<s>[INST] Is it possible to make a dataset given these following mean, range, and standard variance? Given that the:
$Mean = 30$
Range $= X_n - X_1 = 10$
$S^2 = Variance = 40$
is it possible to construct a dataset with those values?
So, to be honest I have no idea how to properly approach this and I have tried "brute forcing" this to find a set of numbers to make this work, but no matter what I do I can't seem to do it. Thus I believe that the answer is "no", however, I'm not sure about this.
Is there a standard procedure I can follow to solve this logically?
 [/INST] No, it is impossible. 
Given any list of $n$ numbers $X_k$ without any constraint of its order. Let
$$\begin{cases}
\overline{X}   &= \frac{1}{n}\sum_{k=1}^n X_k\\
\overline{X^2} &= \frac{1}{n}\sum_{k=1}^n X_k^2
\end{cases}
\quad\text{ and }\quad
\begin{cases}
P         &= \max\{ X_k : 1 \le k \le n \}\\
Q         &= \min\{ X_k : 1 \le k \le n \}
\end{cases}
$$
The (biased) sample variances and unbiased sample variance for the numbers $X_k$ are given by the formulas
$$\begin{cases}
\sigma_X^2 &= \frac{1}{n}\sum\limits_{k=1}^n (X_k - \overline{X})^2 = \overline{X^2} - \overline{X}^2\\
s_X^2      &= \frac{1}{n-1}\sum\limits_{k=1}^n (X_k - \overline{X})^2 = \frac{n}{n-1}\sigma_X^2
\end{cases}$$
Since $(P - X_k)(X_k - Q) \ge 0$ for all $1 \le k \le n$, we have
$$\frac{1}{n}\sum_{k=1}^n (P - X_k)(X_k - Q) \ge 0
\iff  -PQ + (P+Q)\overline{X} - \overline{X^2} \ge 0\\
\implies \sigma_X^2 = (\overline{X^2} - \overline{X}^2) \le (P - \overline{X})(\overline{X} - Q)$$
This inequality is a special case of the Bhatia-Davis inequality. 
Since $P \ge \overline{X} \ge Q$, this leads to
the Popoviciu's inequality on variances:
$$\sigma_X^2 \le \frac14 (P-Q)^2
\quad\iff\quad
s_X^2 = \frac{n}{n-1}\sigma_X^2 \le \frac{n}{4(n-1)}(P-Q)^2$$
Apply this to our problem where $P - Q = 10$, it is clear there is no way for the (biased) sample variances $\sigma_X^2$ to be equal to $40$. For the unbiased sample variances, if we want $s_X^2 = 40$, we need
$$40 \le \frac{n}{4(n-1)}10^2 = 25\frac{n}{n-1}\quad\implies\quad n \le \frac{8}{3}$$
Since $n$ is an integer $\ge 2$, $n$ can only be $2$. However when $n = 2$, 
$$\begin{cases}
\overline{X} &= \frac{P+Q}{2}\\
\overline{X^2} &= \frac{P^2+Q^2}{2}
\end{cases}
\quad\implies\quad
\begin{align}
s_X^2 &= \frac{2}{2-1}\left(\overline{X^2} - \overline{X}^2\right)
= P^2 + Q^2 - \frac{(P+Q)^2}{2}\\
&= \frac12(P-Q)^2 = 50
\end{align}
$$
This doesn't match the given number $40$ again. As a result, there is no data-set which can produces the desired mean, range and sample variances (for both biased and unbiased one).
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