1
Scale 1:
-5  -4  -3  -2  -1  0   1   2   3   4   5
Scale 2:
1   2   3   4   5   6   7   8   9   10  11

How do I calculate the percentage of value X in Scale 1 when 0 is a valid value on the scale?

Ex. 3 in Scale 2 would be 3/11*100 = 27.3%

So whats the function to get -3 in Scale 1 to be 27.3% of its scales total steps?

mdc
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    what is the value x? Sorry but i don't unterstand your question – Dominic Michaelis Mar 14 '13 at 16:23
  • Value x = if x is 3 in scale 2 then X is -3 on scale 1. If its 6 on scale 2 then its 0 on scale 1. – mdc Mar 14 '13 at 16:56
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    Yet another unjustified closure of a question with an obvious interpretation, lacking even the excuse of lack of involvement by the OP. It was ‘reasonably answered in its current form’. I have of course voted to reopen. – Brian M. Scott Mar 14 '13 at 19:47

1 Answers1

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Scale $1$ runs from $-5$ to $5$ in steps of $1$ unit, so it has $11$ values. $-3$ is the third value. Thus, $3$ of the $11$ values are less than or equal to $-3$, so $\frac3{11}$ of the values are less than or equal to $-3$. If you convert this fraction to a percentage, you get $27.3$% when you round to one decimal place.

More generally, if a scale runs from $a$ to $b$ in steps of size $d$, it has $\frac{b-a}d+1$ values. If $v$ is one of the values, there are $\frac{v-a}d$ values smaller than $v$, so $v$ is the $\left(\frac{v-a}d+1\right)$-st value.

In Scale $1$, for instance, $a=-5$, $v=-3$, and $d=1$, so $-3$ is the $\left(\frac{-3-(-5)}1+1\right)$-st value or, after you do the arithmetic, the $3$-rd value. (Of course in this case we don’t need the formula: we can easily see that $-3$ is the $3$-rd value.)

Thus, we have altogether $\frac{b-a}d+1$ values, and $\frac{v-a}d+1$ of them are less than or equal to $v$, so the fraction of them that are less than or equal to $v$ is

$$\frac{\frac{v-a}d+1}{\frac{b-a}d+1}=\frac{v-a+d}{b-a+d}\;,$$

and you can convert this to a percentage simply by multiplying by $100$. In the example with $a=-5$, $b=5$, $d=1$, and $v=-3$, this formula yields the fraction

$$\frac{-3-(-5)+1}{5-(-5)+1}=\frac3{11}\;,$$

as we knew it should.

Brian M. Scott
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