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I have the sampling variance of the sample mean: $V_1=\frac{1}{N-1}\sum_i (x_i-\bar{x})^2$. Now if the largest observation is removed, I have a new variance, $$V_2=\frac{1}{N-2}\sum_{i=0}^{n-1} (x_i-\bar{\bar{x}})^2$$ where $\bar{\bar{x}}$ is the new (reduced) mean of the $N-2$ observations.

Depending on the data, I could see $V_1$ could be greater or lesser than $V_2$. An intuitive example of $V_1>V_2$ is when you consider one anomalous large value and mostly small observations.

Is there an intuitively clear example where I can visually understand the case where $V_1<V_2$?

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    $\begingroup$ For a simple example let $x=(2,20,20)$. $\endgroup$
    – whuber
    Sep 25, 2013 at 16:39
  • $\begingroup$ @whuber: I was teaching some MBA kids there. I was able to give a numerical example, but they wanted something visually understandable, like what they get for $V_1>V_2$... $\endgroup$
    – Bravo
    Sep 25, 2013 at 16:57
  • $\begingroup$ So, more generally, it sounds like you are seeking a way to visualize variance (and changes thereof). Please indicate that in your question so you get appropriate answers. $\endgroup$
    – whuber
    Sep 25, 2013 at 17:37
  • $\begingroup$ When you say "largest sample" do you mean 'largest observation'? In statistics a sample is a collection of observations. Your largest observation is a member of your sample. $\endgroup$
    – Glen_b
    Sep 26, 2013 at 0:21

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Consider ${\bf x}=(1,8,9,10)$: $\bar x = 7$, $s^2 = 16 \frac23$. When you remove the highest value $x_4=10$, it only increases the influence of the low outlier $x_1=1$: $\bar x =6$, $s^2 = 19$. Feel free to play with numbers more to get all the results in integers :).

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