"Is it possible to create a data set where $\bar{x}=30.0$, range $R=10$, and variance $s^2=40.0$?"

I feel sort of dumb asking this question, but I'm not quite sure I'm on the right track. I know that with two data points, the max variance is $50$ if $x_1=25$ and $x_2=35$, and then when adding a third point, variance seems to drop to a maximum of around $33$ if I stay within the range.

Is that correct, and is there a more mathematical way to show what I've said? The most we've been given so far is the equation for sample variance, mean, and a couple paragraphs on what a standard deviation is.

  • $\begingroup$ what does the word Range mean? Why is it a single number? Put another way, what values are allowed for a single data point? $\endgroup$ – Will Jagy May 18 '15 at 0:18
  • $\begingroup$ I believe we are inherently working in the set of real numbers, and the range is taken to be the maximum value minus the minimum value. But I could be mistaken. $\endgroup$ – TomGrubb May 18 '15 at 0:19
  • $\begingroup$ The range is 10, meaning that the maximum-minimum = 10. Any real number is fine. $\endgroup$ – ProbablyNotYourProfessor May 18 '15 at 0:29

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