title: How do I calculate the standard deviation between weighted measurements?
body: <p>I have several weighted values for which I am taking a weighted average.  I want to calculate a weighted standard deviation using the weighted values and weighted average.  How would I modify the typical standard deviation to include weights on each measurement?</p>
  
  <p>This is the standard deviation formula I am using.  </p>
  
  <p><img src="https://i.stack.imgur.com/B1r9p.png" alt="enter image description here"></p>
  
  <p>When I simply use each weighted value for '<em>x</em>' and the weighted average for '\bar{<em>x</em>}', the result seems smaller than it should be.</p>