Simple Weight Question This is a very basic question regarding weights. 
Let us say I have product A that has 100 reviews. Let us say 80 of these reviews are positive and 20 of these reviews are negative. The score for product A is 80%. 
Let us say I have product B that has 1000 reviews. Let us say 800 of these reviews are positive and 200 of these reviews are negative. The score for product B is 80%. 
Even though the scores for product A and B are equal, I think that the score for Product B should be higher because more people filled out feedback then they did for Product A. 
What is the best way to account for the weight of the sample size in a formula that yields a percentage score? 
 A: You have a series of Bernoulli trials and you're estimating the parameter $p$ of a good review.
(+1) @Michael Chernick expresses the most firmly grounded, classic statistical way to think about it. 
An alternative, Bayesian approach:
An alternative is to become a Bayesian, and instead of treating $\rho$ as a parameter (i.e. some number), you can model $\rho$ as a random variable! That is, you can use the tools of probability to model the uncertainty in your head about $\rho$. You will:


*

*Start with some beliefs about $\rho$ (this is your "prior")

*Update those beliefs based upon whether review $1$ is positive or negative. (This is called forming the "posterior.")

*Repeat, do the same thing with review $2$, review $3$, etc....


If we represent our beliefs about $\rho$ using the beta distribution, then our beliefs about $\rho$ after observing a new review will also follow the beta distribution because it is a conjugate prior when the likelihood is Bernoulli.
You can then compute the maximum the maximum a posteriori estimate.
Anyway, the way this work out in simplest terms is before see any actual reviews, you initialize your beliefs about $\rho$ as if there were $\alpha$ positive reviews and $\beta$ negative reviews. Let's say you then observe $x$ actual positive reviews and $y$ actual negative reviews. You would act as if you've seen $\alpha + x$ positive reviews. Your maximum a posteri estimate would be:
$$\hat{p} = \frac{\alpha + x}{\alpha + x + \beta + y}$$
And your posterior (over $\rho$) follows the Beta distribution with parameters $\alpha + x$ and $\beta + y$.
You're effectively incorporating some initial beliefs about whether the product is good or bad.
The weakness of this approach is that your choices of $\alpha$ and $\beta$ may be difficult to justify! Should it be $\alpha =1$ and $\beta=1$? $\alpha = 4.2$ and $\beta = 2.1$? If you have some objective way to set $\alpha$ and $\beta$ based upon outside data, outside information, then the Bayesian approach makes more sense. If you're pulling your choices for $\alpha$ and $\beta$ out of a magic hat, then it can be rather arbitrary.
