Let's consider a simple example to see how this sliding ruler affects the ratings in practice. So, suppose we have an overall overall of the entire population we have 100 reviews. Over all the products, with an average of 2. So the overall average is 2. As opposed to the individual. Which is of five reviews, so, five of those hundred reviews are for this individual. With an average of four. So the idea is that we want to compute this based on adjusted, rating for the individual. So we can draw, a little picture here, a little sliding ruler picture first. And, we start at two, which is the overall, and the individual is at four. And so we want to see where on this ruler the individual is going to fit after we apply the Bayesian adjustment. So, applying the formula we have to take, the we have to factor in first the overall, and the overall average. So the overall average is 2. Times the overall rating, which is 100 or the overall number which is 100. So, this is the overall average times the overall number plus the individual average, which is four times the individual number, which is five. And we want to divide that by the total which is 100. Plus five. So this is just a weighted average of the 2. So we do that out we get on the top 220 and on the bottom 105. Which is 2.1. So as we see this 2.1. Is going to be very close to the two side. And the reason for that is that the individual is only making up 5 out of the 100, so we slide a lot more towards the overall than we do, for the, to the individual. Now, let's, let's, let's say that instead of, five reviews, that this is actually computed based on forty reviews and let's see how, this average is going to change. So we still have 2 times 100. And then what we want to add to that is, instead of 4 times 5, we're going to have 4 times 40. And divide that by 100, plus 40. And doing that out, we get 400 on the top over 140 on the bottom, which is. 2.57 so as we can see we have now slid from 2.1 to 2.57. So we're now moving this is when the average is based on 5 reviews and this is when it was based on 4 reviews. So we're sliding this way and we're sliding more towards the individual as opposed to the overall. And that's the idea behind Bayesian ranking. So now, let's take this and apply it to our DVD player example. So this table right here just shows a summary of what we've computed in terms of the raw or the naive. average. And, what we want to do now is use these raw averages and apply them, apply Bayesian adjustments to them based upon the population. So what we need first and what's not given in the table is what the total number of ratings are, so as we had before. The total number of ratings is 220 ratings. Total. If we just add up these numbers. And the overall average which we had already computed, was 3.564, So we need to use these two numbers in each case. So, let's start with the Panasonic player and let's apply the Bayesian ranking, or the Bayesian ranking formula to it. So, we have the overall, which is 220 times 3.564 and the individual in this case, which is 4 point or 11 for the total number times 4.182. So, the overall number times the overall average, plus the individual number times the individual average, and divide that by 220 plus 11. And if we do the multiplication now, for each of the terms, we get 784.08 plus 46, roughly 46. And divide that by again 220 plus 11 and, we do that out we get 3.593. So we can write that in here, 3.593 for the Panasonic player. So, we can see now that this is pulled much closer towards the average. Of 3.564. It's only 3.593, which is much closer to the average than 4.182 was initially. And the reason for that, again, is that for the small number, the small number in this individual case of 11 as opposed to 220 and the overall. Now we can do, for the Sony, we'll do another example of this application. The overall, in the overall case we have 220 times 3.654, again. And, now for this case, we have 4 times 37, so we have 37 times 4 for the individual and we divide that by 220 plus 37. And we do this out, we have 784.08. On this side plus 148. And you can just look at the two numbers. You see this number's much larger than this number as it should be, but not as much larger as in this case. Divided by 220 plus 37. And we get 3.626. So this is 3.626. So already now we can see that this rating is higher than the Panasonic even though the average rating. Naive case was lower. And the reason is that we're pulling them towards the overall average. But only as much as number gradings will permit us if you will. And we can continue this computation for the, for the Phillips player we're going to get 220 times 3.564. Plus 67 times 3.448 divided by 220 plus 67 and that's going to be 3.537. So, this comes to 3.537. Now in this case, in these other two cases, the the, the individual ratings were higher than the overall. Now in this case, the individual rating is lower than the overall, so we're actually pulling it upward towards that center. And again. We can continue with the Curtis. And if we do that out with the Curtis we'll get 3.539 and for the Toshiba we'll get 3.533. So this is 3.539 and this is 3.533. So what is the result of this? Well, now we're going to have a new order in here. So if we order them from highest to lowest, the top now is going to be the Sony, followed by the Panasonic, followed by the Curtis, then followed by the Phillips, then followed by. The Toshiba. [NOISE] So this summary shown, again, now in this table appear. Which basically just give us ranking. In this case it was 1, 2, 3, 4, 5. Ranking from highest to lowest in the [UNKNOWN] case. Now now has switched to. 2-1-4-3-5, so the first one's not a Sony. Panasonic is moved from 1 to 2. Sony is changed from two to one. Phillips is moved down position to four. The Curtis has moved up to position to three, and the Toshiba stayed where it was before. So the Panasonic and the Phillips players, and well as the Phillips and the Curtis players, have swapped at the basement adjustment supply. And beyond this we note that all of the ratings are pulled closer to the mean as we expect due to the sliding ruler concept, right? So. all of these ratings if we draw them out on this line where this is the overall, all the individuals in each case are going to be, pulled, in, inward towards the mean. So the ordering does change significantly but we note that sometimes Bayesian ranking is not going to alter the ranked order list of the products that much. We you know, we took this example just to show a case where it would change, otherwise it wouldn't have been interesting. It's not always going to change it, but it does in this case. And again, all the ratings are pulled closer to the mean.