A user navigates a service and provides feedback choosing one of the options Very satisfied, Satisfied, Neither satisfied nor dissatisfied, Dissatisfied, Very dissatisfied. Each option is assigned a weight.
Very satisfied ~ 1Satisfied ~ 0.75Neither satisfied nor dissatisfied ~ 0.5Dissatisfied ~ 0.25Very dissatisfied ~ 0
A legacy formula is used to calculate user satisfaction which is not a weighted mean despite being called by peers. The formula is: $\bar{x} = \frac{\sum^n_{i=1} w_ix_i}{\sum^n_{i=1} x_i}$
where $x_i$ is the number of responses for each option and $w_i$ is the weight assigned. Does that formula have a name? Is it the correct way to calculate user satisfaction?
The quantity $\displaystyle \frac{\sum_{i=1}^n w_i x_i}{\sum_{i=1}^n x_i}$ is in fact a weighted average of $w_1,\ldots,w_n$ in which the weights are $\displaystyle \frac{x_j}{\sum_{i=1}^n x_i}$ for $j=1,\ldots,n.$ The quantities $w_1,\ldots,w_n$ are not weights in this weighted-averaging process. Normally one might call this $\overline w$ rather than $\overline x$ since it's an average of $w_1,\ldots,w_n.$