I have an average of 3,5 over 16 data points. If I wish to increase this average to, say, 6, how many future data points must be of a certain value? Assume all future data points are of the same value.
If the current average of 16 data points is 3,5, then four new data points each of value 7 would result in a new average of 4,2.
I know how to calculate this by hand. What I'm looking for is some formula that I can plug in to my spreadsheet.
Would it be possible to calculate how many data points are needed to reach an average of 6, given each new data point has a value of 7? Also, would it be possible to reverse the unknown(say I know there will be 20 more data points. What does each value have to be to reach a new average of 6)?
Is there a term for this type of formula/problem? I tried searching for a solution but I'm not quite sure what I'm supposed to search for.
A simple way of thinking about this is to consider the current total sum in any situation. In the beginning, you have $16$ data points and their total sum is $3.5 \cdot 16 = 56$. Now, you wish to add some terms, let's say $n$ terms (a known number), of value $x$ so that the new average is $\mu_{new}$. This just means that the new total sum is $$ 56 + nx = (16+n)\mu_{new} $$ from which you can solve for $x$: $$ x = \frac{(16+n)\mu_{new} - 56}{n} $$ If instead you know $x$, you can solve for $n$: $$ n = \frac{ 16\mu_{new}-56 }{x -\mu_{new}} $$