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
