Convert a 1-9 scale average to a 1-5 scale average I am new at this and could use some help with a conversion.
Survey participants were given the answer choices on a ranking scale of 1-9 for each question. We then averaged the results for each question to understand what the average answer was for each question. For example, for question B, the average answer was 8.1 out of 9.
How would I convert the average to what the number would be on a 1-5 scale? So using the example above, 8.1, what formula would I have to apply to that number so that the average would be a number on the 1-5 scale? i.e. "the average answer was X.X out of 5.
Thanks for your help. The answer may be very simple, but at this point, I think I have over-thought this problem.
 A: Let's do this by arithmetic: 
Subtract $1$ from the value and range, and work with $7.1/8$. That is the same as $3.55/4$. 
Add $1$ back in and you have $4.55$. 
There is a strong assumption in there about how people would have answered had the scale been different, which you make and are responsible for and which the arithmetic doesn't guarantee.
If you want the algebra, then 
$(\text{new} - 1)/4 = (\text{old} - 1)/8$
so shuffling it all around 
$\text{new} = 0.5\ (1 + \text{old})$
which checks at $0.5\ (1 + 8.1) = 4.55$.  
Geometrically, think about the conversion as given by a line connecting $(1,1)$ and $(9,5)$. 
It often helps to check with extreme cases. Clearly the average can't go below $1$ in both cases, or above $5$ or $9$, depending on the upper limit. So, any conversion must respect those extremes. 
A: The best way is to use a linear transformation. Basically you use 2. First convert the 9 point scale to a 0-1 interval and then convert it to a 5 point interval. This isn't dissimilar to the method used by random number generators, they often start by generating a random number from 0-1 and then transforming it to the desired scale.
So in your case it looks something like this:
Let 
X = score from 9 point scale (I'll use the example of 8.1 below)
y = transformed score to 5 point scale
Step 1: Recode to 0 - 1 scale
Y_1 = (X-min_old scale)/(max_old scale -min_old scale)
Y_1 = (8.1-1)/(9 - 1) = 0.8875
Step 2: transformed to 5 point scale
Y = Y_1 (max_new scale - min_new scale) + min_new scale
Y = 0.8875(5-1)+1 = 4.55
Couple things to keep in mind about this:
1) I would convert your underlying scores first and then find its average, that way you can get its SD too and do any statistical analysis you want on the recoded scores
2) It's much harder going the other way e.g. 5 to 9 point since you are missing some datum e.g. even after recoding you only have 5 scores not 9. This can make direct comparisons a bit "patchy"
