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How can I print the table of contents multiple times in my document? Standard deviation of binned observations I have a dataset of sample observations, stored as counts within range bins. e.g.:  min/max  count 40/44    1 45/49    2 50/54    3 55/59    4 70/74    1   Now, finding an estimate of the average from this is pretty straight forward.  Simply use the mean (or median) of each range bin as the observation and the count as a weight and find the weighted average:  $$\bar{x}^* = \frac{1}{\sum_{i=1}^N w_i} \sum_{i=1}^N w_ix_i$$  For my test case, this gives me 53.82.  My question now is, what's the correct method of finding the standard deviation (or variance)?  Through my searching, I've found several answers, but I'm unsure which, if any, is actually appropriate for my dataset.  I was able to find the following formula both  and .  $$s^{2*} = \frac{ \sum_{i=1}^N w_i (x_i - \bar{x}^*)^2 }{ \frac{(M-1)}{M} \sum_{i=1}^N w_i }$$  Which gives a standard deviation of 8.35 for my test case.  However, the Wikipedia article on  gives both the formula:  $$s^{2*} = \frac{ \sum_{i=1}^N w_i}{(\sum_{i=1}^N w_i)^2 - \sum_{i=1}^N w_i^2} \sum_{i=1}^N w_i(x_i-\bar{x}^*)^2$$  and  $$s^{2*} = \frac{1}{(\sum_{i=1}^N w_i) - 1} \sum_{i=1}^N w_i(x_i-\bar{x}^*)^2$$  Which give standard deviations of 8.66 and 7.83, respectively, for my test case.  Update  Thanks to @whuber who suggested looking into Sheppard's Corrections, and your helpful comments related to them.  Unfortunately, I'm having a difficult time understanding the resources I can find about it (and I can't find any good examples).  To recap though, I understand that the following is a biased estimate of variance:  $$s^{2*} = \frac{1}{\sum_{i=1}^N w_i} \sum_{i=1}^N w_i(x_i-\bar{x}^*)^2$$  I also understand that most standard corrections for the bias are for direct random samples of a normal distribution.  Therefore, I see two potential issues for me:   These are binned random samples (which, I'm pretty sure, is where Sheppard's Corrections come in.)  It's unknown whether or not the data is for a normal distribution (thus I'm assuming not, which, I'm pretty sure, invalidates Sheppard's Corrections.)   So, my updated question is; What's the appropriate method for handling the bias imposed by the "simple" weighted standard deviation/variance formula on a non-normal distribution? Most specifically with regards to binned data.  Note: I'm using the following terms:     $s^{2*}$ is the weighted variance   $N$ is the number of observations.  (i.e. the number of bins) $M$ is the number of nonzero weights.  (i.e. the number of bins with counts) $w_i$ are the weights  (i.e. the counts) $x_i$ are the observations.  (i.e. the bin means) $\bar{x}^*$ is the weighted mean. Resource for Blender addons? Deleting a comment (mine) shouldn't count as a comment vote I had an old question that I revisited in order to link it in a new answer. I noticed that in the comments Steven A Lowe had a funny comment that I -- at the time -- had responded to with "why can't I upvote comments?"  Well, of course, now I can.  So, I voted Steven's comment up then went to delete mine. I got the warning box that I can only submit comment votes once every 5 seconds. This seems like a bug to me. Deleting is not the same as voting. What's so special about standard deviation? Proper use of passive and active voices? Correct usage of 'but for'? Does 'but for' mean:  'If we had X (but we didn't), Y would have been the consequence'?  Or can it also mean; because we had X, as a result Y happened?   Some different examples of but for:  (Case 1)     But for a cup coffee, it was to be a productive day of writing.   Unfortunately, I'd run out, and spent the day craving.    (Case 2)     But for the glass of wine I had dinner, the house work didn't get   done.   (Case 3)     But for the glass of wine I had for dinner, I would have got the   house work done   What is the correct usage of "but for"? Remove Actions/Filters added via Anonymous Functions That's a freaking bad practice I must say. Spent last two hours finding a solution to remove actions and filters added via Anonymous functions.    This is the code used on a Parent Theme, and I need to remove it.  /**  * Add custom columns to admin comments grid  *  * Rate that user set.  */ add_filter( 'manage_edit-comments_columns', function( $default ) {     $columns['smr_comment_rate']  = __( 'Rate', 'txtdmn' );      return array_slice( $default, 0, 3, true ) + $columns + array_slice( $default, 2, NULL, true ); });   Got , played with it heavily, but no help. So, is there any other alternative that will remove actions/filters added via anonymous functions?  Thanks At the moment, I have created a code that retrieves data from the database and displays it. However, for some reason, I cannot see the files I want to retrieve on my page. My goal is that the data gets retrieved from the database, and is displayed on the webpage. I do not need to make a connection with the database since Wordpress does that automatically.  My code:  &lt;?php  global $wpdb; // this adds the prefix which is set by the user upon instillation of wordpress $table_name = $wpdb-&gt;prefix . "wpex_programma"; // this will get the data from your table $retrieve_data = $wpdb-&gt;get_results( "SELECT * FROM wpex_programma; " ); print_r($retrieve_data); ?&gt; &lt;ul&gt; &lt;?php foreach ($retrieve_data as $retrieved_data){ ?&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;column_name;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;another_column_name;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;as_many_columns_as_you_have;?&gt;&lt;/li&gt; &lt;?php  } ?&gt; &lt;/ul&gt;   My question: the data is not shown and I believe it is not retrieved. How can I fix that? The table name is correct, and I have data in my database tabel. The print_r($retrieve_data)returns the following: Array ( [0] =&gt; stdClass Object ( [id] =&gt; 35 [naam] =&gt; /public_html/wp/wp-content/plugins/AbonneerProgrammas/FilesUpload/Country Kickin 1.mp3 ) )  My query log:    My database structure:   New:  &lt;?php  global $wpdb; // this adds the prefix which is set by the user upon instillation of wordpress $table_name = $wpdb-&gt;prefix . "wpex_programma"; // this will get the data from your table $retrieve_data = $wpdb-&gt;get_results( "SELECT * FROM wpex_programma; " ); print_r($retrieve_data); ?&gt; &lt;ul&gt; &lt;?php foreach ($retrieve_data as $retrieved_data){ ?&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;id;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;naam;?&gt;&lt;/li&gt; &lt;?php  } ?&gt; &lt;/ul&gt; launch app, capture stdout and stderr in c++ I dont need this vertices and/or edge. What is the best way to get rid of them? No field properly between $\mathbb Q$ and $E$ iff $G(K/\mathbb Q) \cong A_4$ or $S_4$ Can my friend and I play minecraft together with different mods? What is the most elegant way to check if the directory a file is going to be written to exists, and if not, create the directory using Python? Here is what I tried:  import os  file_path = "/my/directory/filename.txt" directory = os.path.dirname(file_path)  try:     os.stat(directory) except:     os.mkdir(directory)         f = file(filename)   Somehow, I missed os.path.exists (thanks kanja, Blair, and Douglas). This is what I have now:  def ensure_dir(file_path):     directory = os.path.dirname(file_path)     if not os.path.exists(directory):         os.makedirs(directory)   Is there a flag for "open", that makes this happen automatically? On Irix's  4Dwm, there was an ability to minimise windows into a box (contrary to the task bar used by modern window managers). I have seen this also on an old HPUX.  See the "console" square in the linked image:    Is it possible to accomplish on Ubuntu, either with a plugin or maybe some  window manager other than Unity? What's the current state of LaTeX3 (2020)? There are a few questions on this forum asking about why Latex3 is taking so long to come out. However, all of these are from the early 2010s. I'm just wondering what the current state of the project is and whether it's ready (or nearly ready) for prime time? I am following Carroll's GR book. He explain that it is convention to parameterize geodesics of photons by a parameter $\lambda$ such that  $$p^\mu ~=~ \frac{d x^{\mu}}{d \lambda}.\tag{3.62}$$  But this is the definition of 4-velocity for a massive particle in the case of $\lambda=\tau$ equal to proper time. My question then, is $$u^\mu = p^\mu$$ for all massless particles? I ask because it's well-known that proper time $\tau$ freezes for massless particles. Can my commander ever not be my commander? Once and for all: Is there anything that can possibly be done to my commander that would cause it to cease being my commander?    All of these questions ask the same thing, but aren't general enough to be marked as a duplicate.      Is a commander still a commander if it gets turned into a forest?     Is a commander still a commander if it gets turned face down? I have seen both usages.    Stay tuned on our Facebook page to know more  Stay tuned to our Facebook page to know more    But don’t know which one is more appropriate. Please help me figure it out. Thanks Why is a finite integral domain always field? This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse:   let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$,  then as $R$ is closed under multiplication $\prod_{n=1}^k\ x_i=x_j$,  therefore by canceling $x_j$ we get $x_1x_2\cdots x_{j-1}x_{j+1}\cdots x_k=1 $,  by commuting any of these elements to the front we find an inverse for first term, e.g. for $x_m$ we have $x_m(x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k)=1$, where $(x_m)^{-1}=x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k$.   As far as I can see this is correct, so we have found inverses for all $x_i\in R$ apart from $x_j$ if I am right so far. How would we find $(x_{j})^{-1}$?