Re: What is standard deviation exactly?
Glen Barnett wrote: In article [EMAIL PROTECTED], Neil [EMAIL PROTECTED] wrote: I was wondering what the standard deviation means exactly? I've seen the equation, etc., but I don't really understand what st dev is and what it is for. I'm going to take a different tack to that Herman has taken. If I tell you what you already know, my apologies. I assume you're talking about sample standard deviations, not population standard deviations (though interpretation of what it represents is similar). Standard deviation is an attempt to measure how "spread out" the values are - big standard deviation means more spread out, small standard deviation means closer together. A standard deviation of zero means all the values are the same. Note that the standard deviation can't exceed half the range (largest value minus smallest value). Standard deviation is measured in the original units. For example, if you record a set of lengths in mm, their standard deviation is in mm. There is a huge variety of reasonable measures of spread. Standard deviation is the most used. You will get more of a feel for the standard deviation if you compare what it does to some other measures of spread. For example, another common measure is the mean deviation - the average distance of observations from the mean. By contrast, standard deviation is the root-mean-square distance from the mean (as you can see from the formula**). ** At least the n-denominator (maximum likelihood) version is the root-mean-square deviation; the n-1 denominator is just a constant times that. This squaring puts relatively more weight on the larger deviations, and less weight on the smaller deviations than the mean deviation, but it is still a kind of weighted average of the deviations from the mean. Here's a quick (tiny) example to help illustrate some of the points (I am using the n-1 version of the standard deviation here): Sample 1: 4, 6, 7, 7, 8, 10 Mean = 7, mean deviation = 4/3 = 1.333..., std deviation=2 Sample 2: 1, 5, 7, 7, 9, 13 Mean = 7, mean deviation = 8/3 = 2.666..., std deviation =4 Note that Sample 2's values are more 'spread out' than sample 1's, and both of the measures of spread tell us that. Standard deviation is used for a variety of reasons - including the fact that it is the square root of the variance, and variance has some nice properties, both in general and also particularly for normal r.v.'s, but s.d. is measured in original units. Glen This is a useful summary: I'd just like to add one point to it. People sometimes ask, which measure of spread is "best"? Or, why use standard deviation, it seems more complicated than simpler statistics such as mean average deviation. Various measures of spread are useful for different purposes, but the real strength of s.d. is that many other statistical concepts are built upon it. Thus s.d. underpins the notion of a standard (z) score, z score underpins the definition of Pearson product-moment correlation, and hence linear regression; s.d. squared is variance, and this underpins the variance theorem, analysis of variance, F-ratio etc. etc. Thus it's a "big idea", a substantive concept in the structure of statistics, in a way that other measures of spread aren't. There are parallels to this in other branches of science and mathematics. Mass times velocity (momentum) is a useful concept, because it enters into relationships with other concepts. So does (1/2)m v-squared (kinetic energy). But no one uses mass per unit velocity, or mass times the square root of velocity, or m v-cubed, because (as far as I know) these concepts don't enter into any relationships which are useful for describing aspects of the world. Paul Gardner begin:vcard n:Gardner;Dr Paul tel;cell:0412 275 623 tel;fax:Int + 61 3 9905 2779 (Faculty office) tel;home:Int + 61 3 9578 4724 tel;work:Int + 61 3 9905 2854 x-mozilla-html:FALSE adr:;; version:2.1 email;internet:[EMAIL PROTECTED] x-mozilla-cpt:;-29488 fn:Dr Paul Gardner, Reader in Education and Director, Research Degrees, Faculty of Education, Monash University, Vic. Australia 3800 end:vcard
Re: What is standard deviation exactly?
On Mon, 22 May 2000 13:24:25 +1000, "Glen Barnett"
[EMAIL PROTECTED] wrote:
I assume you're talking about sample standard deviations,
not population standard deviations (though interpretation
of what it represents is similar).
...
Note that the standard deviation can't exceed half the range
(largest value minus smallest value).
That's true for the n denominator ("population standard deviation"),
but not for n-1 ("sample standard deviation"). For example, if your
sample is just the two points 0 and 1, the sample standard deviation
is 0.71, and the range is 1.
Duncan Murdoch
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Re: What is standard deviation exactly?
In article [EMAIL PROTECTED], Neil [EMAIL PROTECTED] wrote: I was wondering what the standard deviation means exactly? I've seen the equation, etc., but I don't really understand what st dev is and what it is for. I am not a statistician as you can tell... Even if you were, you might know what it is good for, but not have any idea, other than that, about the name. The standard deviation is the square root of the variance, and variances add for sums of uncorrelated random variables. A more complicated expression results if they are correlated. Sums of large numbers of independent random variables with variances, each of them negligible in the sum, are approximately normal. If a random variable X is normal with mean m and standard deviation s, P((X-m)/s c) depends only on c; in all cases, the standard deviation acts as a scale parameter. I believe the reason it is called the STANDARD deviation is that if the probability distribution is concentrated equally at the two points one standard deviation from the mean, the first two moments agree with that of the original distribution; the deviation from the mean to get this is the standard deviation. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: What is standard deviation exactly?
There are a couple of (practical) features of the standard deviation that are worth noting. First, as a *descriptor* of the variation in a distribution, it is generally not very good. I mean this is the sense that if you want to visualise the amount of variation in a distribution the SD is only useful if the distribution is at leasst symmetric and preferably approximately normal. This appears to me to contribute to the difficulty that students have with it. Second, for a normal distribution, it is easily seen that the variatiion can be described (and measured) by the 'width of the peak'. The question is, at what point do we measure the width? Geometrically, the only two uniquely identifiable points on the curve, other than the maximum, are the two inflexions. (I usually describe these to students by getting them to imagine they are ants riding a motor bike along the curve; they lean into the curve one way, then straighten up and lean the other way.) Consequently, the only measure of the width of the peak that makes sense is the distance between these points - and this is twice the standard deviation. Hence (I think) the word 'standard'. Regards, Alan Herman Rubin wrote: In article [EMAIL PROTECTED], Neil [EMAIL PROTECTED] wrote: I was wondering what the standard deviation means exactly? I believe the reason it is called the STANDARD deviation is that if the probability distribution is concentrated equally at the two points one standard deviation from the mean, the first two moments agree with that of the original distribution; the deviation from the mean to get this is the standard deviation. -- -- Alan McLean ([EMAIL PROTECTED]) Department of Econometrics and Business Statistics Monash University, Caulfield Campus, Melbourne Tel: +61 03 9903 2102Fax: +61 03 9903 2007 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
