Magnifying the outliers in a rating system to rank objects better

Question

1) I am trying to design a product for weekly ratings (1-5) of an entity. The entity's rating will be reset every year. I know most of the weeks the rating will be the same (3) and I want a way to give more weight to 4 (which can occur only 5 to 6 times a year) and a rating of 5, which can happen only 2-3 times a year. Is there a way I can ignore the arithmetic mean and use any other metric / strategy that highlights the outliers so that the entities which get the occasional 5 star stand out far than the regular 3s or 4s?

For I have an Object 1 which has got 3 rating for all 40 weeks and 4 rating for 12 weeks (Avg - 3.23) should be of less value than an Object 2 which has got 3 rating for 48 weeks and 5 rating for 4 weeks (Avg - 3.15). I want Object 2 to be ranked better with higher rating and the difference should be also significant.

2) Will it help my cause if I use a separate scale as in software user story sizing (1,2,3,5,8) instead of (1,2,3,4,5)?

Answer 1

If arithmetic mean is your reference point, try to use its extension - the Ordered Weighted Averaging operator (OWA) defined as:

$$OWA_w(x) = \sum_{i=1}^n w_i*x_i^*$$ $\text{where} \sum_{i=1}^n w_i=1, w_i \in [0,1]$ and $x^*$ is $x$ sorted in the non-increasing order.

OWA is the arithmetic mean when $w_i=\frac{1}{n}$, for $i=1,\dots,n$.

Try to experiment with weight vector $w$ to suit your needs. Assume that the bigger the $x_i$ the bigger weight is used.

For example:

Given $n=5$, $x=(2,3,3,4,4,5)$ define $w=(\frac{6}{21},\frac{5}{21},\frac{4}{21},\frac{3}{21},\frac{2}{21},\frac{1}{21})$.

Then $x^*=(5,4,4,3,3,2)$ and your average is $$OWA_w(x)=\frac{6}{21}*5+\frac{5+4}{21}*4+\frac{3+2}{21}*3+\frac{1}{21}*2.$$