What is the mathematically correct terminology for the contribution of each value of a set to the sum of the set?

Question

In my work I often come across situations where I would like to know how much a particular number contributes to the overall sum of the values of a set. When referring to this value, or the set of numbers that is each individual values contribution to the sum, I usually refer to said values as the "percentage contribution to the sum", and although this conveys the premise, I can't help but think that maybe it's not the correct label for this. My efforts to find the correct term have fallen short repeatedly so I figured I would come to this StackExchange forum to query the math community. Just to clarify my question I present the following situation (and I apologize in advance for this being in code as I am far more proficient in programming than mathematics formatting).

Say I have an array of 4 values

array([16594.38194089, 16833.38403293, 28933.39527259, 28956.02268959])

The sum of which comes to 91317.18393599312

resulting in a "percent composition / percent contribution / this is the term I am looking for" of

array([0.18172245, 0.18433972, 0.31684502, 0.31709281])

And that's it, that's the thing I am looking for the correct term for. It's probably a really simple answer but I like to use the correct names for things whenever possible and this has been eluding me for a while.

Answer 1

If we consider the array as vector of non-negative numbers, we have a vector $x$ (in this case a $4$-tuple): \begin{align*} x&=(x_1,x_2,x_3,x_4)\\ &=(16594.38194089, 16833.38403293, 28933.39527259, 28956.02268959) \end{align*} Denoting with $\|x\|_1:=x_1+x_2+x_3+x_4$ its length, we divide each component of the vector by the length $\|x\|_1$ of it.

We obtain this way \begin{align*} \color{blue}{\frac{x}{\|x\|_1}}&=\frac{1}{91317.18393599312}\cdot x\tag{1}\\ &=(0.18172245, 0.18433972, 0.31684502, 0.31709281) \end{align*} We call this vector (you might call it array) standardised or normalised by its length according to the L1-norm.

Notes:

• The term normalised or standardised is convenient, since the components in (1) sum up to $1$.

• If elements of the vector are negative, we have to take the absolute value $\|x\|_1=|x_1|+|x_2|+|x_3|+|x_4|$ according to the definition of the $L1$-Norm.

• If a vector is already normalised, it will not be changed by an additional normalisation. Since the sum of the components is then already $1$ and division by $1$ does not change anything.

Answer 2

I think proportion is the most correct word for what you are calculating:

the relation of one part to another or to the whole with respect to magnitude, quantity, or degree

We also use proportion to mean an equality of ratios, so this might be a bit ambiguous. One ratio would be the numbers as given $a:b:c$. An equivalent one would be the one scaled so that the sum is one, as in $a/(a+b+c):b/(a+b+c):c/(a+b+c)$. So you could also call this expression a “normalized” ratio.