What formula would you use to cast an average of several numbers into a smaller range?

Question

Say I have four numbers ranging in value from 15-95. If I want to, for a simple example, say that if the average of the four numbers is 15 (lowest possible value), that would relate to 2 on a scale of 2-10. Then instead if the average of the four numbers is 95 (highest possible value), that would relate to 10 on that same scale. What, then, is the formula to figure out where an average of say 57 falls on that scale of 2-10. Does that make sense, and is there a name for what this kind of 'transposition' is called?

Answer 1

If you want "simple", then just do a linear transformation; that is, write down the equation of the line through the points $(15,2)$ and $(95,10)$ and there you go.

Answer 2

Easiest way is to shift the range first to the orign $$\left[15,95\right] \mapsto [0,80]$$ After that scale it to the length you need, here: $$[0,80]\mapsto[0,8]$$

Last but not least shift it back to the position you need

$$\left[0,8\right]\mapsto [2,10]$$

So a funktion to map an $x$ from the frist range to an $y$ of the second range would be given by

$$y = f(x) = (x-15)\cdot\frac{1}{10}+2$$

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Take a look at your testcase $x=15, x=95$ and $x=57$. For those values you will get:

• $f(15) = (15-15)\cdot\frac{1}{10}+2 = 2$
• $f(95) = (95-15)\cdot\frac{1}{10}+2 = 10$
• $f(57) = (57-15)\cdot\frac{1}{10}+2 = 6.2$