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? (The result is referred to as the number of degrees of​ freedom.)

Answer 1

As the problem indicates, degrees of freedom are the number of values ​​that can be independently selected before it is necessary to choose specific values ​​to arrive at the desired result.

The average, on the other hand, results from the sum of a list of values ​​divided by the amount of values ​​in the summed list.

Assume that the mean sought is $x$ and consider that the list is composed of a single element $a$, in that case no random number can be selected, since the mean $x$ must correspond to that number.

If the list were composed of two elements $a$ and $b$, one of the two values ​​could be chosen randomly, and according to the chosen value the second should be the one whose sum with the previous one results in $2x$, this given that the formula of the average $\sum\limits_{i=1}^n \frac{ x_{i}}{n}$.

With three values ​​$a$, $b$ and $c$, it is possible to select two freely, since the thirteen must be the one that balances the sum of $a+b$, that is $(a + b) + c = 3x$.

Thus, in general, with n values, it is possible to select $n-1$ values ​​freely whose sum must be balanced by the last value so that the whole sum is $nx$.

Answer

In a list of $\bf{n}$ values ​​you can assign $\bf{n-1}$ values ​​freely, that is, you have $n-1$degrees of freedom.