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I have a set of percentages {1%, 12.5%, 25%, 50%, 100%} I want to find a way to 'pull up' all of these percentages to be higher in a way that is more advanced than an average mean like ((x+100)/2)

What I would like to achieve is a result set more like: {5%,31.25%,50%,80%,100%} - ie: lower numbers are multiplied by a bigger factorial

I don't need this exact output - I just need a way to increase lower percentages more than higher ones

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It seems like you want this for a computer program. Just draw a function curve $f(x)$ for $x$ between $0$ and $100$ which you are satisfied with, and approximate it piecewise linearly in your program. No need for fancy formulas. – Samuel Jan 17 '12 at 11:50
You are correct - I could do if(0 < x < 10) factor = 5 but I remembered something vaguely from school about getting a curve that would do this for me. I just don't know what this would be called or what to search on – Victoria Jan 17 '12 at 11:58
I have just looked at log(x) which gives me a nice curve - this is a step in the right direction but I'd like the curve to be far steeper so lower values are increased more than log(x) can give – Victoria Jan 17 '12 at 12:11
up vote 2 down vote accepted

How about you try taking the square root. Write each of your percentages as a decimal expansion less than one, example: $\{0.01, 0.125, ..\}$. Then take the square root of each of these values. The nice thing is that 0 will stay zero and all numbers will remain in the proper range. If that is not enough, you can try taking cube roots or $n^{th}$ roots.

Update: getting the square root of your values gives $\{10\%, 25\%, 50\%, 71\%, 100\%\}$, which is close to what you asked for.

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Thanks - cubed root is perfect for my purposes – Victoria Jan 17 '12 at 13:20

One way to do it is to find a linear function $f(x)=ax+b$ such that $f(100)=100$ and another fixed value, e.g. $f(50)=80$. In this case you'll get $f(x)=0.4x+60$.
In this case your list will be: $\{60.4,65,70,80,100\}$.
EDIT: If you fix $f(1)=5$, then you'll get lower values. e.g $f(x)=\frac{95x+400}{99}$. You'll have then $\{5,16.03,28.03,52.02,100\}$

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Thanks for your answer Dennis but I want lower percentages like 1 to be way below 50 still which I don't think I could achieve this way – Victoria Jan 17 '12 at 11:55
you can fix the second value to be $f(1)=5$. This way you'll get lower values. – Dennis Gulko Jan 17 '12 at 12:04

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