Standard deviation and n-1 : degrees of freedom or unbiased? I am trying to understand if the division for n-1 in the calculation of standard deviation related to degrees of freedom, unbiasing of the estimator, or both.   I understand what is a degree of freedom when we talk about t, chi, f distribution and so on.   I was also trying to understand what is a degree of freedom in respect of the standard deviation.   I became aware of another explanation, which relates to the bias of the estimation of $\sigma$.  Now I am confused: why do we divide by $n-1$?  Is it because of the degrees of freedom (and in this case, what does it actually mean?) or is it because we want to have an unbiased estimator? Or are the two explanations connected in some way?  (Is it normal that the concept of DOF is so slippery and difficult to understand? Or is that just me?)  --I have read the suggested topic, and I think that it only apparently answered my question. The OP of that topic asked about degrees of freedom, and the answer talked about biased estimator. I don't understand how they relate to each other.
Degrees of freedom for standard deviation of sample would someone please explain why the degrees of freedom for a random sample is n-1 instead of n ?   I'm looking for an explanation that is intuitive and easily understood by a high school student.
An irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root? Problem: Is there an irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root for $p$ prime? If there is, what is the minimal degree possible?   I can only prove that $x^2-c$ is impossible, by quadratic reciprocity and Chinese remainder theorem. Even the case of $a x^2 - c$ is unknown to me.  Meanwhile, if $f$ is not required  to be irreducible, but only have no root in $\mathbb{Z}$, then $(x^2-a)(x^2-b)(x^2-ab)$ is a solution for non-sqaure $a,b,ab$, since if both $a,b$ are not squares in $\mathbb{Z}/p\mathbb{Z}$, then $ab$ is a square. Yet I still don't know if this is of minimal degree.  Also, it is natural to pose generalizations   When $p$ is not necessarily prime (equivalently for all prime powers). When $p$ is odd prime. When $p$ represents sufficiently large primes (equivalently all but a finite number left). How can I get suffix matching in a search in SharePoint 2013? I'm new to SharePoint 2013 and I'm quite puzzled, that it seems to be impossible to search for the middle part of a string.  Simple example:  I've added a lot of documents (PDFs, ...) from our Windows Server to the SharePoint Library and I want to add metadata. I try to filter the view of the document library, to only get those documents which I currently want to edit.  The names of the documents are very systematic, composed of naming conventions with abbreviations like:  Superthing2000-b_Suppl_REF_13580_note_1108_04en which means e. g.: Document about Product "Superthing2000", variant "b", Supplement to the manual, ID "13580", from 08/2011 in english language.  Now, I'd like for example to filter out all documents containing ID 13580 or "en" for english, but no chance, as suffix matching is not possible, even if I explicitly search for e. g. *en*.  Is there a way to define a search with kind of a "contains" operator which also finds files when the search string is only a part in the middle of the file name? Even the Windows Explorer search feature is able to do that!! Will doing timelapses break my camera? I was planning to do some timelapses in the near future, until I realised that if I leave my camera taking pictures for a week or 2, I will soon reach the max number of shutter count and my camera would break.   Then I read from an unverified source that during timelapse the shutter doesn't move, so there is no way to break it. I am confused.   Is it possible to take timelapses without physically destroying the shutter in a month? In particular, I am interested in Canon 80D model. Every once in a while I see users having issues to update due to errors like this:  E:Type 'ain' is not known on line 1 in source list /etc/apt/sources.list.d/some-ppa.list'   with varying types/line numbers/source list files (often after removing a PPA).  How can such an error be fixed? Finding the limit $\lim_{x\rightarrow \infty} \sqrt[3]{x+1}-\sqrt[3]{x}$ How to use Redstone dust in Minecraft PE? I have Minecraft Pocket Edition and I have collected a lot of redstone dust. For some reason, I couldn't use it for things such as iron doors. Why? Example of a group $G$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ but $G$ is not the internal direct product of them. I was trying to find an example of a group $G$ and normal subgroups $N_1 , \ldots , N_n$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ and yet $G$ is not the internal direct product of  $N_1, \ldots , N_n$ .  It is easy to show that G is the internal direct product of normal subgroups $N_1 , \ldots , N_n$ iff  (i) $G = N_1\cdots N_n$  (ii) $N_i\cap (N_1 \cdots N_{i-1}\cdots N_{i+1} \cdots N_n) = \{e\}$ for $i = 1, 2, \ldots, n$.  So I guess the question boils down to produce an example of a group $G$ and its normal subgroups  $N_1 , \ldots , N_n$ such that   (1) $G = N_1\cdots N_n$  (2) $N_i \cap N_j = \{e\}$ for all $i \neq j$   but     (3) $N_i \cap (N_1 \cdots N_{i-1} N_{i+1} \cdots N_n) \neq \{e\}$ for at least one $1 \le i \le n.$ Thanks in advance for help.... Why do positive and negative variants of the same question elicit the same answer? How does the tester screwdriver work? How does the tester screwdriver work? If I put the tester screwdriver inside the "hot wire" of an electrical socket, it lits up if I press my finger against the metal cap on top of the screwdriver. This happens also if I stand on a surface of isolating material, such as wood. I read elsewhere that this happens due to stray capacitance formed by the "hot wire", the human body, and the ground. One has  $$ Z = R + \dfrac{1}{j \omega C}  $$  for the impedance, so if C  is high enough, the impedance should be close to r , the "effective resistance" of the formed circuit. Here I get lost; why r  is small enough to cause a current in the range of mA  even if I stand on an isolating surface?  So really what I am asking is how one may represent the system hot wire - screwdriver - human body - wooden floor - building - ground as an electrical circuit, and which parts of the physical system contribute to the resistance, capacitance (and inductance?) and in which proportion, even very approximately. Are there any rulesets or laws for Q or Q continuum? Anything, that would prevent them from doing... anything? Or at least limit them from doing something really bad?  In "" episode Q explains to  that they can do, have, get, become anything they want or appear anywhere they want, with just a snap of fingers. If that is true and if there are absolutely no borders or limits, they can't pass, then how can universe work in general? Or, how did it manage to exist so far?  I mean -- any Q could wipe out Borg, entire human population or even entire universe... with just a snap of fingers. What could prevent Q from doing that? $\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$. If $ a$, $ b$, $ x$, $ y$ are integers greater than $1$ such that $ a$ and $ b$ have no common factor except $1$ and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than $1$.  I started this way: $x^a=y^b$ $=&gt;$ $x=y^{b/a}=(y^{1/a})^b$. Thus $x$ is an integer, and so is $y^{1/a}$, so $y=n^a$ for some integer $n$ greater than $1$. Then from the given relation, we get $x=n^b$.   I am not very convinced with my solution, as I feel I did some mistake. Can anyone point out where the mistake is? Any other solution is also welcome :) World edit wood rotation Why is mathematical induction a valid proof technique? Don't we have single sign-on for all Stack Exchange sites? I would like to use many sites in the Stack Exchange network, but I need to login to each site separately. Though it's easy, can't we solve this problem? I know each site has a different url :( but is there any way? Finding an Inverse Function and Composition of Functions? Understanding ROC curve What is wrong with my date regex? Apparently this is a  and should hopefully be fixed in an upcoming update.    I have installed SQL Server Management Studio v17.9 on a clean Windows 10 installation.  This associated all .sql files with SSMS.  I want to associate .sql files with a plain text editor (eg: Sublime) but I am unable to change the default application.  I have tried:   Right click .sql file -> Open With -> Sublime + "Always use this program". Settings -> Default Applications by File Type -> Find .sql -> Change to Sublime Right click .sql file -> Properties -> "Change" button -> Sublime. Using ftype via elevated command prompt as seen    In all 3 cases, my selection is not reflected in any way, anywhere.  The icon remains SSMS, and the file opens with SSMS.  Is there some kind of "File Association retention" setting within SSMS I am unaware of?  Edit: I can change the default application for non-SSMS-associated extensions without issue.  Edit: after deleting registry key HKCU\Software\Microsoft\Windows\CurrentVersion\Explorer\FileExts\.sql, .sql files show a VSCode icon:    If I then double-click New Users.sql, it opens with SSMS and a refresh of regedit shows the .sql entry recreated with ssms.exe associated (the icon remains VSCode):    If I then right-click -> open with -> Choose Another App -> Select Sublime and hit "Always use this to open .sql files", it opens with Sublime and a refresh of regedit shows a new "b" entry created for Sublime.  The icon remains VSCode, and double-clicking opens it in SSMS:    If I then delete the a entry for ssms.exe and rename b -> a such that only sublime exists, then double-click the .sql file, it opens with SSMS and generates a new b entry for itself: Alignment of numbers on the individual bars Computing limit of $(1+1/n)^{n^2}$