My understanding is that you take the square and then the square root so that positive and negative distances from the mean don't cancel each other out. But if that's the case, why don't you immediately take the square root after the square?
Today I taught an introductory class of statistics and a student came up to me with a question, which I rephrase here as: "Why is the standard deviation defined as sqrt of variance and not as the sqrt of sum of squares over N?"  We define population variance: $\sigma^2=\frac{1}{N}\sum{(x_i-\mu)^2}$   And standard deviation: $\sigma=\sqrt{\sigma^2}=\frac{1}{\sqrt{N}}\sqrt{\sum{(x_i-\mu)^2}}$.   The interpretation we may give to $\sigma$ is that it gives the average deviation of units in the population from the population mean of $X$.   However, in the definition of the s.d. we divide the sqrt of the sum of squares through $\sqrt{N}$. The question the student raises is why we do not divide the sqrt of the sume of squares by $N$ instead. Thus we come to competing formula: $$\sigma_{new}=\frac{1}{N}\sqrt{\sum{(x_i-\mu)^2}}.$$ The student argued that this formula looks more like an "average" deviation from the mean than when dividing through $\sqrt{N}$ as in $\sigma$.  I thought this question is not stupid. I would like to give an answer to the student that goes further than saying that the s.d. is defined as sqrt of the variance which is the average squared deviaton. Put differently, why should the student use the correct formula and not follow her idea?  This question relates to an older thread and answers provided . Answers there go in three directions:   $\sigma$ is the root-mean-squared (RMS) deviation, not the "typical" deviation from the mean (i.e., $\sigma_{new}$). Thus, it is defined differently. It has nice mathematical properties. Furthermore, the sqrt would bring back "units" to their original scale. However, this would also be the case for $\sigma_{new}$, which divides by $N$ instead.   Both of points 1 and 2 are arguments in favour of the s.d. as RMS, but I do not see an argument against the use of $\sigma_{new}$. What would be the good arguments to convince introductory level students of the use of the average RMS distance $\sigma$ from the mean?
Inexact measurement and wavefunction collapse As is usually said, measurement of an observable $q$ leads to collapse of wavefunction to an eigenstate of the corresponding operator $\hat q$. That is, now the wavefunction in $q$ representation is $\psi(q)=\delta(q-q_0)$ where $q_0$ is result of measurement.  Now, in reality measurements are never exact, so the wavefunction should not have that sharp peak. Instead it should be broadened depending on precision of measurement. In this case, can we still introduce an operator, an eigenstate of which would the new wavefunction appear? Is it useful in any way? Or does the new wavefunction depend too much on the way it was measured so that each instrument would have its own operator? How would such opeartor look e.g. for a single-slit experiment? How can I uninstall software? Seeing there are various ways to install on Ubuntu. What are the various ways to uninstall items?  Which ways always work? How can I be sure something was uninstalled? Geometric random variables $X_1:G(p_1)$ $X_2:G(p_2)$ $X_3:G(p_3)$ are independent, prove the following : Xdotool using "DISPLAY=:0" not works in Crontab Can we insert or update FieldSets using apex code? Although we can create or update fieldsets using salesforce UI but Is there any way to insert or update FieldSets using apex code (using metadata or API calls) ? How can I know which radio button is selected via jQuery? Sort and count number of occurrence of lines I have Apache logfile, access.log, how to count number of line occurrence in that file? for example the result of cut -f 7 -d ' ' | cut -d '?' -f 1 | tr '[:upper:]' '[:lower:]' is  a.php b.php a.php c.php d.php b.php a.php   the result that I want is:  3 a.php 2 b.php 1 d.php # order doesn't matter 1 c.php Multiple tags search broken when searching as anonymous user? When I input [asdf][lisp] into the search field on , without being logged in, only one tag remains. Is it broken? The reciprocal of many prime numbers p in base 10 have a set of repeating digits p-1. Why? I've noticed that the reciprocal of many prime numbers have a curious number of digits. For example. 1/7 has 6 repeating digits, 1/17 has 16 repeating digits, and 1/47 has 46. There's a rule here that makes this pattern, and I haven't quite figured it out. Does anyone have some insight on this?  There are a few other numbers that follow the pattern subversively. For example, 1/3 has two repeating digits: 33. 1/11 has ten repeating digits: 0909090909. The number 1/13 has a set of six that repeats twice, which makes a set that's 12 digits long.  For reference, here is a website that lists reciprocals of numbers 2 through 70, including non-primes.   The ones that break the rule are 1/2 and 1/5, but they seem to be a matched pair, where the 2 and 5 switch places in the equation 1/X = Y/10  What is the rule that makes this pattern? I'm very curious. Prove or disprove: $S_{10} = \langle (1,3),(1,2, ... ,10) \rangle$ Concatenate multiple files with two blank lines as delimiter? I'm trying to join Markdown files with two newlines between each file.   I've tried using the following command:  paste -sd '\n\n' file1.md file2.md   This joins the files with a newline, but not two newlines, as I need.   What command can I use to join an arbitrary number of files with two \n characters between each join? Cross-site inbookmarks:mine search inbookmarks:mine was fixed for searching within a particular Stack Exchange site, but often I find myself wanting to search for a question I favorited/bookmarked, but not remembering which site it was on when it could conceivably have been on a number of different sites. On the cross-site search at  it does not work, presumably because we have different user ids on each site. Is there a way it could be made to work? Note that this request is for searching, not just viewing, so it's not a duplicate of . Just viewing is not very helpful when you have a huge volume of bookmarks across Stack Exchange sites. I have a table:  table votes (     id,     user,     email,     address,     primary key(id), );   Now I want to make the columns user, email, address unique (together).  How do I do this in MySql?  Of course the example is just... an example. So please don't worry about the semantics. If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$ I was checking my old set of homework problems that I found this one:     If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only   if $a$ is a cube in $G$.   One direction is easy. If $a$ is a cube in $G$ then there exists $y \in G$ such that $a=y^3$. Now with direct manipulations it is shown that $x=y^{-2}$ is a solution to $x^2ax=a^{-1}$.  The other direction is harder though. I first attempted to isolate $x$ and then somehow show that it's necessary for $a$ to be a cube. But since I had no idea how to isolate $x$ I gave up. Then I decided to prove the existence of $y \in G: a=y^3$ by showing that a particular map is bijective so I can come up with a nice variable change to conclude that $a$ must be a cube, but this attempt failed as well. I'm tired now, so I think I'm giving up. I'm looking for a hint or a complete solution if a hint is not possible at this stage. I have used quite some time searching for a solution to my compile problem. The document worked fine untill yesterday. After adding some new references, compiling stoped working properly. I have however used latex only for a few months, so the error might be an easy fix.   My error code in the blg file is:    [836] Utils.pm:209&gt; ERROR - BibTeX subsystem:  C:\Users\andre\AppData\Local\Temp\yb9w7ETEpQ\thesisbiblo.bib_16800.utf8,  line 1375, syntax error: found ",", expected one of: number, name (entry  type, key, field, or macro name), end of entry ("}" or ")") or quoted string  ({...} or "...").    I have biber 2.11 and miktex 2.9 and thought it could have been the cache file as suggested in a previous thread with 2.11 versions, but it did not do the trick to delete that either.  Thanks  This is how my "main" document looks like:  \documentclass[man,12pt,a4paper,babel,english]{apa6}  \usepackage{babel,duomasterforside} \usepackage[utf8]{inputenc} \usepackage[hidelinks]{hyperref}  % Math syntax  \usepackage{siunitx} \usepackage{upgreek} \usepackage{gensymb} \usepackage{amsmath} \usepackage{mathtools} \DeclarePairedDelimiter{\abs}{\lvert}{\rvert}  % Figures \usepackage{graphicx} \graphicspath{{Figures/}} \usepackage{float} \usepackage{subcaption}  % Biblography and citing \usepackage[style=apa,backend=biber,sortcites=true,sorting=nyt,autocite=inline,uniquename=false]{biblatex} \addbibresource{thesisbiblo.bib}  % Frontpage \usepackage{pdfpages}   \begin{document}  % Frontpage      \includepdf[pages=1-]{titlepage.pdf}  % Abstract \input{Abstract} \newpage  % Table of contents \tableofcontents \newpage  % Document \input{Introduction.theory} \newpage \input{Methods} \newpage \input{Results} \newpage \input{Discussion} \newpage \input{Conclusion} \newpage  % References                         \printbibliography  \end{document} I always used to open a new VLC window; go to "open file"...etcetera when I wanted to watch something because when I click on a file in Windows Explorer and chose "open with" or just clicked on the file or right-clicked to choose "play with VLC" it always opened in an old already used VLC window.  Is there any way to get VLC to use a new one without always resorting to: Open VLC, "open file", find file etcetera? Does the Dueling Fighting Style prevent using a shield? The Dueling fighting style doesn't seem to state anything about shields, stating that the character can only wield one weapon while fighting to gain the benefit. Does a shield count as a weapon or otherwise stop the damage bonus from occurring? Another indicator of a review audit: Question has no tags format date in c# How can I format a date as dd/mm/yyyy or mm/dd/yy ?   Like in VB format("dd/mm/yy",now)  How can I do this in C#? Symbol for “such that” (not in set)