Set of numbers given, find mean and SD of the other

Question

A set of eight numbers has mean 5 and SD 4. The number 21 and another integer n are added to the set. The standard deviation of the set is now 6. Find n and the mean of the set now.

Answer 1

One way to do this is to concoct $8$ numbers which have a mean of $5$ and a standard deviation of $4$. For example, take $$ -1,1,3,5,5,7,9,11, $$ and label these points $x_1 , x_2, \dots , x_8$. Then, one can easily check: $$ \frac{1}{8}\sum_{i=1}^8 x_i = 5 $$ and $$ \sqrt{\sum_{i=1}^{8} \frac{(x_i - 5)^2}{8 - 1}} = 4. $$

Adding $21$ and $n$ gives $$ \mu = \frac{8 \cdot \mu_1 + 21 + n}{10} = \frac{61 + n}{10}. $$ The new standard deviation is $$ 6 = \sigma = \sqrt{\sum_{i=1}^{8} \frac{(x_i - \mu)^2}{10 - 1} + \frac{(21 - \mu)^2}{10 - 1} + \frac{(n - \mu)^2}{10 - 1}}. $$ Expanding gives a quadratic equation in $\mu$:

$$\begin{align} 324 = & (-1 - \mu)^2 + (1 - \mu)^2 + (3 - \mu)^2 + 2(5 - \mu)^2 \\ &+ (7 - \mu)^2 + (9 - \mu)^2 + (11 - \mu)^2 + (21 - \mu)^2 + (9 \mu - 61)^2 \end{align}$$ Note that the last term is $(10 \mu - 61 - \mu)^2$ as $\mu_2 = \frac{61 + n}{10}$. Solving this for $\mu$ yields $\mu = 8$. Hence, $n = 19$.

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Edit: For clarification, this does not totally answer the question at hand. My example does not prove that $n = 19$ is the only (integer) solution. However, it is likely that this is the case. The method I used to find the set of integers which worked was by noticing that we needing to find $8$ squares that sum to $112 = 4^2 \cdot (8 -1)$. This gave a natural choice of $$ 112 = 6^2 + 6^2 + 4^2 + 4^2 + 2^2 + 2^2 + 0^2 + 0^2 $$ which implied that $$\begin{array}{c} x_1 = 5 - 6 \\ x_2 = 5 + 6 \\ x_3 = 5 - 4 \\ x_4 = 5+4 \\ x_5 = 5 - 2 \\ x_6 = 5+2 \\ x_7 = 5 - 0 \\ x_8 = 5 + 0 \end{array}$$ This happened to lead to a integral value of $\mu$ and therefore an integral value of $n$.