The variance is: $$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$   I read that $n-1$ is used instead of just $n$ when we are measuring the variance of a sample taken from a bigger population. I don't understand why we're subtracting $1$.   Thank you.
Does anyone have an intuitive explanation (no formulas, just words! :D) about the "$n-1$" instead of "$n$" in the unbiased variance estimator $$S_n^2 = \dfrac{\sum\limits_{i = 1}^n \left(X_i-\bar{X}\right)^2}{n-1}?$$
Iterating through a Collection, avoiding ConcurrentModificationException when removing objects in a loop We all know you can't do the following because of ConcurrentModificationException:  for (Object i : l) {     if (condition(i)) {         l.remove(i);     } }   But this apparently works sometimes, but not always. Here's some specific code:  public static void main(String[] args) {     Collection&lt;Integer&gt; l = new ArrayList&lt;&gt;();      for (int i = 0; i &lt; 10; ++i) {         l.add(4);         l.add(5);         l.add(6);     }      for (int i : l) {         if (i == 5) {             l.remove(i);         }     }      System.out.println(l); }   This, of course, results in:  Exception in thread "main" java.util.ConcurrentModificationException   Even though multiple threads aren't doing it. Anyway.  What's the best solution to this problem? How can I remove an item from the collection in a loop without throwing this exception?  I'm also using an arbitrary Collection here, not necessarily an ArrayList, so you can't rely on get. How do I escape a single quote in SQL Server? Find $\int \frac{x^3}{x^2-1} \:\mathrm{d}x$ Find $\displaystyle \int \dfrac{x^3}{x^2-1} \:\mathrm{d}x$.   I think there are a few ways to do this. We could just break this up using long division or break it up into partial fractions then use long division. My homework says to break this up into its roots so that's why I brought up partial fractions. Is that the best way to do it here? Should you declare methods using overloads or optional parameters in C# 4.0? I was watching , and it got me thinking about when optional parameters are available in C# what is going to be the recommended way to declare methods that do not need all parameters specified?  For example something like the FileStream class has about fifteen different constructors which can be divided into logical 'families' e.g. the ones below from a string, the ones from an IntPtr and the ones from a SafeFileHandle.  FileStream(string,FileMode); FileStream(string,FileMode,FileAccess); FileStream(string,FileMode,FileAccess,FileShare); FileStream(string,FileMode,FileAccess,FileShare,int); FileStream(string,FileMode,FileAccess,FileShare,int,bool);   It seems to me that this type of pattern could be simplified by having three constructors instead, and using optional parameters for the ones that can be defaulted, which would make the different families of constructors more distinct [note: I know this change will not be made in the BCL, I'm talking hypothetically for this type of situation].  What do you think? From C# 4.0 will it make more sense to make closely related groups of constructors and methods a single method with optional parameters, or is there a good reason to stick with the traditional many-overload mechanism? C++11: defining function on std::array When I try to start Runescape I get an error message that says: CLassNotFoundException and lists Rs2Applet.class. How do I fix this so I can play? Increasing real valued function whose image set is connected Let $S = [0,1) \cup [2,3]$ and $f\colon S \rightarrow \mathbb R$ be such that $f(S)$ is connected . Which of the following are true:  a) $f$ is discontinuous exactly at one point.  b) $f$ is discontinuous exactly at two points.  c) $f$ is discontinuous at infinite many points.  d) $f$ is continous.  I think $f$ is continuous . We need to show that $\lim_{x \to a^+}f(x) = f(a)$ for any $a \in S$. Since $f(x)$ is increasing, $\lim_{x \to a^+}f(x) \leq f(a)$. Suppose $\lim_{x \to a^+}f(x) &lt; f(a)$. Then there is $l$ such that $\lim_{x \to a^+}f(x) &lt; l &lt;f(a)$, which contradicts connectedness of $f(S)$. Similarly we can show that $\lim_{x \to a^-}f(x) = f(a)$.  I would be thankful if anyone would check my solution and correct me if there are any errors. How do I prove equality of x and y? If $0\leq x,y\leq\frac{\pi}{2}$ and $\cos x +\cos y -\cos(x+y)=\frac{3}{2}$, then how can I prove that $x=y=\frac{\pi}{3}$? Your help is appreciated.I tried various formulas but nothing is working. Where can I find good ConTeXt documentation Is there a way to find the number of accepted answers you have, other than manually counting each one? How should we handle downvote retaliation? Avoiding labeling features if overlapped by another layer in QGIS? With QGIS 2.12.2, how can I set up layer labeling to avoid placing labels where features from another layer already exist?  For example, if I have a stream/river polyline layer that contains lake "centerlines", and I place a "lake" polygon layer above it in the drawing order, I don't want the river layer to place a label inside the lake.  Instead, I would rather have the river labeled outside of the lake (as needed).  That way, I can place labels from the lakes layer and I don't run into label collisions.  Here is an example, where (I have intentionally put the lines on top for visual purposes) what I am hoping to achieve is no river center-line labels shown inside the lake polygon: How can I check if share certificates I found have value? I have been clearing my parents flat and found share certificates from    P and O  European Ferries   Can I find out if they are still valid? If not, is there a market for them as collectibles or the like? What the matter is vs. what is the matter used in the affirmative I want to know what the matter is with her.  I want to know what's the matter with her. I want to know what's her problem.   Is "I want to know what's the matter with her" and 'what's the matter'  incorrect as Cambridge dictionary suggests (). In their explanation it is suggested that the word 'matter' be replaced with 'problem' as is written in the 3rd example.    Thanks for whatever help you can provide on this.  I've read Rathony's link and agree that his link answered the question about inversion. Now my question is specifically about the correctness of using this wording and if it is correct is there a rule in its usage. How to manage a redirect request after a jQuery Ajax call Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? Story about a trip to a Valley of Sound I read a short story, can’t remember if it was part of a larger narrative or just on its own.  It was about a crew that was taking a bunch of space tourists to a planet that had plants with seeds in glass-like structures that make certain sounds. The main character in this story feels connected to this planet because there’s a Valley of Sound, where there’s a large number of these glass plants that make all sounds imaginable.  People can almost pick out a loved one’s voice for instance. I absolutely love this concept but can’t for the life of me remember what the story/book was called! Is there a way to avoid "+canned.response" in the email address using Gmail's canned response feature If I set up a filter and send it a canned response, Gmail appends +canned.response to my username in the reply.  For example, given the filter "from:foo@example.com to:me@example.com" set up to send a canned response, when foo@example.com sends me an email the canned response comes from me+canned.response@example.com instead of me@example.com, exposing that I'm using a canned response.  Is there a way to avoid this? Make a file pointer read/write to an in-memory location I can run faster than _____. (1) him (2) he?