a normal distribution. About 68.27% of the values lie within one standard deviation of the mean. Similarly, about 95.45% of the values lie within two standard deviations of the mean. Nearly all (99.73%) of the values lie within three standard deviations of the mean." According to your thesis, the above claim is obvious nonsense, because in point of fact, sigma has no association with the normal distribution. But you know what? Wikipedia can be edited by anyone, and you can correct this egregious error today, if you like. Correct this widespread erroneous thinking -- fix these errors, everywhere you find them. Let's perform a little test. Let's submit "5 sigma" to Wolfram Alpha and see whether it makes the same mistake you say I have been making, and that the above linked article is making: ------------------------------------------------------ Link: https://www.wolframalpha.com/input/?i=5+sigma https://www.wolframalpha.com/input/?i=5+sigma Quote: Input Interpretation: 5 sigma (standard deviations) z-score : 5 Probabilities: zɝ (left-tailed p-value) | 1-2.867×10^-7 zɱ (right-tailed p-value) | 2.867×10^-7 abs(z)ɱ (two-tailed p-value) | 5.733×10^-7 abs(z)ɝ (confidence level) | 1-5.733×10^-7 Associated two-sided confidence level: 100-5.733×10^-5% ------------------------------------------------------ So it seems that, without prompting, Wolfram Alpha draws the same conclusion that Wikipedia does, and every other online reference does when confronted by terms such as "standard deviation" or "sigma" -- that, without providing a specific context, the default context is that sigmas refer to positions on a standard distribution and have statistical meanings associated with that default assumption. > Of course, even with a Gaussian assumption, standard deviation does not have a particular numerical value. When a scientist uses the term "5 sigma", she is in most cases using a context of a normalized normal distribution, one that follows the 68-95-99.7 rule described above. Therefore yes, 5 sigma does lead to a a particular value, by being applied to a definite integral of a normalized normal distribution that uses it as an argument. By this reasoning, 5 sigma means: (Sage notation) integrate(e^(-x^2/2),x,a,b) / sqrt(2 * pi) Picture: http://i.imgur.com/YhC302b.gif http://i.imgur.com/YhC302b.gif If I try to compute the value used in physics, the result for 5 sigma, I can use this: (Sage notation) N(integrate(e^(-x^2/2),x,5,infinity) / sqrt(2 * pi)) With this result: 2.86651571870318e-7 Which is correct, and is the expected one-tailed value for 5 sigma, even though according to your argument, 5 sigma has no connection to normal distributions. > Sorry 'bout that. What, for your specious argument? No problem, it comes with the territory. But surely you realize I have more than answered your objection.

graycat 13y ago
> That context is the normal distribution. No. Here is an important use of standard deviation not related at all to the Gaussian distribution: The set of all real valued random variables that have a standard deviation and one that is finite form a Hilbert space. The crucial part of the argument is completeness. Of course we like to use Hilbert space for projections and converging sequences, so it's super nice that those random variables form a Hilbert space.
- lutusp 13y ago
  >> That context is the normal distribution. > No. Here is an important use of standard deviation not related at all to the Gaussian distribution ... Before you troll through the world's imaginary problems again, I am going to ask you one more time to remember how this conversation got started (the context), and why I answered as I did. Someone nontechnical asked what the significance of 5σ was in the context of a physics experiment. I replied by saying that 5σ was mapped to a p-value this way: http://i.imgur.com/pcjr6cN.gif http://i.imgur.com/pcjr6cN.gif And the p-value was the point, leading to a discussion of the high level of discipline in experimental physics compared, say, to the social sciences, which accept p-values of .01-.05, values I included in my original reply for comparison. The context is the normal distribution. Wake up and smell the Cappuccino.
  - graycat 13y ago
    Maybe this will clear it up for you: That the physicists are making a normal distribution assumption is from shaky down to sloppy down to silly down to absurd. Making this assumption for experimental data, e.g., the LHC data or the gravitational wave data, where they are hanging on by their fingernails anyway, is close to treating the normal distribution as some gift from God. And, if some field makes a normal, or mean 0, variance 1 normal, assumption, then they should say so and, hopefully, justify that assumption. Your examples from Wolfram and Wikipedia show that a lot of people have guzzled that old Kool Aid on the normal distribution. A few years ago I had a date with a high school teacher, and she assumed that test scores are normal. My father had an MS in education and my brother was a ugrad psych major, so, when my brother's psych material got to statistics, Dad taught him about the normal distribution. It appears that around 1900 and then for 80 years or so, and still in some parts of some fields, a normal assumption, without mention or justification, was common and remains in what you call the "context". "Context" or not, common or not, popular or not, it's still an assumption, needs mention and justification, and usually is not at all well justified or justifiable. Sure, with their normal assumption and, say, five sigma, experimental physicists get particular probabilities of Type I error, so called p values. No question here. The issue is the normal assumption. There are places where a normal assumption is quite solid. The most common justification is from the central limit theorem. So, take random variables X(1), X(2), ... and assume that they are independent and all have the same distribution with finite standard deviation. Then the central limit theorem says essentially that as positive integer n grows to infinity the distribution of the sum X(1) + ... + X(n) will converge to normal (really should divide the sum by square root of n). So, where might we get some such X(1) + ... + X(n)? Sure, from Brownian motion where there are many little bumps and the bumps come close enough to satisfying the hypothesis. So can use this also in thermodynamics. So, there are places where a normal assumption is justified. But just making a normal assumption for essentially all experimental errors as needed for the accepted five sigma criterion is close to a weird religion and not flattering to a modern science. What physics is doing with their five sigma criterion is a statistical hypothesis test. My examples of distribution free hypothesis tests via resampling is a better justified and more conservative and robust way to do an hypothesis test. Physicists might consider using such. Such distribution-free statistical methods form a large field. My example of the role of Hilbert space is quite practical. If you want to explain to the common man in the street why physics likes their five sigma criterion, then, sure, you need the normal assumption. Then you should mention that there physics is making a normal assumption. There you might not confuse the common man in the street with a claim that in making this normal assumption physics is close to drinking some swill of boiled tails of rats and bats. But with the now high interest in computer science of big data, machine learning, etc. I was assuming that the HN audience could and should hear the real stuff -- that in what physics is doing, there's a normal (Gaussian) assumption in there.

by lutusp 13y ago

graycat 13y ago

> That context is the normal distribution. No. Here is an important use of standard deviation not related at all to the Gaussian distribution: The set of all real valued random variables that have a standard deviation and one that is finite form a Hilbert space. The crucial part of the argument is completeness. Of course we like to use Hilbert space for projections and converging sequences, so it's super nice that those random variables form a Hilbert space.

lutusp 13y ago
>> That context is the normal distribution. > No. Here is an important use of standard deviation not related at all to the Gaussian distribution ... Before you troll through the world's imaginary problems again, I am going to ask you one more time to remember how this conversation got started (the context), and why I answered as I did. Someone nontechnical asked what the significance of 5σ was in the context of a physics experiment. I replied by saying that 5σ was mapped to a p-value this way: http://i.imgur.com/pcjr6cN.gif http://i.imgur.com/pcjr6cN.gif And the p-value was the point, leading to a discussion of the high level of discipline in experimental physics compared, say, to the social sciences, which accept p-values of .01-.05, values I included in my original reply for comparison. The context is the normal distribution. Wake up and smell the Cappuccino.
- graycat 13y ago
  Maybe this will clear it up for you: That the physicists are making a normal distribution assumption is from shaky down to sloppy down to silly down to absurd. Making this assumption for experimental data, e.g., the LHC data or the gravitational wave data, where they are hanging on by their fingernails anyway, is close to treating the normal distribution as some gift from God. And, if some field makes a normal, or mean 0, variance 1 normal, assumption, then they should say so and, hopefully, justify that assumption. Your examples from Wolfram and Wikipedia show that a lot of people have guzzled that old Kool Aid on the normal distribution. A few years ago I had a date with a high school teacher, and she assumed that test scores are normal. My father had an MS in education and my brother was a ugrad psych major, so, when my brother's psych material got to statistics, Dad taught him about the normal distribution. It appears that around 1900 and then for 80 years or so, and still in some parts of some fields, a normal assumption, without mention or justification, was common and remains in what you call the "context". "Context" or not, common or not, popular or not, it's still an assumption, needs mention and justification, and usually is not at all well justified or justifiable. Sure, with their normal assumption and, say, five sigma, experimental physicists get particular probabilities of Type I error, so called p values. No question here. The issue is the normal assumption. There are places where a normal assumption is quite solid. The most common justification is from the central limit theorem. So, take random variables X(1), X(2), ... and assume that they are independent and all have the same distribution with finite standard deviation. Then the central limit theorem says essentially that as positive integer n grows to infinity the distribution of the sum X(1) + ... + X(n) will converge to normal (really should divide the sum by square root of n). So, where might we get some such X(1) + ... + X(n)? Sure, from Brownian motion where there are many little bumps and the bumps come close enough to satisfying the hypothesis. So can use this also in thermodynamics. So, there are places where a normal assumption is justified. But just making a normal assumption for essentially all experimental errors as needed for the accepted five sigma criterion is close to a weird religion and not flattering to a modern science. What physics is doing with their five sigma criterion is a statistical hypothesis test. My examples of distribution free hypothesis tests via resampling is a better justified and more conservative and robust way to do an hypothesis test. Physicists might consider using such. Such distribution-free statistical methods form a large field. My example of the role of Hilbert space is quite practical. If you want to explain to the common man in the street why physics likes their five sigma criterion, then, sure, you need the normal assumption. Then you should mention that there physics is making a normal assumption. There you might not confuse the common man in the street with a claim that in making this normal assumption physics is close to drinking some swill of boiled tails of rats and bats. But with the now high interest in computer science of big data, machine learning, etc. I was assuming that the HN audience could and should hear the real stuff -- that in what physics is doing, there's a normal (Gaussian) assumption in there.