Talk:Standard deviation
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Accuracy of this article?
Correct if I'm wrong, but I always thought the std dev equation had (n-1) in the denominator, not n: where n is the number of data pts.
- This is kind of explained in the article, but to put it differently: The standard deviation of a (finite or infinite) population is defined as the square root of the population variance. The variance of a finite population is defined as the sum of the squared mean deviations, divided by the population size. If one is trying to estimate the population variance based on a (small) sample, then it would be wrong in general to equate the sample with the population. In particular, estimating the population variance as the sum of the squared mean deviations divided by the sample size yields a biased estimator of the population variance. Dividing by n−1 instead of n yields an unbiased estimator. However, in the extreme case where n equals the population size (of a finite population), it would be wrong to divide by n−1, since one has seen the entire population, hence there is no uncertainty about its variance and one can simply calculate it instead of estimating it. It's the distinction between the variance of a finite population and its unbiased estimator that leads to the terms n and n−1, respectively, in the denominator (and to widespread confusion). --MarkSweep 07:23, 19 May 2005 (UTC)
But what does the result of the standard deviation tell us?
Why perform the equation? What do the results say? Kingturtle 07:43, 6 Nov 2003 (UTC)
- To say it differently.....this article needs a section explaining the application of standard deviation. When is it used? And what can the results tell us? Kingturtle 06:19, 10 Feb 2004 (UTC)
sigma level
when you read a quote like this "To see a very light Higgs (say, 115 GeV) at the 5 σ level will require a year of running", does that mean that the signal is within 5 standard deviations or something? I hear that usage a lot in experimental physics, and I am unsure what it means. is a higher sigma confidence level good or bad? I once heard that things at the 3σ level are suspect, and one can't trust the result. So what's 5 σ? anyone can explain this? i think it would be a nice addition to this article... Lethe
- If I understand correctly, such quotes mean "if we assume that this data is the result of random statistical fluctuation rather than an actual signal, this is how many sigmas away from the mean our results are." In other words, the sigma level gives a measure of how likely it is that you are just seeing a fluke. In your Higgs example, the meaning is that if they ran the detector for a year, a false detection would require a statistical fluctuation of five sigmas away from the mean. Higher sigma means more confidence, because it means your signal would be an increasingly large fluke if it weren't real. Isomorphic 03:54, 12 Jul 2004 (UTC)
Thanks for the explanation dude. I'm going to try adding it to the article. -Lethe
What about underlying assumptions?
This is quite a short and incomplete treatment of standard deviation.
It should be mentioned that a Normal distribution about the true mean is assumed. The article doesn't differentiate between the true mean and the sample mean, which in turn explains the difference between the true standard deviation and the sample standard deviation (and why the normalisation constant is n-1 and not n, i.e. the sample mean takes away one degree of freedom as it is computed from the data set and only estimates the true mean).
Nothing is said about confidence intervals, which is what standard deviations are useful for: 1-sigma is the 67% interval, i.e. 67% of the data is within 1 standard deviation, 2-sigma is 90%, 3-sigma is 95% and so on.
Again, the standard deviation can only be interpreted if the data is distributed normally about the mean. Before I add more criticism, let me have a look at the article(s) for mean and normal distributions.
- The standard deviation exists for any population, and always has meaning regardless of whether the distribution is normal. The confidence interval interpretation you're giving only works for normal distributions, but nothing about the standard deviation itself assumes normality. Isomorphic 15:01, 26 Oct 2004 (UTC)
- It is nonsense to say "a Normal distribution about the true mean is assumed" or that confidence intervals are the only thing that standard deviations are useful for, or that "the standard deviation can only be interpreted if the data is distributed normally about the mean". If you're doing certain things with confidence intervals, then "the standard deviation can only be interpreted if the data is distributed normally about the mean" if you're doing certain other things with confidence intervals, or many other things not involving confidence intervals but for which the standard deviation is useful, then normality should not be assumed. Michael Hardy 20:31, 26 Oct 2004 (UTC)
Making it easier to read
26 Oct 2004 : FROM
Simply put, the standard deviation tells us how far a typical member of a sample or population is from the mean value of that sample or population.
TO
Simply put, the standard deviation tells us how far a typical member of the population (or sample) is from the mean value of that population (or sample).
Interpretation and application
An anon just changed the 5 to a 4 in the sentence "For example, the three samples (0, 0, 14, 14), (0, 6, 8, 14), and (6, 6, 8, 8) each have an average of 7. Their standard deviations are 7, 5 and 1, respectively" as his first edit. I'm paranoid against vandalism, so is he right? -- Kizor 10:46, 21 May 2005 (UTC)
- No he is not - I have just reverted it.--Niels Ø 14:19, May 21, 2005 (UTC)
- Thank you, my good fellow! -- Kizor 19:24, 21 May 2005 (UTC)