First of all I must state that I am not a mathematician, so please correct me if I use wrong terminology.
I am building a web application which needs to calculate the rating for each entity based on both the quantity and score of the reviews for that entity.
In other words, I don't want to just calculate the rating based on the average score as that would make an entity with one hundred 9 score (review score can be from 0 to 10) reviews rate lower than an entity with only one 9.5 score review.
I need an algorithm to calculate rating and add rating "weight" to the final rating based on how many reviews the entity has, so that for instance in the above example the entity with 100 9 score reviews would get a rating that is higher than the entity with only one 9.5 score review. In other words, the final entity rating score will be based on the "relationship" between quality and quantity.
There is another important thing to note: an entity can not have a rating higher than 10, so the rating "weight" added by the quantity can not be linear.
In the algorithm we can use any data about the reviews/rating, that is individual review score, total number of reviews, sum of all reviews, number of good reviews (score 8 or higher) so far, etc, in each iteration of the rating calculation process.
Any kind of help or info regarding this would be appreciated. Thank you.
What you can do, for instance, is take the rate of reviews (w weighted mean), divide it by two (to reduce the scoring to a scale of $[0,5]$ and add this value to $5(1-e^{-q})$. So the formula becomes $$\text{score}=5p/10+5(1-e^{-q/Q})$$ where $p$ is the review rating and $q$ is the number of ratings, and you chose for $Q$ an appropriate number that shows what importance you attach to the notion "quantity."
An example: An item has $3$ times a revision score of $6$ and $2$ times a revision score of $7$.
Then $p=(3⋅6+2⋅7)/5=6.4$
If we take $Q=10$, then $5(1-e^{-5/10})\approx 3.88$, so the total score is $3⋅2+3⋅9=7.1$ rounded $7$.
On the other hand, if somebody has $20$ scorings of $6$, then $p=6$ and $5(1-e^{-20/10})\approx 4.58$, so the final score is $3+4.6$ rounded giving $8$.
The choice of $Q$ depends on what you call "few," "moderate," "many."
As a rule of thumb, consider a value $M$ that you consider "moderate", and take $Q=-M/\ln(1/2)\approx 1.44M$.
So if you think $100$ is a moderate value, then take $Q=144$.
Finally, you can also replace the equal weight on quantity and quality by a skewed one so that the final formula becomes:$$\text{score}=Pp+10(1-P)(1-e^{-q/Q}))$$ where $P\in [0,1]$ (in the original formula we had $P=0.5$).