I’m trying to compute the variance of how much pumpkins weigh across pumpkins from 125 farms. However, I only have, for each farm, three pieces of data: (1) the variance of weights for pumpkins from that farm (2) how many pumpkins the farm has (3) the total weight of the pumpkins from the farm.  How can I compute a “weighted variance” of how much a pumpkin weighs across farms? Can I just compute a weighted average of the individual variances?
Say there are $m+n$ elements split into two groups ($m$ and $n$). The variance of the first group is $\sigma_m^2$ and the variance of the second group is $\sigma^2_n$. The elements themselves are assumed to be unknown but I know the means $\mu_m$ and $\mu_n$.  Is there a way to calculate the combined variance $\sigma^2_{(m+n)}$?   The variance doesn't have to be unbiased so denominator is $(m+n)$ and not $(m+n-1)$.
Adding vertical space at the start of a page I'd like to, first thing before writing anything on a page, add some vertical space.  Minimal (non)working example:  \documentclass{article} \begin{document}  \vspace{250px} I want to be further down on the page :(  \vspace{250px} Haha! Sucker :D \end{document}     How can I get the first \vspace to work, just like the second one? Is there really a way to make a true toon shader material in Cycles? Is there an English word meaning "the use of uncommon words"? Is there an English word meaning "the use of uncommon words" or similar? Cutting strands of wire to fit plug I am a hvac tech, but my question is I have 2awg stranded and my plug is 6awg, will it matter much if I cut some strands to fit my plug? the plug is 60amp 250vac for a piece of kitchen equipment if the matters. What is this rhetorical scheme or literary device called? I can't recall the name of the rhetorical scheme or literary device involved with using the same word more than once within the same sentence but with different meanings:  "The only thing we have to fear is fear itself."    fear - verb fear - noun   "Turn right right here."   right - adv. directional right - adv. in this place   "My feet are but feet away."   feet - noun lower appendage feet - noun distance The junction has a stop sign on each of the four entrances.   The junction has a stop sign on each of the 4 entrances.   The first is preferred, for some reason, by many English texts. Why? I haven't seen this phenomenon in other languages. It is well known that compactness implies pseudocompactness; this follows from . I know that the converse does not hold, but what is a counterexample?  (A  is a topological space $S = \langle X,{\mathfrak I}\rangle$ such that every continuous function $f:S\to\Bbb R$ has bounded range.) Are the tags shown in a particular order? I noticed that, sometimes, the tags I add to a question are not shown in the same order I wrote them, when the question is saved. Are they shown in a particular order? If they are, in which order they are shown? Why is it difficult to differentiate between  and ? Is it because we don't clearly understand how both of these phenomenon takes place?   My thoughts: From an answer to one of my earlier question  I could understand that light waves behave differently when it strikes a slit which is of several times bigger than its wavelength and when it strikes a slit of size comparable to it its wavelength.In the former case the different points on the slit acts as wave-front sources producing circular wave fronts which interfere to form the diffraction pattern but in the later case plane wave-front striking a slit after passing through it forms circular wave-fronts and two such wave-fronts from two different slits interfere to form the interference pattern. But, I'm not satisfied by that because I don't get any intuition about points on the slit acting as wavefront sources. So, what is actually happening? How can I display a short chapter name in the header and a long chapter name in the ToC? I am writing a book and one of my chapters has a very long title. As it can't be displayed correctly on the top of even pages, I provide a "short title name" via the   \chapter[short title]{long title}   command. But then, "short title" appears in the table of contents instead of the original/true/long one.  How can I have the "short title" in the header of even pages, and the "long title" in the ToC and in the "Title header"?  For reference, here is the code I use:  \documentclass[11pt,a4paper]{book} \begin{document} \tableofcontents  \chapter[Short title]{Very vey very very very very very very long title     i can't display in the header} Lorem ipsum dolor sit amet, consectetur adipiscing elit. \end{document} Word or phrase that can mean something and its opposite In the movies, the pack of Uruk-Hai capture Merry and Pippin, and are bringing them back to Saruman as quickly as possible.  Why didn't they use Wargs to carry the Hobbits instead of packing them on their backs?    Or, better yet, why didn't the Nazgul fly down and take them on their Fell Beasts? Intuition behind the definition of a measurable set Note: Suppose $f(z)$ is a polynomial or a proper/strictly proper rational function.   Can someone demonstrate why if all coefficient of $f(z)$ are real then relation $f(\bar z)$ = $\bar {f(z)}$ hold  Is there a theorem for this result? When trying to mount an exfat filesystem, I get the following error:  Error mounting /dev/sda6 at /media/gkp/Backup: Command-line  `mount -t "exfat" -o "uhelper=udisks2,nodev,nosuid" "/dev/sda6" "/media/gkp/Backup"'  exited with non-zero exit status 32: mount: unknown filesystem type 'exfat'   Exfat is used on some USB sticks and camera sd cards. What can I do to mount this type of filesystem? I've found a couple of references () that suggest using final as much as possible and I'm wondering how important that is. This is mainly in the the context of method parameters and local variables, not final methods or classes. For constants, it makes obvious sense.  On one hand, the compiler can make some optimizations and it makes the programmer's intent clearer. On the other hand, it adds verbosity and the optimizations may be trivial.  Is it something I should make an effort to remember? I'm new to this website. I've checked the other topics and couldn't find the answer to my question. I've also tried to find the result with TI-CAS but the only thing it does is simplify the inequality which really doesn't help at all.  the equation is : $1000n^3 &lt; 2^n$ Now I've been trying to use log but it gets me nowhere (I haven't seen this form before), the answer is supposed to be 24 but I'm not sure. So my question is how to solve step by step this inequality for n. Could you guys help me with this one? How to make a complete factory reset, without anyone being able to retrieve my data? I will have a few Android phones that I will have to let go.   I would like to make a proper clean wipe of the data, without no one being able to retrieve them, and do a factory reset.  I have read about the , encrypting the device before doing a factory reset, doing both, etc.  Since I am not an Android/adb guru, I would like to know if some of you would have some idea on how to achieve this with certainty.  I know "certainty" is a big word when it comes to proper data erasure on flash memory but still.  I have found some answers here and somewhere else, saying that encrypting and doing a factory reset on your Android device is enough, but I could not find a reliable source.  Just a few information that could be useful:   I'm running Linux as my OS and, therefore, I'm not looking for a Windows-based solution My devices are non-rooted and (almost) identical (Samsung A510F + 1 A510FD) Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$, with $f: (k_1,k_2)\to\Bbb R$ differentiable at $a\in (k_1,k_2)$, $\lim b_n=\lim a_n=a$, and $a_n&lt;a&lt;b_n$.   My attempt: Im trying to write the above equality as a particular case of this other equality that hold by any interior point $a$  $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a-h)}{2h}$$  Because $a_n&lt;a&lt;b_n$ then I can write  $$f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}=\lim_{n\to\infty}\frac{f(h_n+a)-f(a-j_n)}{h_n+j_n}$$  Where $h_n=b_n-a&gt;0$ and $j_n=a-a_n&gt;0$, where these two new sequences tends monotonically to zero.  Now my problem: my thought is that $h_n$ and $j_n$ are particular sequences then I can build a sequence $k_n$ such that $j_n$ and $h_n$ are subsequences so I can write:  $$\{h_n:n\in\Bbb N\}\subset\{k_n:n\in\Bbb N\}\text{ and }\{j_n:n\in\Bbb N\}\subset\{k_n:n\in\Bbb N\}$$  (I can build this $k_n$ by example doing $k_{2n}=h_n$ and $k_{2n-1}=j_n$) and then I have the desired expression  $$f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}=\lim_{n\to\infty}\frac{f(k_n+a)-f(a-k_n)}{2k_n}$$  My question:   Is this proof right? In particular Im unsure of the implicit statement "if the limit exist for $k_n$ then it must exist too for the particular case with $h_n$ and $j_n$". The use of two sequences, $h_n$ and $j_n$ in the limit make me unsure about the validity of this manipulation and assumption. There is a simpler/different way to prove the same?   Thank you in advance. how to decode this JSON string?