Question

Suppose that you need to create a list of n values that have a specific known mean. Some of the n values can be freely selected. How many of the n values can be freely assigned before the remaining values are​ determined

Answer 1

Answer: a maximum of n-1 can be freely selected

Explanation:

The fact that we already know the mean means that, if we have n values:

X = (x₁ + x₂ + ....... + xₙ)/n

where X is known and n is known.

Now, suppose we can assign n freely therms: this is not the case, because if X is different than 0, we could assign the n values equal to zero and the equality would be false.

Now suppose we can assign n-1 values freely, then we would have the equation:

X = (x₁ + x₂ +....)/n + xₙ/n

where the term (x₁ + x₂ + ..) is conformed with the random values and xₙ must be chosen in order to satisfy the equation. So we would have the equation:

(X - (x₁ + x₂ +....)/n)*n = n*X - (x₁ + x₂ +....) = xₙ

The equation that can be solved, so a maximum of n-1 can be freely selected.

Answer 2

n-1

Explanation:

All but one of the values may be freely chosen. The last one must be chosen to bring the sum to the a value that is n times the mean.